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Doctoral Thesis

Morse homology of the loop space on the moduli space of flatconnections and Yang-Mills theory

Author(s): Janner, Remi

Publication Date: 2010

Permanent Link: https://doi.org/10.3929/ethz-a-006118513

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DISS. ETH NO. 19025

MORSE HOMOLOGY OF THE LOOP SPACE ON THEMODULI SPACE OF FLAT CONNECTIONS AND

YANG-MILLS THEORY

A dissertation submitted to

ETH ZURICH

for the degree of

Doctor of Sciences

presented by

REMI JANNER

Dipl. Math. ETH Zurich

born December 23, 1981

citizen of

Bosco Gurin TI, Switzerland

Prof. Dr. Dietmar A. Salamon, examiner

Prof. Dr. Alberto Abbondandolo, co-examiner

Prof. Dr. Michael Struwe, co-examiner

2010

Acknowledgements

First I want to express my gratitude to my supervisor Professor Dietmar Salamon forhis guidance throughout the last years as well as for his enthusiasm for the subject.

I would like to thank the co-examiners Professor Alberto Abbondandolo and ProfessorMichael Struwe for their interest in my work.

I thank my colleagues and former colleagues from the ETH for the enjoyable workingenvironment and pleasant moments spent together; as well as the ETH for the financialsupport.

Above all, I would like to thank my friends, my parents Graziana and Boris, my sisterValentina and my wife Tiziana for all their constant support and understanding.

3

Contents

Abstract iii

Riassunto v

Introduction vii

I Bijection between perturbed closed geodesics and perturbedYang-Mills connections 1

1 The moduli spaceMg(P ) and the equations of the critical points 31.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Gauge group . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Inner product on Ω∗(Σ, gP ) . . . . . . . . . . . . . . . . . . 41.1.3 Curvature and Yang-Mills functional . . . . . . . . . . . . . . 5

1.2 The moduli spaceMg(P ) . . . . . . . . . . . . . . . . . . . . . . . 61.2.1 Tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Mg(P ) and the moment map theory . . . . . . . . . . . . . . 8

1.3 Estimates on the surface . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Perturbed geodesics onMg(P ) . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Unperturbed energy functional . . . . . . . . . . . . . . . . . 111.4.2 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.3 The map F0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.4.4 The Jacobi operator of a perturbed geodesic . . . . . . . . . . 14

1.5 Perturbed Yang-Mills connections . . . . . . . . . . . . . . . . . . . 161.5.1 The map F ε . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.2 The linear operator Dε . . . . . . . . . . . . . . . . . . . . . 17

1.6 Norms I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Elliptic estimates 212.1 Proof of the theorem 15 for p = 2 . . . . . . . . . . . . . . . . . . . 252.2 Proof of the theorem 15 for p ≥ 2 . . . . . . . . . . . . . . . . . . . 28

3 Quadratic estimates I 33

4 The map T ε,b between the critical connections 374.1 Definition and properties of T ε,b . . . . . . . . . . . . . . . . . . . . 374.2 Proof of the existence theorem . . . . . . . . . . . . . . . . . . . . . 394.3 Proof of the local uniqueness theorem . . . . . . . . . . . . . . . . . 444.4 Local uniquess modulo gauge . . . . . . . . . . . . . . . . . . . . . 45

i

ii CONTENTS

5 A priori estimates for the perturbed Yang-Mills connections 495.1 L2(Σ)-estimates for the curvature term FA . . . . . . . . . . . . . . . 495.2 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3 L2(Σ)-estimates for the curvature term ∂tA− dAΨ . . . . . . . . . . 575.4 L∞-bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6 Surjectivity of T b,ε 67

7 Bijection between the critical connections 81

II Isomorphism between the Morse homologies 85

8 Flow equations and Morse homologies 878.1 Geodesic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 878.2 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.3 Yang-Mills flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.4 Norms II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.5 Morse homologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9 Linear estimates for the Yang-Mills flow operator 959.1 Proof of the theorem 55 . . . . . . . . . . . . . . . . . . . . . . . . 1039.2 Proof of the theorem 56 . . . . . . . . . . . . . . . . . . . . . . . . 108

10 Quadratic estimates II 115

11 The map Kε2: A first Approximation 119

12 The mapRε,b between flows 125

13 A priori estimates for the Yang-Mills flow 12913.1 Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

14 Exponential convergence 157

15 L∞-bound for a Yang-Mills flow 165

16 Relative Coulomb gauge 173

17 Surjectivity ofRε,b 179

18 The main theorem 195

Curriculum Vitae 199

Abstract

In this dissertation we consider a non-trivial principal SO(3)-bundle π : P → Σ on asurface Σ with genus bigger than one and we study some aspects of the moduli spaceM given by the quotient between the space of the flat connectionsA0(P ) and the iden-tity component of the gauge group. On the one side, onM we look at the perturbedgeodesics that can be seen as equivalence classes of connections on a principal SO(3)-bundle π : P × S1 → Σ × S1. On the other side, we study the perturbed Yang-Millsconnections of the principal SO(3)-bundle πε : P × S1 → Σ × S1 where the metricon Σ is rescaled by a small parameter ε2. In his dissertation, Ying-Ji Hong explainedhow we can define a map T ε between the representatives of the perturbed geodesicsand the latter perturbed Yang-Mills connections.

In the first part of this thesis, we show that the map T ε is a bijection if we choose anenergy bound and provided that ε is sufficiently small and that the Morse indices ofa perturbed geodesic and of the correspondent perturbed Yang-Mills connection coin-cide.

In the second part, we compare the heat flow between two perturbed geodesics andthe Yang-Mills flow between the correspondent Yang-Mills connections. More pre-cisely, first, for a representative of a heat flow A : S1 × R → A0(P ) between therepresentatives, A− and A+, of two perturbed geodesics with Morse index differenceone, we construct a Yang-Mills L2-flow between the two connections T ε(A−) andT ε(A+). In this way we can prove the existence of a map Rε from the set of the heatflows betweenA− andA+ into the space of the Yang-Mills flows between T ε(A−) andT ε(A+). Then, we show that Rε is a bijection provided that we choose ε sufficientlysmall.

With these ingredients we can prove the main result of this thesis: The bounded Morsehomology of the loop space of M is isomorph to the bounded Morse homology ofthe space of connections A(P × S1) modulo the gauge group provided that ε is smallenough where the first homology is defined using the heat flow and the second oneusing the ε-dependent Yang-Mills flow.

iii

Riassunto

In questa tesi di dottorato consideriamo un fascio principale non triviale π : P → Σcon fibra SO(3) e con una superficie Σ, con genere maggiore di uno, come varietadi base e studiamo alcuni aspetti dello spazio modulare M dato dal quoziente tra lospazio delle connessioni piatte A0(P ) e i suoi isomorfismi. Da un lato, osserviamo legeodetiche perturbate suM che possono essere viste come delle connessioni di un fas-cio principale π : P×S1 → Σ×S1. Dall’altro lato studiamo le connessioni perturbatedi Yang-Mills di un fascio principale, sempre con fibra SO(3), πε : P ×S1 → Σ×S1

dove la metrica di Σ viene riscalata con un parametro ε2. Ying-Ji Hong nella sua tesi didottorato ha spiegato come definire una mappa T ε tra i rappresentanti delle geodetichee le connessioni di Yang-Mills considerate.

Nella prima parte di questa tesi, dimostriamo che la mappa T ε e biiettiva, se si sceglieun limite di energia e il parametro ε sufficientemente piccolo, e che gli indici di Morsedi una geodetica e della corispondente connessione di Yang-Mills coincidono.

Nella seconda parte, confrontiamo il flusso di calore tra due geodetiche perturbate e ilflusso di Yang-Mills tra le corrispondenti due connessioni di Yang-Mills. Piu precisa-mente, dapprima scegliamo due rappresententi,A− eA+, di due geodetiche perturbate,con differenza uno tra i loro indici di Morse, e un rappresentante di un flusso di caloreA : S1 × R → A0(P ) tra A− e A+; poi costruiamo un flusso L2 di Yang-Mills trale connessioni T ε(A−) e T ε(A+). In questo modo possiamo mostrare l’esistenza diuna mappa Rε dall’insieme dei flussi di calore tra A− e A+ allo spazio dei flussi diYang-Mills tra T ε(A−) e T ε(A+). Inoltre riscontriamo che Rε e biiettiva ammessoche il parametro ε sia sufficientemente piccolo; infatti non solo le connessioni criticheT ε(A−) e T ε(A+) ma anche i flussi dipendono dal parametro ε.

Infine dimostriamo il risultato principale di questa tesi: vale a dire che l’omologia diMorse dello spazio delle curve chiuse di M e isomorfa all’omologia di Morse dellospazio delle connessioni di P × S1, modulo isometrie, se si considera un limite dienergia e ε sufficientemente piccolo; la prima omologia e definita con il flusso di calorementre la seconda con il flusso di Yang-Mills che dipende dal parametro ε.

v

Introduction

The moduli space of flat connections for a principal bundle over a surface Σ with genusg is an infinite dimensional analogue of a symplectic reduction and was investigatedfor the first time in 1983 by Atiyah and Bott in [4] where they showed that, on thisparticular moduli space, one can define an almost complex structure induced by theHodge-*-operator acting on the 1-forms over Σ and hence induced by its conformalstructure; with the almost complex structure and the inner product on the 1-forms onecan also obtain a symplectic form. Furthermore, if we choose a principal non trivialSO(3)-bundle, then the moduli space Mg(P ), defined as the quotient between thespace of the flat connections A0(P ) ⊂ A(P ) and the identity component of the gaugegroup G0(P ), is a smooth compact symplectic manifold of dimension 6g − 6 (cf. [6]).In the nineties some aspects of the topology ofMg(P ) were investigated by Dostoglouand Salamon in [7], where they proved an isomorphism between the symplectic andthe instanton Floer homology related to this moduli space, and in the work of Hong (cf.[9]) which is the starting point of this thesis. Hong took an oriented compact manifoldB with a Riemannian metric gB and a harmonic map φ : B →Mg(P ) and he showedthat if the Jacobi operator of φ is invertible, then there exist a constant ε0 and, for0 < ε < ε0, a family Aε of Yang-Mills connections of the principal SO(3)-bundlesP × B → Σ × B, where the base manifold has a partial rescaled metric gΣ ⊕ 1

ε2gB,

which converges to the connection that generates φ. In this thesis we choose B = S1

and a slightly different rescaling of the metric and we look at same problem; moreprecisely the setting is the following one.

The critical connections

On the one hand, we consider the loop space onMg(P ) and his elements can be seenas connectionsA(t)+Ψ(t)dt on a the manifold Σ×S1, whereA(t) ∈ A0(P ) and Ψ(t)is a 0-form in Ω0(Σ, gP ), satisfying the condition d∗A (∂tA− dAΨ) = 0. The 1-form∂tA− dAΨ corresponds to the speed vector of our loop and thus the perturbed energyfunctional is

EH(A) =1

2

∫ 1

0

(‖∂tA− dAΨ‖2

L2(Σ) −Ht(A))dt (1)

where Ht : A(P ) → R is a generic equivariant Hamiltonian map which is introducedin order to obtain an invertible second variational form and the time-dependent Hamil-tonian vector field Xt is defined such that, for any connection A ∈ A(P ) and any1-form α, dHt(A)α =

∫Σ〈Xt(A) ∧ α〉.

The critical points of (1) has to satisfy the equation

πA (−∇t(∂tA− dAΨ)− ∗Xt(A)) = 0, (2)

vii

viii Introduction

where ∇t := ∂t + [Ψ, .] and where πA denote the projection of the 1-forms in to thelinear space of the harmonic 1-forms which corresponds to the tangent space of themanifoldMg(P ) at the point [A]. The equation (2) can be therefore also written as

−∇t(∂tA− dAΨ)− ∗Xt(A)− d∗Aω = 0; (3)

the 2-form ω(t) ∈ Ω2(Σ, gP ) is defined uniquely by the identity

dAd∗Aω = [(∂tA− dAΨ) ∧ (∂tA− dAΨ)]− dA ∗Xt(A)

and thus, d∗Aω corresponds to the non-harmonic part of −∇t(∂tA− dAΨ)− ∗Xt(A).

On the other hand, we can take the 3-manifold Σ× S1 with the metric ε2gΣ ⊕ gS1 fora positive parameter ε and consider the principal SO(3)-bundle P × S1 → Σ × S1.In this case, for a connection Ξ = A + Ψ dt ∈ A(P × S1), where A(t) ∈ A(P ),Ψ(t) ∈ Ω0(Σ, gP ) the curvature is FΞ = FA−(∂tA−dAΨ)∧dt and thus the perturbedYang-Mills functional can be written as

YMε,H(Ξ) =1

2

∫ 1

0

(1

ε2‖FA‖2

L2(Σ) + ‖∂tA− dAΨ‖2L2(Σ) −Ht(A)

)dt; (4)

one can easily remark that this functional has the property that for a connection Ξflat in the Σ-direction YMε,H(Ξ) = EH([A]). A perturbed Yang-Mills connectionΞε ∈ A(P × S1) has therefore to satisfy the two conditions

1

ε2d∗AεFAε −∇t(∂tA

ε − dAεΨε)− ∗Xt(Aε) = 0, (5)

d∗Aε(∂tAε − dAΨε) = 0. (6)

We can remark that the two equations (3) and (5) are similar and in the second one thetwo form 1

ε2FAε plays the role of −ω. If we choose a perturbed geodesic A+ Ψdt and

we write write −ω = dAα0 then the connection A + ε2α0 is an approximate solutionof the equation (5) because

1

ε2d∗A+ε2α0

FA+ε2α = −d∗Aω +O(ε2);

in addition A + ε2α0 turn out to be also an approximative solution of (6) by the def-inition of Ψ. Then, by a contraction argument one can define a map between theperturbed geodesics below an energy level b, denoted by CritbEH , and the set of theperturbed Yang-Mills connections CritbYMε,H with energy less than b provided that theparameter ε is small enough. Furthermore, this map can also be defined uniquely, itis surjective and maps perturbed geodesics to perturbed Yang-Mills connections withthe same Morse index. Summarizing, in the first part of this thesis, we will prove thefollowing theorem.Theorem A. We assume that the Jacobi operators of all the perturbed geodesics areinvertible and we choose a regular value b of the energy EH and p ≥ 2. Then thereare two positive constants ε0 and c such that the following holds. For every ε ∈ (0, ε0)there is a unique gauge equivariant map

T ε,b : CritbEH → CritbYMε,H

Introduction ix

satisfying, for Ξ0 ∈ CritbEH ,

d∗εΞ0

(T ε,b(Ξ0)− Ξ0

)= 0,

∥∥T ε,b(Ξ0)− Ξ0∥∥

Ξ0,2,p,ε≤ cε2. (7)

Furthermore, this map is bijective and indexEH (Ξ0) = indexYMε,H (T ε,b(Ξ0)).

The result of HongHong could assume that the harmonic map φ has an invertible Jacobi operator because,you can reach this condition for example for a 2-dimensional manifold B and eventu-ally slightly perturbing the metric gB. For B = S1 the Jacobi operator of a geodesicis never invertible and for this reason we need to introduce a perturbation in our func-tional EH as we will discuss in the chapter 1.

Another important point worth to be remarked is the different choice of the rescaling.On the one side, both choices give the same equations for the Yang-Mills connections,for B = S1, if we don’t consider the Hamiltonian perturbation, and hence his meth-ods work also in our case; we can therefore say that Hong proved the existence of themap T ε,b. However, he did not prove its uniqueness and its surjectivity. On the otherside, the different choice of the metric give two different Yang-Mills energy function-als; in fact, using the metric gΣ ⊕ 1

ε2gS1 one obtain the Yang-Mills energy functional

εYMε, 1εH instead of YMε,H and the properties of YMε,H will play a major role in

the proof of the surjectivity of T ε,b and in particular, to obtain the a priori estimates forthe curvature of the perturbed Yang-Mills connections.

The flows

On the one side, every map [Ξ] : S1 × R → Mg(P ) can be seen as a connectionΞ = A+ Ψdt+ Φds ∈ A(P × S1 × R) which satisfies

FA = 0, ∂tA− dAΨ ∈ H1A, ∂sA− dAΦ ∈ H1

A (8)

and if we have a map A : S1 × R→ A0(P ), the second and the third condition of (8)yield to unique 0-forms Ψ,Φ ∈ Ω0(P ×S1×R). In order to achieve the transversalitycondition for the heat flow we need to choose a generic abstract perturbation on theloop space instead of the Hamiltonian one. Furthermore, [Ξ] is a heat flow betweenthe perturbed geodesics Ξ± ∈ CritbEH , b ∈ R, if it satisfies the flow equation for thefunctional EH , i.e.

∂sA− dAΦ− πA(∇t (∂tA− dAΨ) + ∗Xt(A)

)= 0,

lims→±∞

Ξ(s) = Ξ±(9)

where the first equation can be written as

∂sA− dAΦ−∇t (∂tA− dAΨ)− ∗Xt(A) + d∗Aω = 0,

dAd∗Aω = − [(∂tA− dAΨ) ∧ (∂tA− dAΨ)]

x Introduction

where ω(t, s) ∈ Ω2(Σ, gP ). In the following we denote by M0(Ξ−,Ξ+) the modulispace of the solutions of (9).

On the other side, a perturbed, ε-dependent, Yang-Mills flow between two perturbedYang-Mills connections Ξ± ∈ CritbYMε,H can be consider as a connection Ξ := A +Ψdt + Φds on the 4-manifold Σ × S1 × R, where Φ ∈ Ω0(S1 × R, gP ) makes theequations gauge invariant, and it satisfies the equations

∂sA− dAΦ +1

ε2d∗AFA −∇t (∂tA− dAΨ)− ∗Xt(A) = 0,

∂sΨ−∇tΦ−1

ε2d∗A (∂tA− dAΨ) = 0, lim

s→±∞Ξ = Ξ±.

(10)

In this case, we denote byMε(Ξ−,Ξ+) the moduli space given by the solutions of (10).

If we let ε go to 0, we can expect that in the limit we have the following equations

FA = 0, d∗A (∂tA− dAΨ) = 0, (11)

∂sA− dAΦ−∇t (∂tA− dAΨ) + d∗Aω = 0, (12)

where ω should be seen as the limit of 1ε2FA for ε → 0 and the two equations (11)

and (12) are exactly the conditions for the perturbed geodesic flow. We can thereforeexpect a bijective relation also between the flows of the two functionals for ε smallenough.Theorem B. We assume that the energy functionalEH is Morse-Smale and we choosep > 2 and a regular value b > 0 of EH . There are constants ε0, c > 0 such that thefollowing holds. For every ε ∈ (0, ε0), every pair Ξ0

± := A0± + Ψ0

±dt ∈ CritbEH withindex difference 1, there exists a unique map

Rε,b :M0(Ξ0−,Ξ

0+

)→Mε

(T ε,b(Ξ0

−), T ε,b(Ξ0+))

satisfying for each Ξ0 ∈M0(Ξ0−,Ξ

0+

)d∗εΞ0

(Rε,b(Ξ0)−Kε2(Ξ0)

)= 0, Rε,b(Ξ0)−Kε2(Ξ0) ∈ im

(Dε(K2(Ξ0))

)∗,∥∥Rε,b(Ξ0)−Kε2(Ξ0)

∥∥1,2;p,1

≤ cε2.

Furthermore,Rε,b is bijective.In the last theorem the connection Kε2(Ξ0) should be seen as a first approximation ofthe Yang-Mills flow.

The isomorphism between the Morse homologiesThe theorem A assures a bijection between the critical connections with the same indexand the theorem B between the flows and thus we can compare the Morse homologiesdefined using the L2-flow of the two functionals below a energy level b. In the loopspace case the homology is well defined by the works of Salamon and Weber (cf. [17],

Introduction xi

[24]) and in the Yang-Mills case we know that the flow exists when the base manifoldis two or three dimensional (cf. [15]) or when we have a symmetry of codimension 3on a base manifold of higher dimension (cf. [16]), but any results about the Morse-Smale transversality or about the orientation of the unstable manifolds are known andtherefore a priori HM∗

(Aε,b (P × S1) /G0 (P × S1)

)might even not be defined; in

our case, it makes sense because the unstable manifolds ofAε,b (P × S1) /G0 (P × S1)inherit these properties from the unstable manifolds of LbMg(P ). Here LbMg(P ) ⊂LMg(P ) and Aε,b (P × S1) ⊂ A (P × S1) denote respectively the subsets whereEH ≤ b and YMε,H ≤ b.Theorem C. We assume that the energy functional EH is Morse-Smale. For everyregular value b > 0 of EH there is a positive constant ε0 such that, for 0 < ε < ε0, theinclusion LbMg(P )→ Aε,b (P × S1) /G0 (P × S1) induces an isomorphism

HM∗(LbMg(P ),Z2

) ∼= HM∗(Aε,b

(P × S1

)/G0

(P × S1

),Z2

).

Another way to approach this problem could have been to consider the W 1,2-flows; inthis case both homologies are well defined since the Palais-Smale condition is satisfiedin both cases (cf. [26]) and thus by the general Morse theory (cf. [1]) the transverval-ity may be achieved. It is also interesting to remark that the (Morse) homology ofLbMp(P ) correspond to the Floer homology of the cotangent bundle T ∗Mg(P ) usingthe Hamiltonian HV given by the kinetic plus the potential energy and consideringonly orbits with the action bounded by b. This result was proves by Viterbo (cf. [21]),by Salamon and Weber (cf. [17]) and by Abbondandolo and Schwarz (cf. [3], [2]) inthree different ways. Furthermore, Weber (cf. [24]) proved that the Morse homologyof the loop space defined by the heat flow is isomorphic to its singular homology. Wehave therefore the following identities

HM∗(Aε,b/G0

) ?∼= H∗(Aε,b/G0

)∼=

HM∗(LbMg(P )

) ∼= H∗(LbMg(P )

)∼=

HF b∗ (T ∗Mg(P ), HV )

where we denotedAε,b (P × S1) /G0 (P × S1) byAε,b/G0. We would like to concludethis discussion remarking that it is an open question whether the Morse homology ofAε,b/G0 defined using the L2-flow is isomorphic to its singular homology.

OutlineThis thesis is divided in two parts. In the first one we prove the bijective relation statedin theorem A (theorem 49) between the perturbed geodesics on the moduli space of flatconnections and the perturbed, ε-dependent, Yang-Mills connections for any ε smallenough. The first chapter is of preliminary nature; in fact, first, we briefly introducethe moduli spaceMg(P ) := A0(P )/G0(P ) of flat connections of a non-trivial prin-cipal SO(3)-bundle P over a surface (Σ, gΣ) of genus g. Then, on the one hand, wecompute the equations of the perturbed closed perturbed geodesics onMg(P ) and its

xii Introduction

Jacobi operator and on the other hand, we discuss for a given ε > 0 the equations forthe perturbed Yang-Mills connections of the principal SO(3)-bundle P×S1 → Σ×S1

where the metric on Σ is rescaled by a factor ε2. Next, we also define the norm whichwill play a fundamental role in the proof of the theorem A and at the end of the chap-ter we cite some useful estimates for connections on a surface. In the successive twochapters we compute the linear (chapter 2) and the quadratic (chapter 3) estimates andin chapter 4, for each perturbed geodesic we construct, using a Newton’s iteration anda contraction argument, a perturbed Yang-Mills connection defining in this way aninjective map T ε,b and furthermore, we prove that this map is unique under the con-dition (7). In the next chapter, we show some a priori estimates (chapter 5) that weneed to prove the surjectivity of the map T ε,b (chapter 6). We prove the surjectivity ofthe map T ε,b indirectly: We assume that there is a sequence of perturbed Yang-Millsconnections Ξεν , εν → 0, which is not in the image of T εν ,b, and we show that thissequence has a subsequence which converges to a geodesic Ξ0; then using the unique-ness property of T ε,b this subsequence turn out to be in the image of T εν ,b(Ξ0) yieldinga contradiction. In the last chapter, we conclude the proof of the theorem A provingthat T ε,b maps perturbed geodesics to perturbed Yang-Mills connections with the sameindex (theorem 50); in fact the theorem A follows directly from the definition 26 of themap T ε,b, its surjectivity (theorem 40) and the index theorem 50.

In part II we prove the theorems B (theorem 102) and C (theorem 103). In the chapter8 we discuss the geodesic flow and the Yang-Mills flow equations and we define theε-dependent norm that is needed in this part of the thesis. In the chapters 9 and 10we show respectively some linear and quadratic estimates that we need in the chap-ter 11 to construct an approximation of a perturbed Yang-Mills flow starting from aperturbed geodesic flow and in the chapter 12 to define uniquely the map Rε,b using acontraction argument. The next four chapters are of preparatory nature for the proofof the surjectivity ofRε,b (chapter 17); in fact in the chapter 13 we show some a prioriestimates for the perturbed Yang-Mills flow and then we prove the uniformly expo-nential convergence of the flows (chapter 14), an estimate for the L∞-norm of theircurvature terms (chapter 15) and two theorems that allows to choose the right relativeCoulomb gauge (chapter 16). Also in this case, the surjectivity (17) is showed using anindirectly argument; in fact we prove that any sequence of perturbed Yang-Mills flowsΞεν , εν → 0, which is not in the image of the mapRε,b, has a subsequence which con-verges (modulo gauge) by the implicit function theorem to a perturbed geodesic flowand thus, by the uniqueness of Rε,b, it is in the image of this map. In the last chapterof the second part we finally prove first the theorem B, which follows easily from thedefinition 77 of the mapRε,b and its surjectivity (theorem 98), and then the theorem C.

Outlook 1

A possibility to extend the result of this thesis is to consider the following setting (cf.[6]). We choose a principal non trivial SO(3)-bundle π : P → Σ with a surface Σ,with genus g bigger than 1, as the base manifold and we pick an orientation preservingdiffeomorphism h : Σ → Σ and an automorphism f : P → P on P which descends

Introduction xiii

to h, i.e. h π = π f ; f induces a symplectomorphism φf : Mg(P ) → Mg(P )defined by φf ([A]) = [f ∗A]. Furthermore, we consider the principal SO(3)-bundlePf → Σh generated by f ; where Pf denotes the mapping cylinder of P defined as theset of equivalence classes [p, t] ∈ P ×R given by the relation [p, t+ 1] ≡ [f(p), t] andwhere Σh is the mapping cylinder with the equivalence classes [x, t] ∈ Σ×R given by[x, t+ 1] ≡ [h(x), t]. In addition, we pick a gauge invariant map H : R×A(P )→ Rwhich satisfies Ht+1(A) = Ht(f

∗A).

On the one hand, we can consider the spaceLbφf (Mg(P )) which contains all the curves

onMg(P ) that satisfy the condition γ(t+1) = φf (γ(t)) and have energy smaller thanb on the time intervall [0, 1]; the energy functional is given in the same way as (1).Thus instead of perturbed closed geodesics on Mg(P ), we can study the perturbedgeodesics γ : R →Mg(P ) that satisfy the condition γ(t + 1) = φf (γ(t)); a lift of γcan be considered as connection A+ Ψdt ∈ A(Pf ) which satisfies the conditions

FA = 0, d∗A(∂tA− dAΨ) = 0, (13)

A(t+ 1) = f ∗A(t), Ψ(t+ 1) = Ψ(t) f (14)

and the equation (2).

On the other hand, we can look at the space A(Pf )/G0(Pf ) and consider the Yang-Mills functional (4) defined rescaling the metric on Σ by a parameter ε2; where thegauge group G0(Pf ) contains all the maps g : R → G0(P ) that satisfy g(t + 1) =g(t) h. With this setting, the perturbed Yang-Mills connections A + Ψdt ∈ A(Pf )have to satisfy (5) and (6) and the condition (14).

Also in this case, below an energy level b, we expect a bijection between the lifts of theperturbed geodesics onMg(P ) and the perturbed Yang-Mills connections on A(Pf )provided that ε is sufficiently small. Furthermore, we may prove an isomorphismbetween the homologies of Lbφf (M

g(P )) and Aε,b(Pf )/G0(Pf ) defined with the L2-flows for ε sufficiently small and thus we may obtain an analogous result as in thetheorem C; where Aε,b(Pf ) is the subspace of all the connections of A(Pf ) with theε-dependent Yang-Mills energy smaller than b.

Outlook 2

The manifoldMg(P ) = A0(P )/G0(P ) can be also interpreted as a symplectic quo-tient defined with the moment map µ : A(P ) → Ω0(gP ), µ(A) = ∗FA. A naturalquestion which therefore arise is whether there is a finite dimensions analogue of thecorrespondence discussed so far; the answer is positive and we can approach the prob-lem in the following way.

We choose a finite dimensional symplectic manifold X and a Lie group G acting freeon it; we assume in addition that a Hamiltonian action is generated by an equivariantmoment map µ : X → g := Lie(G), where g denotes the Lie algebra of G, with reg-ular value 0 and that the compatible almost complex structure J on X is G-invariant.

xiv Introduction

Furthermore, we choose a time dependent andG invariant potential Vt : X → R. Thuswe can compare the two setting we have the following table.

Connections case Finite dimensions caseSpace A(P ) X

Lie group G0(P ) G

Lie algebra Ω0(Σ, g) g

Moment mapµ : A(P ) → Ω0(Σ, g)

A 7→ ∗FAµ : X → g

0-section A0(P ) µ−1(0)

Symplectic quotient Mg(P ) µ−1(0)/G

On the one side, we can study the perturbed geodesics on the symplectic quotientM := µ−1(0)/G, that we assume compact, and hence the critical points of

Eµ,V (x, ξ) :=1

2

∫ 1

0

(|x+ Lxξ|2 − Vt(x)

)dt (15)

for (x, ξ) ∈ L(µ−1(0))×L(g) and where Lx(t)ξ(t) ∈ Tx(t)X denotes the fundamentalvector field generated by ξ(t) ∈ g and evaluated at x(t).. The critical points of Eµ,V ,where the set of those with energy less than b is denoted by CritbEµ,V , are the solutionsof the equations

µ(x) = 0, ∇t,ξ(x+ Lxξ) +∇Vt(x) ∈ im JLx, L∗x(x+Xξ) = 0 (16)

where∇t,ξv := ∇tv+∇vLxξ for any vector field v along x. The third equality impliesthat x + Lxξ is the velocity of x perpendicular to the action of G and hence it is thevelocity of x projected on the tangent space of the symplectic quotient. This conditioncould also naturally be given to define an energy Eµ,V independent on ξ as we didfor the functional EH and the condition d∗A (∂tA− dAΨ) = 0. The time dependentpotential, since it is G invariant, is well defined also on the symplectic quotient andit is introduced in order to make the Jacobi operator of Eµ,V for any perturbed closedgeodesic invertible using the results of Weber (cf. [23]).

On the other side, we choose on the loop space of X × g the twisted energy functional

Eµ,V,ε(x, ξ) :=1

2

∫ 1

0

(1

ε2|µ(x)|2 + |x+ Lxξ|2 − Vt(x)

)dt (17)

for (x, ξ) ∈ L(X) × L(g). The critical points of Eµ,V,ε, denoted by CritbEµ,V,ε , are thesolutions of the equations

1

ε2JLxµ(x) = ∇t,ξ(x+ Lxξ) +∇Vt(x), L∗x(x+ Lxξ) = 0. (18)

This last energy functional is the analogous of the perturbed Yang-Mills energy func-tional YMε,H . Also for the finite dimensional case, we can prove a bijection betweenthe critical loops below a given energy level b and for ε small enough (cf. [10]).

Introduction xv

Theorem D. We assume that the Jacobi operator is invertible for every geodesic andwe choose a regular value b of the energy functional Eµ,V . Then there are two positiveconstants ε0 and c such that the following holds. For every ε ∈ (0, ε0) there is a uniquemap

Sε,b : CritbEµ,V → CritbEµ,V,ε

satisfying, for (x, ξ) ∈ CritbEµ,V and (α, ψ) defined by (expx(α), ξ + ψ) = Sε,b(x),

1

ε2L∗xα +∇tψ = 0, ||(α, ψ)||ξ,2,2,ε ≤ cε2, (19)

where∇t := ∂t + [ξ, ·]. Sε,b is bijective and Sε,b(g∗(x, ξ)) = g∗Sε,b(x, ξ) for any loopg ∈ LG and any (x, ξ) ∈ CritbEµ,V .The first condition of (19) can be seen as a Coulomb gauge condition for the finitedimensional case and the norm ‖ · ‖ξ,2,2,ε is an ε-dependent norm defined consideringthe properties of the equations (18). In the theorem exp denotes the exponential mapon X and the action of L(G) on L(X) × L(g) is defined by g∗(x, ξ) = (gx, gξg−1 +g−1∂tg) for g ∈ L(G), (x, ξ) ∈ L(X) × L(g); on the Lie algebra part this is quiteunusual, in fact, we identify L(g) with the 1-forms of a trivial principal G-bundle overS1. Analogously as before, one could also expect a bijection between the L2-flows andthen an isomorphism between the Morse homologies:Conjecture E. We assume that the energy functional Eµ,V is Morse-Smale. For everyregular value b > 0 of Eµ,V there is a positive constant ε0 such that, for 0 < ε < ε0, theinclusion LbM→ Lε,b(X×g)/LG induces an isomorphism of the Morse homologies

HM∗(LbM

) ∼= HM∗(Lε,b(X × g)/LG

).

Here LbM ⊂ LM denotes the subset where Eµ,V ≤ b and Lε,b(X × g)/LG ⊂L(X)× L(g)/LG denotes the subset where Eµ,V,ε ≤ b.Therefore, this finite dimensional setting should have the same structure as the problemstudied in this thesis.

Part I

Bijection between perturbed closedgeodesics and perturbed Yang-Mills

connections

1

The moduli spaceMg(P ) and the

equations of the criticalpoints 1

1.1 Preliminaries

In order to introduce the moduli space of flat connections for a non-trivial principalSO(3)-bundle over a surface Σ, we first explain some facts about a principal G-bundleπ : P → Σ where G is a compact Lie group with Lie algebra g and P and Σ aresmooth manifolds. The action of G on P defines a vertical space

V :=

(p, pξ :=

d

dt

∣∣∣∣t=0

p exp(tξ)

)∣∣∣∣ p ∈ P, ξ ∈ g

⊂ TP

in the tangent bundle and hence a choice of a connection, i.e. an equivariant functionA : TP → g which satisfies

i) A(p, pξ) = ξ ∀p ∈ P, ∀ξ ∈ g,ii) A(pg, vg) = g−1A(p, v)g ∀p ∈ P, ∀v ∈ TpP,

could also be seen as a choice of an equivariant horizontal distribution H ⊂ TPwhich corresponds to the kernel of A and at each point p ∈ P induces the short exactsequence

0 −→ Hp = kerA(p, .)ι−→ TpP−→Vp −→ 0,

where ι is the inclusion of Hp in TpP and Hpg = Hpg and Vp the restriction of V atthe point p. In addition, since Vp = ker(dπ(p)) and TpP = Hp ⊕ Vp, dπ(p) inducesan isomorphism between Hp and Tπ(p)Σ, hence the horizontal distribution is isomorphto the pullback π∗TΣ and this observation implies that a vectorfield X on Σ has aunique horizontal lift X ⊂ H on P such that X(p) ∈ Hp and dpπ(X(p)) = X(π(p)).The set of all the connections of a principal bundle is denoted by A(P ) and it is anaffine space; in fact, for every connection A0 ∈ A(P ), A(P ) = A0 + Ω1

Ad,H(P, g)where Ω1

Ad,H(P, g) denotes the set of all equivariant functions α : TP → g such thatV ⊂ kerα, i.e. α is horizontal. Similarly, Ωk

Ad,H(P, g) is the space of equivariant andhorizontal k-forms, i.e for an ω ∈ Ωk

Ad,H(P, g) we have

ω(pg; v1g, v2g, ..., vkg) =g−1ω(p; v1, v2, ..., vk)g,

ω(p; v1, ..., vk) =0, if vi = pξ for an i ∈ 1, ..., k,

3

4 1. The moduli spaceMg(P ) and the equations of the critical points

where p ∈ P, g ∈ G, ξ ∈ g, vi ∈ TpP, 1 ≤ i ≤ k. Therefore, the equivariant andhorizontal k-forms Ωk

Ad,H(P, g) correspond to the k-forms over Σ with values in theadjoint bundle, i.e. Ωk

Ad,H(P, g) = Ωk(Σ, gP ), where gP := P ×Ad g is the associatedbundle defined by the equivalence classes [pg, ξ] ≡ [p,Adgξ] ≡ [p, gξg−1].

1.1.1 Gauge groupThe Lie group G(P ) of equivariant smooth maps u : P → G is called the gauge groupof P , i.e.

G(P ) := u ∈ C∞(P,G) | u(pg) = g−1u(p)g, ∀p ∈ P, ∀g ∈ G.

Since G acts on P , every element of the gauge group induces a gauge transformationof the bundle P , i.e. u : P → P ; p 7→ pu(p) which is a G-bundle isomorphism.Conversely, since the group G acts free, a G-bundle isomorphism descends from agauge transformation. Moreover, G(P ) is isomorph to the group of the sections ofthe associated bundle P ×c G, where we have the equivalence [p, g] ≡ [pq, q−1gq] forevery p ∈ P , g, q ∈ G. In fact, for a u ∈ G(P ) the section is defined as Σ→ P ×c G;π(p) 7→ [p, u(p)]; conversely, a section u : π(p) 7→ [p, u(p)] of P ×cG induces a gaugetransformation u(p) = pu(p). This last remark implies that G(P ) = C∞(Σ, P×cG) =Ω0(Σ, P ×c G) and hence the Lie algebra of G(P ) is the space of the equivariant,horizontal 0-forms over P , i.e TId(G)G(P ) = Ω0(Σ, gP ) where Id(G) : P → G; x 7→ eis the identity of G(P ). A gauge transformation acts on the space of connections by

u∗A = u−1Au+ u−1du

and hence we can consider u as a change of coordinates. Furthermore, in order to com-pute the infinitesimal gauge transformation on a connection A, we choose an elementφ of the Lie algebra Ω0(Σ, gP ) and we set ut = exp(tφ) = 1 + tφ+O(t2), then

d

dt

∣∣∣t=0

(u∗tA) = − d

dt

∣∣∣t=0

(u−1t Aut + u−1

t dut) = −[A, φ]− dφ = −dAφ. (1.1)

In fact, choosing a connection A ∈ A(P ), we can define the covariant derivative

dA : Ω0(Σ, gP )→ Ω1(Σ, gP ); φ 7→ dAφ = dφ+ [A, φ]

and the exterior derivative

dA : Ωk(Σ, gP )→ Ωk+1(Σ, gP );ω 7→ dAω = dω + [A ∧ ω]

where [ω1 ∧ ω2] := ω1 ∧ ω2 − (−1)lkω2 ∧ ω1 denotes the super Lie bracket operatorfor ω1 ∈ Ωl(Σ, gP ) and ω2 ∈ Ωk(Σ, gP ).

1.1.2 Inner product on Ω∗(Σ, gP )

The Hodge operator acts not only on Ωk(Σ), but on Ωk(Σ, gP ), too; in fact, since1

Ωk(Σ, gP ) = Γ(∧k T ∗Σ ⊗ gP

), for all ω ∈ Ωk(Σ), and all ξ ∈ Ω0(Σ, gP ), we define

1Γ(∧k T ∗Σ⊗ gP

)denotes the sections of the bundle ∧kT ∗Σ⊗ gP → Σ.

1.1 Preliminaries 5

∗(ω⊗ ξ) := ∗ω⊗ ξ. Therefore, using two inner products, one on Ωk(Σ) defined usingthe Hodge operator and an invariant inner product on the Lie algebra on Ω0(Σ, gP ), wehave an inner product on the k-forms Ωk(Σ, gP )

〈a, b〉 =

∫Σ

〈a ∧ ∗b〉 ∀a, b ∈ Ωk(Σ, gP ); (1.2)

for two vectorfields X , Y on Σ, 〈a ∧ b〉(X, Y ) = 〈a(X), b(Y )〉 − 〈a(Y ), b(X)〉. Theadjoint operator of the exterior derivative dA : Ωk(Σ, gP )→ Ωk+1(Σ, gP ) is

d∗A : Ωk+1(Σ, gP )→ Ωk(Σ, gP ); θ 7→ d∗Aθ

which satisfies 〈dAω, θ〉 = 〈ω, d∗Aθ〉 for every ω ∈ Ωk(Σ, gP ) and every θ ∈ Ωk+1(Σ, gP ).

1.1.3 Curvature and Yang-Mills functional

For any connection A ∈ A(P ), the two form FA := dA + 12[A ∧ A] ∈ Ω2(Σ, gP )

is called curvature of A and the gauge group acts by Fu∗A = u−1FAu for every u ∈G(P ). With this last definition it is possible to introduce the set of flat connections

A0(P ) := A ∈ A(P ) | FA = 0

and for an A ∈ A0(P ), since dA dA = 0, the cohomology groups

HkA(Σ, gP ) := ker dA/im dA

∣∣∣Ωk(Σ,gP )

= ker dA ∩ ker d∗A

∣∣∣Ωk(Σ,gP )

, ∀k ∈ N

are well defined. Moreover, we have the splitting

Ωk(Σ, gP ) = dAΩk−1(Σ, gP )⊕HkA(Σ, gP )⊕ d∗AΩk+1(Σ, gP ). (1.3)

We denote the canonical projection in to the harmonic forms by πA, i.e.

πA : Ωk(Σ, gP )→ HkA(Σ, gP ).

If ω is a closed curve on the k-forms, i.e. ω(t) ∈ Ωk(Σ, gP ) for every t ∈ S1,and if A is a loop on the space of the connections, the operator dA is defined by(dAω)(t) = dA(t)ω(t) ∈ Ωk+1(Σ, gP ); thus, if A is flat, the projection πA is defined sothat (πAω)(t) = πA(t)ω(t) ∈ Hk

A(t)(Σ, gP ).

Lemma 1. Let A0 ∈ A0(P ). Then TA0A0(P ) = ker dA0 .

Proof. Let At = A0 +∑∞

i=1 tiαi a curve in A0(P ) with α1 = d

dt|t=0 At ∈ TA0A0(P )

for t ∈]− ε, ε[, ε > 0. Since 0 = FA0 = FAt , we have that

0 =d

dt

∣∣∣t=0FAt =

d

dt

∣∣∣t=0

(FA0 + tdA0α1 +O(t2)) = dA0α1

thus, α1 ∈ ker dA0 .


The inner product on the 2-forms of the adjoint bundle defines the Yang-Mills func-tional

YM : A(P )→ R; A 7→ YM(A) :=1

2‖FA‖2

L2(Σ)

and since the inner product on the Lie algebra is invariant under the adjoint action, foran u ∈ G(P ), ∀A ∈ A(P ),

YM(u∗A) :=1

2〈Fu∗A, Fu∗A〉 =

1

2〈u−1FAu, u

−1FAu〉 =1

2〈FA, FA〉 = YM(A)

and, as a consequence, the Yang-Mills functional is invariant under a gauge transfor-mation. In addition, a connection A ∈ A(P ) is called Yang-Mills connection if it is acritical point of the Yang-Mills functional.

Lemma 2. A connection A ∈ A(P ) is Yang-Mills if and only if d∗AFA = 0.

Proof. Let At = A+ tα with α ∈ Ω1(Σ, gP ) a variation of A, then

0 =1

2

d

dt〈FAt , FAt〉

∣∣∣t=0

= 〈 ddtFAt

∣∣∣t=0, FA〉

= 〈 ddt

(FA + tdAα +1

2t2[α ∧ α])

∣∣∣t=0, FA〉 = 〈dAα, FA〉 = 〈α, d∗AFA〉

and thus d∗AFA = 0.

1.2 The moduli spaceMg(P )

For the following, we choose a compact oriented Riemann surface Σ of genus g and anon-trivial principal SO(3)-bundle π : P → Σ. Next, we define the even gauge groupG0(P ) as the elements of G(P ) that can be lifted to a map P → SU(2); equivalently, itcan be seen as the unit component of G(P ) and for more details we refer to [6]. Finallywe can introduce the moduli space

Mg(P ) := A0(P )/G0(P )

which is a compact smooth manifold of dimension 6g − 6; this was proved Dostoglouand Salamon in [6] using the works of Newstead in [13] in the following way. Bythe holonomy, the above definition of G0(P ) and the fact that we work with a non-trivial bundle, the moduli space Mg(P ) can be identified with the space of odd ho-momorphisms θ : π1(Σ) → SU(2) modulo SU(2), these homomorphisms are calledodd because they have to satisfy the condition

∏gi=1[θ(ai), θ(bi)] = −1, where ai, bi,

1 ≤ i ≤ g, are the generators of π1(Σ), i.e.

Mg(P ) := A0(P )/G0(P ) ∼= homodd(π1(Σ),SU(2))/SU(2). (1.4)

Since an element of homodd(π1(Σ),SU(2)) identifies the 2g-tuple of generetors a1, b1,..., ag, bg of π1(Σ) with a 2g-tuple of matrices A1, B1, ..., Ag, Bg ∈ SU(2) which sat-isfies

∏g1[Ai, Bi] = −1, if we define the surjective homomorphism µg : SU(2)2g →

SU(2), (A1, B1, ..., Ag, Bg) 7→∏g

i=1[Ai, Bi], we have thatMg(P ) ∼= µ−1g (−1)/SU(2)

1.2 The moduli spaceMg(P ) 7

where SU(2) acts by conjugation on µ−1g (−1).

This construction gives us a non-abelian holonomy group and hence, an even gaugeelement u, which is an element of the isotropy group2, maps P to the identity; in fact itslift u : P → SU(2) has to commute, at each point p ∈ P , with a non-abelian subgroupof SU(2), hence u(p) ∈ centre(SU(2)) = ±1 and u(p) is, therefore, equal 1. Inaddition, dA : Ω0(Σ, gP )→ Ω1(Σ, gP ) is injective; assume not, then by equation (1.1),there would be a curve ut = exp(tφ) ⊂ G0(P ), dAφ = 0, such that u∗tA = A for t near0; hence the isotropy group would not be discrete and thus, we have a contradiction.Moreover, d∗AdA : Ω0(Σ, gP ) → Ω0(Σ, gP ) is invertible, because the fact that dA isinjective implies that d∗A is surjective (by the decomposition of Ω0(Σ, gp), see equation(1.3)) and in addition im d∗A = im d∗AdA by the decomposition of Ω1(Σ, gp).

Theorem 3. Suppose π : P → Σ is a non-trivial principal SO(3)-bundle and Σ is acompact oriented Riemann surface of genus g ≥ 1. ThenMg(P ) is a smooth (6g−6)-manifold.

This theorem was proved by Dostoglou and Salamon in [6] combining (1.4) with theresults of Newstead in [13] that imply also

Theorem 4. Let g ≥ 2.Mg(P ) is connected and simply connected.

Remark. If g = 2, then the moduli spaceM2(P ) can be seen as an intersection ofquadrics in P5 (cf. [14]) .

1.2.1 Tangent space

The infinitesimal gauge transformation for Ψ ∈ Ω0(Σ, gP ) acts on a connection by

A(P )→ TA(P ); A 7→ −dAΨ

and thus, the tangent space at [A] ∈ Mg(P ), with A ∈ A0(P ), can be identified withthe first homological group H1

A(Σ, gP ), in fact because of equation (1.1) and of theorthogonal splitting ker dA = im dA ⊕H1

A(Σ, gP ), we have

TAA0(P )/im dA = ker dA/im dA = H1A(Σ, gP ). (1.5)

Hence if we choose a conformal structure on Σ, then we have a complex structure onMg(P ) which is not, but the Hodge-*- operator acting on H1

A(Σ, gP ). We refer to [6]and [11] for more details.

Moreover, since the tangent space of [A] ∈ Mg(P ), for every A ∈ A0(P ), can beidentified with H1

A(Σ, gP ), we have a symplectic form ωA(a, b) =∫

Σ〈a ∧ b〉, for

a, b ∈ H1A(Σ, gP ), and a complex structure defined by the Hodge-∗-operator. Since the

symplectic 2-form does not depend on the base connection A, it is constant and thus,closed. Hence,Mg(P ) is a Kahler manifold; this symplectic approach of the space ofconnections was introduced by Atiyah and Bott in [4].

2An u ∈ G(P ) is an element of the isotropy group of a connection A if and only if u∗A = A.


1.2.2 Mg(P ) and the moment map theoryThe infinitesimal gauge transformation can be seen as an Hamiltonian vector field ofthe Hamiltonian function

HΨ : A(P )→ R; A 7→ HΨ(A) = 〈∗FA,Ψ〉 = −ω(FA,Ψ);

where Ψ ∈ Ω0(Σ, gP ); in fact, for an α ∈ TAA(P ), then

dHΨ(A)α =d

dt

∣∣∣t=0HΨ(A+ tα) =

d

dt

∣∣∣t=0〈∗(FA + tdAα +

1

2t2[α ∧ α]),Ψ〉

=〈dAα, ∗Ψ〉 = 〈α, ∗dAΨ〉 = ω(α, dAΨ),

and the moment map is clearly µ : A(P ) → Ω2(gP ); A 7→ FA. Since µ−1(0) =A0(P ), we have that the Mardestein-Weinstein quotient is in this case Mg(P ) =A0(P )/G0.

Lemma 5. We choose two flat connections A′, A′′ ∈ A0(P ), then

minu∈G(P )

‖A′ − u∗A′′‖L2(Σ) ≤ d([A′], [A′′])

where d(·, ·) denotes the distance between [A′] and [A′′] on the smooth compact mani-foldMg(P ).

Proof. First, we choose a curve Ξ : [0, 1] → M such that Ξ(0) = [A′], Ξ(1) = [A′′]and ∫ 1

0

‖∂tΞ‖L2(Σ) dt = d ([A′], [A′′]) (1.6)

then we pick a differentiable lift A : [0, 1] → A0(P ) of Ξ with A(0) = A′ and[A(t)] = Ξ(t). A(t) is a curve on space of flat connections and thus its derivative∂tA is an element in the kernel of dA. Hence we can decompose its derivative in theharmonic part πA (∂tA) and in the image of a 0-form dAξ := ∂tA − πA (∂tA). Next,we define η := −

∫ t0ξ ds and then

exp(η)∗A = exp(−η)A exp(η) + exp(η)−1d exp(η)

and hence

∂t (exp(η)∗A) = exp(−η) [A, ∂tη] exp(η) + exp(η)∗∂tA

+ exp(−η) d(∂tη) exp(η)

= exp(η)∗ (∂tA+ dA (∂tη)) .

Thus we can conclude that

∂t (exp(η)∗A) = exp(η)∗ (∂tA− dAξ) = exp(η)∗πA(∂tA),

which implies

d∗exp(η)∗A (∂t (exp(η)∗A)) =d∗exp(η)∗A (exp(η)∗πA(∂tA))

= exp(η)∗ (d∗AπA(∂tA)) = 0.

1.3 Estimates on the surface 9

Therefore

minu∈G(P )

‖A′ − u∗A′′‖L2(Σ) ≤ ‖A′ − exp(η(1))∗A(1)‖L2(Σ)

≤∫ 1

0

‖∂t (exp(η)∗A)‖L2(Σ) dt =

∫ 1

0

‖πA (∂tA)‖L2(Σ) dt

=

∫ 1

0

‖∂tΞ‖L2(Σ) dt = d ([A′], [A′′]) .

1.3 Estimates on the surface

In this section we list some estimates that will be needed all along this exposition. Thefirst two lemmas were proved in [7] (lemma 7.6 and lemma 8.2) for p > 2 and q =∞;the proofs in the case p = 2 and 2 ≤ q <∞ is similar.

Lemma 6. We choose p > 2 and q = ∞ or p = 2 and 2 ≤ q < ∞. Then there existtwo positive constants δ and c such that for every connection A ∈ A(P ) with

‖FA‖Lp(Σ) ≤ δ

there are estimates

‖ψ‖Lq(Σ) ≤ c‖dAψ‖Lp(Σ), ‖dAψ‖Lq(Σ) ≤ c‖dA ∗ dAψ‖Lp(Σ),

for ψ ∈ Ω0(Σ, gP ).

Lemma 7. We choose p > 2 and q = ∞ or p = 2 and 2 ≤ q < ∞. Then there existtwo positive constants δ and c such that the following holds. For every connectionA ∈ A(P ) with

‖FA‖Lp(Σ) ≤ δ

there exists a unique section η ∈ Ω0(Σ, gP ) such that

FA+∗dAη = 0, ‖dAη‖Lq(Σ) ≤ c‖FA‖Lp(Σ).

The following lemma is a symplified version of the lemma B.2. in [18] where Salamonallows also to modify the complex structure on Σ if it is C1-closed to a fixed one.

Lemma 8. Fix a connection A0 ∈ A0(P ). Then, for every δ > 0, C > 0, andp ≥ 2, there exists a constant c = c(δ, C,A0) ≥ 1 such that, if A ∈ A(P ) satisfy‖A− A0‖L∞(Σ) ≤ C then, for every ψ ∈ Ω0(Σ, gP ) and every α ∈ Ω1(Σ, gP ),

‖ψ‖pLp(Σ) ≤ δ‖dAψ‖pLp(Σ) + c‖ψ‖pL2(Σ), (1.7)

‖α‖pLp(Σ) ≤ δ(‖dAα‖pLp(Σ) + ‖dA ∗ α‖pLp(Σ)

)+ c‖α‖pL2(Σ). (1.8)


Lemma 9. We choose p ≥ 2. There is a positive constant c such that the followingholds. For any connection A ∈ A0(P ) and any α ∈ Ω1(Σ, gP )

‖α‖Lp(Σ) + ‖dAα‖Lp(Σ) + ‖d∗Aα‖Lp(Σ) + ‖d∗AdAα‖Lp(Σ) + ‖d∗Ad∗Aα‖Lp(Σ)

≤c‖(dAd∗A + d∗AdA)α‖Lp + ‖πA(α)‖Lp(Σ).(1.9)

Proof. For any flat connectionA, the orthogonal splitting of Ω1(Σ) = im dA⊕im d∗A⊕H1A(Σ, gP ) implies that there is a positive constant c0 such that

‖dAd∗Aα‖Lp(Σ) + ‖d∗AdAα‖Lp(Σ) ≤ c0‖(dAd∗A + d∗AdA)α‖Lp(Σ);

thus, we can conclude the proof applying the lemma 6.

Lemma 10. We choose p ≥ 2. There is a positive constant c such that the followingholds. For any δ > 0, any connection A ∈ A0(P ), α ∈ Ω1(Σ, gP ) and ψ ∈ Ω0(Σ, gP )

‖dAα‖Lp(Σ) ≤ c(δ−1 ‖α‖Lp(Σ) + δ ‖d∗AdAα‖Lp(Σ)

),

‖d∗Aα‖Lp(Σ) ≤ c(δ−1 ‖α‖Lp(Σ) + δ ‖dAd∗Aα‖Lp(Σ)

),

‖dAψ‖Lp(Σ) ≤ c(δ−1 ‖ψ‖Lp(Σ) + δ ‖d∗AdAψ‖Lp(Σ)

).

(1.10)

Furthermore, for any δ > 0, any connection A + Ψdt ∈ A(P × S1), α + ψdt ∈Ω1(Σ× S1, gP )

ε ‖∇tα‖Lp(Σ×S1) ≤ c(δ−1 ‖α‖Lp(Σ×S1) + δε2 ‖∇t∇tα‖Lp(Σ×S1)

),

ε2 ‖∇tψ‖Lp(Σ×S1) ≤ c(δ−1ε ‖ψ‖Lp(Σ×S1) + δε3 ‖∇t∇tψ‖Lp(Σ×S1)

).

(1.11)

Proof. The last two estimates follow analogously to the lemma D.4. in [17]. The firstcan be proved as follows. We choose q such that 1

p+ 1

q= 1 then

‖dAα‖Lp(Σ) = supα

〈dAα, δ−1α + δdAd∗Aα〉

‖δ−1α + δdAd∗Aα‖Lq(Σ)

≤ supα

c〈δ−1α + δd∗AdAα, d∗Aα〉

δ−1 ‖α‖Lq(Σ) + δ ‖dAd∗Aα‖Lq(Σ) + ‖d∗Aα‖Lq (Σ)

≤(δ−1 ‖α‖Lp(Σ) + δ ‖d∗AdAα‖Lp(Σ)

)supα

c ‖d∗Aα‖Lq(Σ)

‖d∗Aα‖Lq(Σ)

=c(δ−1 ‖α‖Lp(Σ) + δ ‖d∗AdAα‖Lp(Σ)

).

where the supremum is taken over all non-vanishing 1-forms α ∈ Lq with dAd∗Aα ∈Lq. The norm ‖δ−1α + δdAd

∗Aα‖Lq(Σ) is never 0 because∥∥δ−1α + δdAd

∗Aα∥∥2

L2(Σ)= δ−2 ‖α‖2

L2(Σ) + δ2 ‖dAd∗Aα‖2L2(Σ) + 2 ‖d∗Aα‖

2L2(Σ) 6= 0,

otherwise we would have a contradiction by the Holder inequality and the operatorδ−1 + δdAd

∗A is surjective. The second and the third estimate of the lemma can be

shown exactly in the same way.

1.4 Perturbed geodesics onMg(P ) 11

1.4 Perturbed geodesics onMg(P )

1.4.1 Unperturbed energy functionalThe goal is to find a loopA ⊂ C∞(R/Z,A0) such that the projection Π(A) onMg(P )is a geodesic; Π is defined by

Π : A0(P )→Mg(P ); B 7→ [B] = B mod G0

and since ∂tA ∈ TAA0 = H1A(Σ, gP ) ⊕ im dA and dΠ(A)∂tA ∈ TΠ(A)Mg(P ) which

corresponds to H1A(Σ, gP ),

0 = d∗A(∂tA− dAΨ) = dA(∂tA− dAΨ)

for a Ψ such that Ψ(t) ∈ Ω0(Σ, gP ) for all t ∈ S1. Hence, since d∗AdA is invertible,

Ψ = (d∗AdA)−1d∗A∂tA

and Ψ is uniquely determined; thus,

πA(∂tA) = ∂tA− dA(d∗AdA)−1d∗A∂tA = ∂tA− dAΨ.

The unperturbed energy of our curve is, therefore,

E(A) =1

2

∫ 1

0

|dΠ(A)∂tA|2dt =1

2

∫ 1

0

|∂tA− dAΨ|2dt. (1.12)

1.4.2 PerturbationWe consider a time dependent Hamiltonian map

H : R/Z×A0(P )→ R; (t, A) 7→ Ht(A)

which is invariant under G0(P ) and constructed using the holonomy (see [6]); then wecan perturbe the energy functional subtracting from E the integral of Ht(.), i.e.

EH(A) =1

2

∫ 1

0

|∂tA− dAΨ|2dt−∫ 1

0

Ht(A)dt. (1.13)

The equivariance of Ht(·) means that we indroduce a perturbation on the energy func-tional on the loop space of the smooth manifold Mg(P ). Weber ([23]) using theThom-Smale transversality proved that the set

νreg := H ∈ C∞(S1 ×Mg(P ),R) |The Jacobi operator for EH is bijective for all critical loops

is open and dense in C∞(S1 ×Mg(P ),R) endoved with the compact-open topologyand νreg is residual. Therefore we can choose Ht near the zero function as we like suchthat the Jacobi operator of EH for all the perturbed geodesics is invertible and fromnow our perturbation is choosen with this property. Furthermore, in the same paper


Weber proved that below a given energy level we have only finite perturbed geodesics.

Next, we define a perturbation Ht : A(P ) → R, where Ht(A) = H(A) for everyA ∈ A0(P ). A first approach is to pick a gauge invariant holonomy perturbation onA(P ) since every Hamiltonian Ht can be constructed in this way (cf. [6]). Since Ht isconstant along G(P )∗A for a given connection A ∈ A(P ) and

TA(G(P )∗A) = im dA,

for any ψ ∈ Ω0(Σ, gP )

0 =dHt(A)dAψ =

∫Σ

〈Xt(A) ∧ dAψ〉 =

∫Σ

〈dAψ, ∗Xt(A)〉

=

∫Σ

〈ψ, d∗A ∗Xt(A)〉 =

∫Σ

〈ψ, ∗dAXt(A)〉

and hencedAXt(A) = 0. (1.14)

Another possibility is the following. We pick a smooth map ρ : [0,∞) → [0, 1] withthe property that ρ(x) = 0 if x ≥ δ2

0 and ρ(x) = 1 if x ≤(

2δ03

)2 for a δ0 which satisfiesthe conditions of the lemmas 6 and 7 for p = 2 and q = 4. Then we define Ht(A) = 0for every A with ‖FA‖L2 ≥ δ0 and

Ht(A) := ρ(‖FA‖2

L2

)Ht (A+ ∗dAη(A))

otherwise, where η is the unique 0-form given by the theorem 7 for the connection A.In this case, ifA is flat thenHt(A+∗sdAη) is constant for every 0-form η ∈ Ω0(Σ, gP )and every s ∈ (−ε, ε) with ε sufficiently small. Thus

0 =∂sHt(A+ ∗sdAη)∣∣s=0

= dHt(A) ∗ dAη =

∫Σ

〈Xt(A) ∧ ∗dAη〉

=

∫Σ

〈∗dAη, ∗Xt(A)〉 =

∫Σ

〈∗η, dA ∗Xt(A)〉

and hence we can conclude that dA ∗Xt(A) = 0.

In both cases, the time-dependent Hamiltonian vector field Xt : A(P ) → Ω1(gP ) isdefined such that, for any 1-form α and any connectionA, dHt(A)α =

∫Σ〈Xt(A)∧α〉.

1.4.3 The map F0

In this section, we will show that we can characterise the perturbed geodesics usingthe map

F0(A,Ψ) :=

(−∇t(∂tA− dAΨ)− ∗Xt(A)−d∗A(∂tA− dAΨ) ∧ dt

)=

(F0

1 (A,Ψ)F0

2 (A,Ψ)

)(1.15)

defined for two loops A(t) ∈ A(P ) and Ψ(t) ∈ Ω0(Σ, gP ). In fact, a closed curve A,A(t) ∈ A0(P ) for all t ∈ S1 ∼= R/Z, descends to a perturbed geodesic if and only ifF0(A,Ψ) ∈ im d∗A × 0 by the next theorem.


Theorem 11. A closed curve A, A(t) ∈ A0 for all t ∈ S1 ∼= R/Z, descends to aperturbed geodesic if and only if there are Ψ(t) ∈ Ω0(Σ, gP ) and ω(t) ∈ Ω2(Σ, gP )such that

−∇t(∂tA− dAΨ)− ∗Xt(A)− d∗Aω = 0, (1.16)

d∗A(∂tA− dAΨ) = 0, (1.17)

where∇t := ∂t + [Ψ, .]. If this holds ω is the unique solution of

dAd∗Aω = [(∂tA− dAΨ) ∧ (∂tA− dAΨ)]− dA ∗Xt(A). (1.18)

Remark. In the section 1 we defined the moduli space of flat connectionsMg(P ) bytaking the quotient A0(P )/G0(P ) where G0(P ) is the even gauge group and thus ageodesic γ(t) ⊂Mg(P ) lifts to a closed path in A0(P ) which is unique modulo

G0

(P × S1

):= g ∈ G(P × S1) | g(t) ∈ G0(P ) ∀t ∈ S1.

The group G0 (P × S1) acts clearly also on the connections A(P × S1) of a principalbundle P × S1 → Σ× S1.

Next, we denote the set of perturbed geodesics below a energy level b by

CritbEH :=A+ Ψdt ∈ L(A0(P )⊗Ω0(Σ, gP ) ∧ dt)|EH(A) ≤ b, (1.16), (1.17)

.

Proof. Let As and Ψs, s ∈]−ε, ε[, ε > 0, be two variations of A and Ψ such that A0 =A, As ∈ A0(P ) and d

ds

∣∣s=0

As = α, Ψ0 = Ψ, Ψs ∈ Ω0(Σ, gP ) and dds

∣∣s=0

Ψs = ψ. It isimportant to remark that, since Ψs depend on As, we have not a choice of the variationof Ψ, i.e. Ψs := (d∗AsdAs)

−1d∗As∂tAs. Then A is a critical point if for each variation ofthis kind the derivative of the energy respect to s vanishes in 0. i.e.

0 =d

ds

∣∣∣s=0

EH(As) =d

ds

∣∣∣s=0

(1

2

∫ 1

0

|∂tAs − dAsΨs|2dt−∫ 1

0

Ht(As)dt

)=

∫ 1

0

〈 dds

(∂tAs − dAsΨs)∣∣∣s=0

, ∂tA− dAΨ〉dt−∫ 1

0

d

dt(Ht(As))

∣∣∣s=0

dt

=

∫ 1

0

〈∂td

dsAs

∣∣∣s=0− dA

d

dsΨs

∣∣∣s=0− d

ds(d+ [As ∧ .])

∣∣∣s=0

Ψ, ∂tA− dAΨ〉dt

−∫ 1

0

dHt(A)α dt

=

∫ 1

0

〈∂tα− dAψ − [α,Ψ], ∂tA− dAΨ〉dt−∫ 1

0

〈Xt(A) ∧ α〉dt

next, since∇t := ∂t + [Ψ, .] and its dual satisfies∇∗t = ∗∇t∗ = −∇t on the 1-forms,

=

∫ 1

0

〈∇tα− dAψ, ∂tA− dAΨ〉dt−∫ 1

0

〈α ∧ ∗(∗Xt(A))〉dt

and since 〈α ∧ ∗(∗Xt(A))〉 = 〈α, ∗Xt(A)〉, we have

=

∫ 1

0

〈α,−∇t(∂tA− dAΨ)− ∗Xt(A)〉dt−∫ 1

0

〈ψ,=0︷︸︸︷

d∗A(∂tA− dAΨ)〉dt.


This implies that−∇t(∂tA−dAΨ)−∗Xt(A) ∈ Ω1(Σ, gP ) is orthogonal toH1A(Σ, gP )

and as concequence −∇t(∂tA− dAΨ)− ∗Xt(A) is an image of d∗A because

d∗A (∇t(∂tA− dAΨ) + ∗Xt(A))

=∇td∗A(∂tA− dAΨ) + ∗ [(∂tA− dAΨ) ∧ ∗(∂tA− dAΨ)] + ∗dAXt(A) = 0

(1.19)

where in the last step we used the lemma 1.14 and that [α ∧ ∗α] = 0 for any 1-formα. Thus, there exists a loop ω on the 2-forms, i.e. ω(t) ∈ Ω2(Σ, gP ), with the propertyd∗Aω = −∇t(∂tA−dAΨ)−∗Xt(A) and hence, πA(−∇t(∂tA−dAΨ)−∗Xt(A)) = 0.Moreover, we have

dAd∗Aω =− dA∇t(∂tA− dAΨ)− dA ∗Xt(A)

=− ([dA,∇t] +∇tdA)(∂tA− dAΨ)− dA ∗Xt(A)

=[(∂tA− dAΨ) ∧ (∂tA− dAΨ)]− dA ∗Xt(A)

where the last identity follows from dA(∂tA− dAΨ) = 0 and from the first commuta-tion formula of the next lemma.

Lemma 12. We have the following two commutation formulas:

[dA,∇t] = −[(∂tA− dAΨ) ∧ · ]; (1.20)

[d∗A,∇t] = ∗[(∂tA− dAΨ) ∧ ∗ · ]. (1.21)

Proof. The lemma follows from the definitions of the operators using the Jacobi iden-tity for the super Lie bracket operator.

1.4.4 The Jacobi operator of a perturbed geodesicLemma 13. The Jacobi operator of a loop A ⊂ A0, which descends to a perturbedgeodesic onMg(P ), is

D0(A)(α, ψ) =πA (2[ψ, (∂tA− dAΨ)] + d ∗Xt(A)α +∇t∇tα)

+ πA(∗[α ∧ ∗dA(d∗AdA)−1 (∇t(∂tA− dAΨ) + ∗Xt(A))

]) (1.22)

where α(t) ∈ H1A(t)(Σ, gP ), Ψ is defined uniquely by

d∗A(∂tA− dAΨ) = 0 (1.23)

and ψ(t) ∈ Ω0(Σ, gP ) by

−2 ∗ [α ∧ ∗(∂tA− dAΨ)]− d∗AdAψ = 0. (1.24)

Proof. Let As ⊂ A0(P ), Ψs ⊂ Ω0(Σ, gP ) and βs ⊂ Ω1(Σ, gP ), s ∈]− ε, ε[, ε > 0, bevariations of A, Ψ and of a β, β(t) ∈ H1

A(Σ, gP ) for all t ∈ S1, such that

d

dsAs

∣∣∣s=0

= α,d

dsΨs

∣∣∣s=0

= ψ, ∇t = ∂t + [Ψs, . ], β0 = β.


We have to compute the second variational formula and hence

d

ds

∣∣∣s=0

(∫ 1

0

〈πAs(∇t(∂tAs − dAsΨs) + ∗Xt(As)), βs〉dt)

=

∫ 1

0

〈 dds

∣∣∣s=0

πAs(∇t(∂tAs − dAsΨs) + ∗Xt(As)), β〉dt

because 〈πA(∇t(∂tA− dAΨ) + ∗Xt(A)), dds

∣∣s=0

βs〉 = 0; since β(t) ∈ H1A(Σ, gP ), the

Jacobi operator is

πA

(d

ds

∣∣∣s=0

πAs(∇t(∂tAs − dAsΨs) + ∗Xt(As))

)=πA

( dds

∣∣∣s=0

(∇t(∂tAs − dAsΨs) + ∗Xt(As)

− d∗AsdAs(d∗AsdAs)

−1(∇t(∂tAs − dAsΨs) + ∗Xt(As)

)))we compute separatly the derivative of the summands and with the Leibniz rule we get

=πA

(d

ds

(∇t(∂tAs − dAsΨs)

)∣∣∣s=0

+ d ∗Xt(A)α

)− πA

(d

dsd∗As

∣∣∣s=0

dA(d∗AdA)−1 (∇t(∂tA− dAΨ) + ∗Xt(A))

)− πA

(d∗A

d

ds

∣∣∣s=0

(dAs(d

∗AsdAs)

−1(∇t(∂tAs − dAsΨs) + ∗Xt(As)

)))the third term is 0 because of im d∗A ⊂ kerπA, i.e.

=πA ([ψ, (∂tA− dAΨ)] + d ∗Xt(A)α) + πA

(∇t

d

ds

∣∣∣s=0

(∂tAs − dAsΨs)

)+ πA

(∗[α ∧ ∗dA(d∗AdA)−1 (∇t(∂tA− dAΨ) + ∗Xt(A))]

)Since d

dsdAs∣∣s=0

= [α ∧ .], we obtain

=πA ([ψ, (∂tA− dAΨ)] + d ∗Xt(A)α) + πA (∇t(∂tα− [α,Ψ]− dAψ))

+ πA(∗[α ∧ ∗dA(d∗AdA)−1 (∇t(∂tA− dAΨ) + ∗Xt(A))]

)Using the commutation formula (1.20) and ∂tα− [α,Ψ] = ∇tα, we have (1.22), i.e

D0(A)(α, ψ) = πA (2[ψ, (∂tA− dAΨ)] + d ∗Xt(A)α)

+ πA(∗[α ∧ ∗dA(d∗AdA)−1 (∇t(∂tA− dAΨ) + ∗Xt(A))] +∇t∇tα

).

The identity d∗A(∂tA− dAΨ) = 0 yields to the condition (1.24):

0 =d

dt

∣∣∣t=0

(d∗At(∂tAt − dAtΨt)

)=− ∗[α ∧ ∗(∂tA− dAΨ)] + d∗A(∂tα− [α,Ψ]− dAψ)


Since ∂tα− [α,Ψ] = ∇tα, we get

=− ∗[α ∧ ∗(∂tA− dAΨ)] + d∗A∇tα− d∗AdAψ=− 2 ∗ [α ∧ ∗(∂tA− dAΨ)] +∇td

∗Aα− d∗AdAψ

where the last step follows from the commutation formula (1.21).

1.5 Perturbed Yang-Mills connections

1.5.1 The map F ε

Now, we choose a Riemann metric gΣ on the surface Σ and we consider the manifoldΣ×S1 with the partial rescaled metric (ε2gΣ⊕ gS1) for a given ε ∈]0, 1]; furthermore,we denote by πε : P × S1 → Σ × S1 the principal SO(3)-bundle over Σ × S1 andwe assume that the restriction P × s → Σ × s is non-trivial. If we choose aconnection Ξ = A + Ψ dt ∈ A(P × S1) where A(t) ∈ A(P ), Ψ(t) ∈ Ω0(Σ, gP )for all t ∈ S1, then the L2-norm induced by the metric (ε2gΣ ⊕ gS1) of the curvatureFΞ = FA − (∂tA− dAΨ) ∧ dt is given by

‖FΞ‖2L2 =

∫ 1

0

(1

ε2‖FA‖2

L2(Σ) + ‖∂tA− dAΨ‖2L2(Σ)

)dt

where the norm ‖ · ‖L2(Σ) := 〈·, ·〉 is the norm over Σ given by (1.2); if we add thesame perturbation as in (1.13), we can define the perturbed Yang-Mills functional

YMε,H(Ξ) :=1

2

∫ 1

0

(1

ε2‖FA‖2

L2(Σ) + ‖∂tA− dAΨ‖2L2(Σ)

)dt−

∫ 1

0

Ht(A) dt.

(1.25)A critical connection Ξε = Aε + Ψεdt ∈ A(P × S1) of YMε,H is called a perturbedYang-Mills connection and has to satisfy the equation d∗εΞεFΞε − ∗Xt(A) = 0 that isequivalent to the two conditions

1

ε2d∗AεFAε = ∇t(∂tA

ε − dAεΨε) + ∗Xt(Aε), (1.26)

− 1

ε2d∗Aε(∂tA

ε − dAΨε) = 0. (1.27)

In the following, if we write a perturbed Yang-Mills connection as Ξε = Aε + Ψεdtwith apex ε, then we mean that Ξε is a critical point of the functional YMε,H and wedenote the set of perturbed Yang-Mills connections below an energy level b by

CritbYMε,H :=

Ξε ∈A(P × S1)| YMε,H (Ξε) ≤ b, (1.26), (1.27).

If we fix a connection Ξ0 = A0 + Ψ0dt, then we can define an ε-dependent map F ε,for ε > 0, by F ε(A,Ψ) = F ε1(A,Ψ) + F ε2(A,Ψ) and

F ε1(A,Ψ) =1

ε2d∗AFA −∇t(∂tA− dAΨ)− ∗Xt(A)

+1

ε2dAd

∗A(A− A0)− dA∇t(Ψ−Ψ0),

(1.28)

1.5 Perturbed Yang-Mills connections 17

F ε2(A,Ψ) =

(− 1

ε2d∗A(∂tA− dAΨ) +

1

ε2∇td

∗A(A− A0)−∇2

t (Ψ−Ψ0)

)∧ dt;

(1.29)then the zeros of F ε are perturbed ε-Yang-Mills connections and they satisfy the localgauge condition

d∗εΞ0

(Ξ− Ξ0

)=

1

ε2d∗A0(A− A0)−∇Ψ0

t (Ψ−Ψ0) = 0

respect to the reference connection A0 + Ψ0dt by the following remark already con-sidered by Hong (cf. [9]).

Remark. We remark that Ξε = Aε + Ψεdt is a perturbed Yang-Mills connection onP×S1 and satisfies the gauge condition d∗εΞε(α

ε+ψεdt) = 0 with αε+ψεdt := Ξε−Ξ0

if and only ifdΞεd

∗εΞε(α

ε + ψεdt) + d∗εΞεFΞε − ∗Xt(Aε) = 0. (1.30)

In fact, if we derive the equation (1.30) by d∗εΞε , we obtain

d∗εΞεdΞεd∗εΞε(α

ε + ψεdt) + d∗εΞεd∗εΞεFΞε − d∗εΞε ∗Xt(A

ε) = 0

then, since ∗Xt(Aε) does not contain 1-forms over S1 and since d∗εΞεd

∗εΞεFΞε = − ∗

[FΞε ∧ ∗FΞε ] = 0,

d∗εΞεdΞεd∗εΞε(α

ε + ψεdt)− ∗ 1

ε2dAεXt(A

ε) = d∗εΞεdΞεd∗εΞε(α

ε + ψεdt) = 0

where the last step follows from (1.14). The last computation implies that d∗εΞε(αε +

ψεdt) = 0 and thus it follows also that d∗εΞεFΞε = 0.

Remark. If we choose 32< p < ∞ and b > 0, then for every perturbed Yang-Mills

connection Ξε = Aε + Ψεdt ∈ A1,p(P × S1), there exists a gauge transformationu ∈ G2,p

0 (P × S1) such that u∗Ξε is smooth. Katrin Wehrheim proved this statementfor weak Yang-Mills connections (cf. [25] theorem 9.4) and her proof holds also forperturbed Yang-Mills connections.

1.5.2 The linear operator Dε

If we linearise the equations (1.26) and (1.27) we obtain the two components of theJacobi operator

Jacε,H(Ξε) : Ω1(Σ× S1, gP )→ Ω1(Σ× S1, gP )

of a perturbed Yang-Mills connection:

Jacε,H(Aε + Ψεdt)(α, ψ) =1

ε2d∗AεdAεα +

1

ε2∗ [α ∧ ∗FAε ]

− d ∗Xt(Aε)α−∇t∇tα + dAε∇tψ − 2[ψ, (∂tA

ε − dAεΨε)]

+

(1

ε2∗ [α ∧ ∗(∂tAε − dAεΨε)]− 1

ε2∇td

∗Aεα +

1

ε2d∗AεdAεψ

)dt,

(1.31)


for any α(t) ∈ Ω1(Σ, gP ) and ψ(t) ∈ Ω0(Σ, gP ). In 1982, Atiyah and Bott (cf. [4])showed that the Jacobi operator of a Yang-Mills connection Ξε = Aε + Ψεdt is Fred-holm of index 0; for the perturbed case we have the same result. First, we recall thatthe gauge group acts on the 1-forms adding the image of dΞε and hence α+ ψ dt is anelement of Ω1(Σ× S1, gP )/GΣ(P × S1) if and only if

0 = 〈α + ψ dt, dΞεφ〉 = 〈d∗εΞε(α + ψ dt), φ〉

for every φ ∈ Ω0(Σ × S1, gP ) and consequently, if and only if α + ψ dt ∈ ker d∗εΞε .Therefore, under the condition d∗εΞε(α + ψ dt) = 0 we have that

Jacε,H(Ξε)(α + ψ dt) = Jacε,H(Ξε)(α + ψ dt) + dΞεd∗Ξε(α + ψ dt) (1.32)

which can be written as

(d∗εΞεdΞε + dΞεd∗εΞε) (α+ψ dt)+∗[(α+ψ dt)∧∗FΞε ]−ε2d∗Xt(A

ε)(α+ψ dt) (1.33)

where the first term is the Laplace operator of α + ψ dt and the second one is oforder zero and thus, we have a selfadjoint elliptic operator and therefore, a Fredholmoperator with index 0. In addition, this can allow us to work with (1.33) instead ofusing the Jacobi operator and (1.33) can be written as the operator Dε(A + Ψdt) :=Dε1(A+ Ψdt) +Dε2(A+ Ψdt) dt given by

Dε1(A+ Ψdt)(α, ψ) :=1

ε2(d∗AdAα + dAd

∗Aα + ∗[α ∧ ∗FA])− d ∗Xt(A)α

−∇t∇tα− 2[ψ, (∂tA− dAΨ)]

Dε2(A+ Ψdt)(α, ψ) :=1

ε2(2 ∗ [α ∧ ∗(∂tA− dAΨ)] + d∗AdAψ)−∇t∇tψ.

(1.34)

Moreover, the operator Dε is almost linearisation of F ε; to be precise Dε does notcontain the derivatives of dA, d∗A and ∇t of the last two terms in both components(1.28) and (1.29), because these can be treated like quadratic terms as we will see inthe lemma 23. If the reference connection A+ Ψ dt is clear from the context, then wewill write the operators without indicating it (e.g. Dε1(α, ψ) := Dε1(A,Ψ)(α, ψ)).

1.6 Norms I

If we fix a connection Ξ0 = A0 + Ψ0dt ∈ A(Σ × S1), then we can define a normon its tangential space and since A(Σ × S1) is an affine space, we can use it as ametric on A(Σ × S1). Let ξ(t) = α(t) + ψ(t) ∧ dt such that α(t) ∈ Ω1(Σ, gP ) andψ(t) ∈ Ω0(Σ, gP ) or α(t) ∈ Ω2(Σ, gP ) and ψ(t) ∈ Ω1(Σ, gP ). Then we define thefollowing norms3

‖ξ‖p0,p,ε :=

∫ 1

0

(‖α‖pLp(Σ) + εp‖ψ‖pLp(Σ)

)dt,

‖ξ‖∞,ε := ‖α‖L∞(Σ×S1) + ε‖ψ‖L∞(Σ×S1)

3We recall that ‖ · ‖2L2(Σ) = 〈·, ·〉 is the inner product (1.2) induced by the metric gΣ.

1.6 Norms I 19

and

‖ξ‖pΞ0,1,p,ε:=

∫ 1

0

(‖α‖pLp(Σ) + ‖dA0α‖

pLp(Σ) + ‖d∗A0

α‖pLp(Σ) + εp‖∇tα‖pLp(Σ)

)dt

+

∫ 1

0

εp(‖ψ‖pLp(Σ) + ‖dA0ψ‖

pLp(Σ) + εp‖∇tψ‖pLp(Σ)

)dt,

‖ψ‖p0,p,ε :=

∫ 1

0

‖ψ‖pLp(Σ) dt.

Inductively,

‖ξ‖pΞ0,k+1,p,ε :=

∫ 1

0

(‖α‖pΞ0,k,p,ε

+ ‖dA0α‖pΞ0,k,p,ε

)dt

+

∫ 1

0

(‖d∗A0

α‖pΞ0,k,p,ε+ εp‖∇tα‖pΞ0,k,p,ε

)dt

+

∫ 1

0

(‖ψ dt‖pΞ0,k,p,ε

+ ‖dA0ψ ∧ dt‖pΞ0,k,p,ε

+ εp‖∇tψ dt‖pΞ0,k,2,ε

)dt.

Also in this case, if the reference connection is clear from the context we write thenorms without mentioning it.

Remark. For i = 1, 2, we can define by W k,p(Σ × S1,ΛiT ∗(Σ × S1) ⊗ gP×S1) theSobolev space of the sections of ΛiT ∗(Σ× S1)⊗ gP×S1 → Σ× S1 as the completionof4

Γ(ΛiT ∗(Σ× S1)⊗ gP ) = Ωi(Σ× S1, gP×S1)

respect to the norm ‖ · ‖Ξ0,k,p,1. Furthermore, we can define the Sobolev space of theconnections on P × S1 as5

Ak,p(P × S1) = Ξ0 +W k,p,

where W k,p = W k,p(Σ× S1, T ∗(Σ× S1)⊗ gP×S1), Ξ0 ∈ A(P × S1).

Remark. The Sobolev space of gauge transformation G2,p0 (P × S1) is the completion

of G0(P × S1) with respect the Sobolev W 1,p-norm on 1-forms, i.e. g ∈ G2,p0 (P × S1)

if g−1dΣ×S1g ∈ W 1,p and hence g : A1,p(P × S1)→ A1,p(P × S1).

Remark. The gauge condition d∗Ξε(Ξε−Ξ0) = 0 assures us that if the perturbed Yang-

Mills connection Ξε is an element of A1,2(P × S1), then, for any k ≥ 2, there is anu ∈ G2,p

0 (P × S1) such that u∗Ξε ∈ Ak,2(P ) (cf. [25], Chapter 9).

Remark. We choose a reference connection Ξ0. Analogously as for the lemma 4.1in [7], if we define ξ = α + ψ dt where α(t) = α(εs) and ψ(t) = εψ(εs), then‖ξ‖k,p,ε = ε

1p‖ξ‖Wk,p for 0 ≤ t ≤ ε−1. In addition, all the Sobolev inequalities hold

as they are stated in the following lemma.

4Let E →M a vector bundle, then ΓE denotes the space of section of the bundle.5For more information, see appendix B of [25].


Theorem 14 (Sobolev estimates I). We choose 1 ≤ p, q <∞ and l ≤ k. Then there isa constant cs such that for every ξ ∈ W k,p(Σ× S1,ΛiT ∗(Σ× S1)⊗ gP×S1), i = 1, 2,and any reference connection Ξ0:

1. If l − 3q≤ k − 3

p, then

‖ξ‖Ξ0,l,q,ε ≤ csε1/q−1/p ‖ξ‖Ξ0,k,p,ε. (1.35)

2. If 0 < k − 3p, then

‖ξ‖Ξ0,∞,ε ≤ csε−1/p ‖ξ‖Ξ0,k,p,ε. (1.36)

Proof. This lemma follows from the last remark and the Sobolev embedding theorem(see theorem B2 in [25]).

Elliptic estimates 2The aim of this chapter is to estimate (theorem 16) the ‖ · ‖2,p,ε-norm of a 1-formξ = α+ ψ dt using the Lp-norm of the operator Dε(Ξ) when Ξ = A+ Ψdt representsa perturbed closed geodesic onMg(P ). We recall that we assume that Jacobi operatoris invertible for every perturbed geodesic.

Hong in [9] proved a weaker estimate which, in our setting, can be identified with

‖α + ψ dt− πA(α)‖1,2,ε + ε‖πA(α)‖1,2,ε

≤cε2‖Dε(Ξ + αε0)(α, ψ)‖0,2,ε + cε‖πADε(Ξ + αε0)(α, ψ)‖L2

where αε0 ∈ im d∗A is the unique solution of

d∗AdAαε0 = ε2∇t(∂tA− dAΨ) + ∗Xt(A

0);

in addition he estended the last estimate to

‖α + ψ dt‖k,2,ε ≤ c‖Dε(Ξ + αε0)(α, ψ)‖k−1,2,ε

and with this inequality he proved the existence of a map from the perturbed geodesicsCritbEH to the perturbed Yang-Mills connections CritbYMε,H , but he did not show itsuniqueness and its surjectivity. With the last two estimates is not possible to obtainthe uniqueness statement of the theorem 25 even for p = 2 and, as we have alreadydiscussed, the surjectivity could not be established using his rescaling of the metric, inparticular because you can not expect that the norms of the curvature ∂tA− dAΨ havea uniforme bound for all the Yang-Mills connections below a given energy level.

For this chapter we choose a regular value b of the energy EH , we fix a perturbedclosed geodesic Ξ = A+ Ψ dt ∈ CritbEH and we define every operator and every normusing this connection. Since the perturbed geodesic Ξ is smooth, there is a positiveconstant c0 which bounds the L∞-norm of the velocity and its derivatives, in particular

‖∂tA− dAΨ‖L∞ + ‖∇t (∂tA− dAΨ)‖L∞ ≤ c0. (2.1)

Notation. In general, we denote a constant, which is needed to fulfill an estimate, byc; it can therefore indicate different constants also in a single computation.

Theorem 15. We choose a constant p ≥ 2. If p = 2 we set j = 0 otherwise j = 1.There exist two constants ε0 > 0 and c > 0 such that

‖ξ‖2,p,ε ≤ c (ε‖Dε(ξ)‖0,p,ε + ‖πA(α)‖Lp) , (2.2)

21

22 2. Elliptic estimates

‖(1− πA)ξ‖2,p,ε ≤ c(ε2‖Dε(ξ)‖0,p,ε + ε‖πA(α)‖Lp + jε2‖∇2

tπA(α)‖Lp), (2.3)

‖α− πA(α)‖2,p,ε ≤cε2(‖Dε1(ξ)‖Lp + ε2‖Dε2(ξ)‖Lp

)+ cε2

(‖πA(α)‖Lp + ‖∇j+1

t πA(α)‖Lp),

(2.4)

for every ξ = α + ψ dt ∈W2,p and 0 < ε < ε0.

We will prove the last theorem in the next sections of the chapter, first for p = 2 andthen for a general p. We want also to remark that the estimates for p = 2 are enough toprove the bijection between the critical connections, but for the identification betweenthe flows between the critical points, that we will explain in the second part of thisthesis, we need the theorem also for p > 2. We recall that by the lemma 11 forperturbed geodesic Ξ = A+ Ψdt we can associate a two form ω defined as the uniquesolution of

dAd∗Aω = [(∂tA− dAΨ) ∧ (∂tA− dAΨ)]

which is equivalent to

ω = dA(d∗AdA)−1(∇t(∂tA− dAΨ) + ∗Xt(A)).

Theorem 16. We choose p ≥ 2 and we assume that there is a constant c0 such that

|〈D0 (α) , α〉| ≥ c0 (‖α‖L2 + ‖∇tα‖L2)2 (2.5)

for every α ∈W2,p. Then there are two constants c > 0 and ε0 > 0 such that

‖πA(α)‖Lp + ‖∇tπA(α)‖Lp + ‖∇2tπA(α)‖Lp

≤c(ε‖Dε (α, ψ) dt‖0,p,ε + ‖πA (Dε1 (α, ψ) + ∗[α ∧ ∗ω]) ‖Lp

),

(2.6)

‖α + ψ dt‖2,p,ε ≤ c(ε ‖Dε(α, ψ)‖0,p,ε + ‖πA (Dε1(α, ψ) + ∗[α ∧ ∗ω]) ‖Lp

), (2.7)

‖α + ψ dt− πA(α)‖2,p,ε

≤c(ε2‖Dε(α, ψ)‖0,p,ε + ε‖πA (Dε1(α, ψ) + ∗[α ∧ ∗ω]) ‖Lp

),

(2.8)

‖α− πA(α)‖2,p,ε ≤cε2‖Dε1(α, ψ)‖Lp + cε4‖Dε2(α, ψ)‖Lp+ cε2‖πA (Dε1(α, ψ) + ∗[α ∧ ∗ω]) ‖Lp

(2.9)

for every α + ψ dt ∈ W 2,p and 0 < ε < ε0.

Remark. The condition (2.5) is always satisfied whenever the Jacobi operator D0 isinvertible because there is a positive constant c such that ‖α‖2

L2 ≤ c 〈D0(α), α〉L2 and

‖∇tα‖2L2 = |〈πA (∇t∇tα) , α〉L2| ≤ c

∣∣⟨D0(α), α⟩L2

∣∣+ c‖α‖2L2

where the last estimate follows from the definition of D0(α) and (2.1).

Proof. We first prove the theorem for p = 2. If we define ψ1 using the condition(1.24), i.e.

d∗AdAψ1 = −2 ∗ [πA(α) ∧ ∗(∂tA− dAΨ)], (2.10)

23

then by the definitions (1.22) and (1.34) we have

‖πA(Dε1(α, ψ) + ∗[α ∧ ∗ω]

)‖L2 (‖πA(α)‖L2 + ‖∇tπA(α)‖L2)

≥|〈πA(Dε1(α, ψ) + ∗[α ∧ ∗ω]

), πA(α)〉|

≥|〈D0 (πA(α), ψ1) , πA(α)〉|− |〈πA

(Dε1(α, ψ) + ∗[α ∧ ∗ω]

)−D0 (πA(α), ψ1) , πA(α)〉|

≥|〈D0 (πA(α), ψ1) , πA(α)〉|− |〈πA(2[ψ − ψ1, (∂tA− dAΨ)]), πA(α)〉|− |〈πA(−d ∗Xt(A)(α− πA(α))−∇t∇t(α− πA(α)), πA(α)〉|≥|〈D0 (πA(α), ψ1) , πA(α)〉|− |〈2[ψ − ψ1, (∂tA− dAΨ)], πA(α)〉|− c (‖α− πA(α)‖L2 + ‖∇tπA(α)‖L2) ‖πA(α)‖L2

where for the last inequality we use the fact that

‖πA(∇t∇t(α− πA(α)))‖L2 ≤ c‖∇t(α− πA(α))‖L2 + c‖α− πA(α)‖L2

because for a 0-form γ and a 2-form ω we can write α−πA(α) = dAγ+d∗Aω and thususing the commutation formulas (1.20) and (1.21) and the estimate (2.1)

‖πA(∇t∇t(α− πA(α)))‖L2 = ‖πA(∇t∇t(dAγ + d∗Aω))‖L2

≤2‖[(∂tA− dAΨ),∇tγ]‖L2 + ‖[∇t(∂tA− dAΨ), γ‖L2

+ 2‖[(∂tA− dAΨ), ∗∇tω]‖L2 + ‖[∇t(∂tA− dAΨ), ∗ω]‖L2

≤c‖∇t(α− πA(α))‖L2 + c‖α− πA(α)‖L2 .

Next, we need to estimate the term |〈2[ψ − ψ1, (∂tA− dAΨ)], πA(α)〉| and in order todo this, by (1.34) and (2.10), we observe that

d∗AdA(ψ − ψ1) = ε2Dε2(α, ψ)− 2 ∗ [(α− πA(α)) ∧ ∗(∂tA− dAΨ)] + ε2∇t∇tψ

and hence, where E denotes 2 ∗ [(α− πA(α)) ∧ ∗(∂tA− dAΨ)], we have

−|〈2[ψ − ψ1, (∂tA− dAΨ)], πA(α)〉|≥ −

∣∣⟨2 [ε2 (d∗AdA)−1Dε2(α, ψ), (∂tA− dAΨ)], πA(α)

⟩∣∣−∣∣⟨2 [(d∗AdA)−1E, (∂tA− dAΨ)

], πA(α)

⟩∣∣−∣∣⟨2 [(d∗AdA)−1 (ε2∇t∇tψ

), (∂tA− dAΨ)

], πA(α)

⟩∣∣≥− c

(‖α− πA(α)‖L2 + ε2‖Dε2(α, ψ)‖L2

)(‖πA(α)‖L2 + ‖∇tπA(α)‖L2)

−(ε2‖∇tψ‖L2 + ε2‖ψ‖L2

)(‖πA(α)‖L2 + ‖∇tπA(α)‖L2) .

Since

|〈D0 (πA(α), ψ1) , πA(α)〉| ≥ c (‖πA(α)‖L2 + ‖∇tπA(α)‖L2)2


we have that

‖πA(α)‖L2 + ‖∇tπA(α)‖L2

≤c‖πA(Dε1(α, ψ) + ∗[α ∧ ∗ω]

)‖L2

+ c (‖α− πA(α)‖L2 + ‖∇t(α− πA(α))‖L2)

+ cε2 (‖∇tψ‖L2 + ‖ψ‖L2 + ‖Dε2(α, ψ)‖L2)

≤c(‖πA (Dε1 (α, ψ) + ∗[α ∧ ∗ω]) ‖L2 + ε‖Dε (α, ψ) ‖0,2,ε

+ ε‖πA(α)‖L2 + ε‖∇tπA(α)‖L2

)(2.11)

where the second inequality follows from theorem 15. Therefore

‖πA(α)‖L2 + ‖∇tπA(α)‖L2

≤c (ε‖Dε (α, ψ) ‖0,2,ε + ‖πA (Dε1 (α, ψ) + ∗[α ∧ ∗ω]) ‖L2) .(2.12)

In addition, we have that, for q ≥ 2,

‖∇t∇tπA(α)‖Lq ≤c ‖(dA + d∗A)∇t∇tπA(α)‖Lq + ‖πA∇t∇tπA(α)‖Lq≤c ‖∇tπA(α)‖Lq + c ‖πA(α)‖Lq

+ ‖πA (Dε1(α + ψdt)− ∗[α, ∗ω])‖Lq+ c‖α‖Lq + c‖ψ‖Lq + c‖∇t(1− πA)α‖Lq≤c (ε‖Dε (α, ψ) ‖0,2,ε + ‖πA (Dε1 (α, ψ) + ∗[α ∧ ∗ω]) ‖L2)

(2.13)

where the second inequality follows from the commutation formulas, the definition ofDε1 and the triangular inequality and the third by the theorem 15 and (2.12). Finally, inthe case p = 2, the theorem 16 follows from the theorem 15 and from the inequalities(2.12) and (2.13) for q = 2.For 2 < p < 6 we use the Sobolev’s theorem 14 for ε = 1:

‖πA(α)‖Lp + ‖∇tπA(α)‖Lp≤c(‖πA(α)‖L2 + ‖∇tπA(α)‖L2 + ‖∇2

tπA(α)‖L2

)≤c(ε‖Dε (α, ψ) dt‖0,2,ε + ‖πA (Dε1 (α, ψ) + ∗[α ∧ ∗ω]) ‖L2

),


),

(2.14)

where the third step follows from the Holder identity. (2.13), (2.14) and the theorem15 yield now to the estimates (2.7), (2.8) and (2.9). The estimate (2.6) follows thenfrom (2.13) with q = p, (2.7) and (2.8).

In order to prove the estimates for p ≥ 6 we proceed in the same way. By the Sobolev’stheorem 14 for ε = 1 and the Holder inequality:

‖πA(α)‖Lp + ‖∇tπA(α)‖Lp≤c(‖πA(α)‖L3 + ‖∇tπA(α)‖L3 + ‖∇2

tπA(α)‖L3

)≤c(ε‖Dε (α, ψ) dt‖0,3,ε + ‖πA (Dε1 (α, ψ) + ∗[α ∧ ∗ω]) ‖L3

),


).

(2.15)

The estimates (2.7), (2.8) and (2.9) are a consequense of (2.15) and the theorem (15);(2.6) follows then from (2.13) with q = p, (2.7) and (2.8).

2.1 Proof of the theorem 15 for p = 2 25

2.1 Proof of the theorem 15 for p = 2

First we define the norm ‖ · ‖−1,2,ε in the following way for a i-form (i = 1, 2) ξ =α + ψ dt:

‖ξ‖−1,2,ε := supa+b dt∈W 1,2

〈α, a〉+ ε2〈ψ, b〉‖a+ b dt‖1,2,ε

where W 1,2 denotes the Sobolev space W 1,2(Σ×S1,ΛiT ∗(Σ×S1)⊗ gP×S1), and fora 0-form ψ

‖ψ‖−1,2,ε := supb∈W 1,2

〈ψ, b〉‖b‖1,2,ε

.

Lemma 17. There exist two constants ε0, c > 0 such that

‖α + ψdt‖1,2,ε ≤ c(ε2‖Dε1(α, ψ)‖−1,2,ε + ‖πA(α)‖L2

)(2.16)

for every α + ψ dt ∈W1,2 and 0 < ε < ε0.

Proof. We denote by 〈·, ·〉 the ε-independent inner product over Σ × S1, then on theone hand, by integration by parts

ε2〈Dε1(α, ψ), α〉 =〈d∗AdAα + dAd∗Aα− ε2d ∗Xt(A)α, α〉

+ 〈−ε2∇t∇tα− ε22[ψ, (∂tA− dAΨ)], α〉≥‖dAα‖2

L2 + ‖d∗Aα‖2L2 + ε2‖∇tα‖2

L2

− ε2c1

(‖α‖2

L2 + ‖ψ‖L2‖α‖L2

)≥c2

(‖α‖2

1,2,ε − ‖πA(α)‖2L2

)− ε2δc1‖ψ‖2

L2 −ε2c1

δ‖α‖2

L2

where in the first inequality we use (2.1), and on the other hand

ε4〈Dε2(α, ψ), ψ〉 =ε2〈d∗AdAψ − ε2∇t∇tψ + 2 ∗ [α, ∗(∂tA− dAΨ)], ψ〉≥ε2‖dAψ‖2

L2 + ε4‖∇tψ‖2L2 − c1ε

2‖α‖L2‖ψ‖L2

≥c2

(ε2‖ψ‖2

L2 + ε2‖dAψ‖2L2 + ε4‖∇tψ‖2

L2

)− c1ε

2

δ‖α‖2

L2 − c1ε2δ‖ψ‖2

L2 ;

therefore, since ‖(1− πA)α‖L2 ≤ c‖dAα‖L2 + ‖d∗Aα‖L2 by the theorem 6

ε2〈Dε1(α, ψ), α〉+ ε4〈Dε2(α, ψ), ψ〉

≥2c(‖α + ψ dt‖21,2,ε − ‖πA(α)‖2

L2)− c1δε2‖ψ dt‖2

L2 −c1ε

2

δ‖α‖2

L2

≥c(‖α + ψ dt‖21,2,ε − ‖πA(α)‖2

L2)

(2.17)

for c1δ ≤ c and ε sufficiently small. Finally, since by definition

ε2〈Dε1(α, ψ), α〉 ≤ ε2‖Dε1(α, ψ)‖−1,2,ε‖α‖1,2,ε


andε4〈Dε2(α, ψ), ψ〉 ≤ ε2‖Dε2(α, ψ)dt‖−1,2,ε‖ψdt‖1,2,ε,

we obtain that there are two positive constants c and ε0 sucht that

‖α + ψdt‖1,2,ε ≤ c(ε2‖Dε1(α, ψ) +Dε2(α, ψ)dt‖−1,2,ε + ‖πA(α)‖L2

)holds for any positive ε < ε0.

Lemma 18. There exist two constants ε0 > 0 and c > 0 such that

‖ψdt‖1,2,ε ≤ c(ε2‖Dε(α, ψ)‖−1,2,ε + ε‖πA(α)‖L2 + ε2‖∇t(πA(α))‖L2

)‖α− πA(α)‖1,2,ε ≤ cε2 (‖Dε(α, ψ)‖−1,2,ε + ‖πA(α)‖L2 + ‖∇t(πA(α))‖L2)

(2.18)

for every α + ψ dt ∈W1,2 and 0 < ε < ε0.

Proof. Analogously to the previous lamma, if we estimate ε2〈Dε(α, ψ), α− πA(α) +ψdt〉, we obtain that

ε2〈Dε1(α, ψ),α− πA(α)〉 ≥ c‖α− πA(α)‖21,2,ε − c1ε

2‖ψ dt‖L2‖α− πA(α)‖L2

− c1ε2‖πA(α)‖L2‖α− πA(α)‖L2 − 2ε2|〈∇tπA(α),∇t(α− πA(α))〉|

ε4〈Dε2(α, ψ),ψ〉 ≥ c‖ψ dt‖21,2,ε − c1ε

2‖ψ‖L2‖α‖L2

and the following claim yields to

‖ψdt‖1,2,ε ≤c(ε2‖Dε2(α, ψ)dt‖−1,2,ε + ε‖α‖L2

),

‖α− πA(α)‖1,2,ε ≤cε2 (‖Dε1(α, ψ)‖−1,2,ε + ‖πA(α)‖L2)

+ cε2 (‖∇t(πA(α))‖L2 + ‖ψ‖L2) .

Combining the last two estimates and choosing ε sufficiently small, we conclude theproof of the lemma.

Claim 19. There is a positive constant c such that

|〈∇tπA(α),∇t(α− πA(α))〉| ≤ c‖α− πA(α)‖L2‖∇tπA(α)‖L2

+ c‖α− πA(α)‖L2‖πA(α)‖L2 .(2.19)

Proof. The claim follows from the fact that α − πA(α) = dAγ + d∗Aω for a 0-form γand a 2-form ω using the commutation formulas (1.20) and (1.21):

|〈∇tπA(α),∇t(dAγ + d∗Aω)〉| ≤ |〈∇tπA(α), dA∇tγ + d∗A∇tω〉|+ ‖∂tA− dAΨ‖L∞‖∇tπA(α)‖L2(‖γ‖L2 + ‖ω‖L2)

≤ |〈∇t ∗ [(∂tA− dAΨ) ∧ ∗πA(α)], γ〉|+ |〈∇t[(∂tA− dAΨ) ∧ πA(α), ω〉|+ c‖∇tπA(α)‖L2(‖(1− πA)α‖L2)

≤ c‖α− πA(α)‖L2‖∇tπA(α)‖L2

+ c‖α− πA(α)‖L2‖πA(α)‖L2 .

where we use the estimate (2.1) and that

‖γ‖L2 + ‖ω‖L2 ≤ c‖(1− πA)α‖L2

by the lemma 6.

2.1 Proof of the theorem 15 for p = 2 27

Proof of theorem 15 for p = 2. By definition we have the following estimates for any1-form α and any 0-form ψ.

‖dAα + dAψdt‖−1,2,ε = supω+βdt∈W 1,2

〈dAα, ω〉+ ε2〈dAψ, β〉‖ω + βdt‖1,2,ε

= supω+βdt∈W 1,2

〈α, d∗Aω〉+ ε2〈ψ, d∗Aβ〉‖ω + βdt‖1,2,ε

≤ ‖α‖L2 + ε‖ψ‖L2 ,

‖d∗Aα‖−1,2,ε = supb∈W 1,2

〈d∗Aα, b〉‖b‖1,2,ε

≤ ‖α‖L2 ,

‖∇tα‖−1,2,ε = supβ∈W 1,2

〈∇tα, β〉‖β‖1,2,ε

= supβ∈W 1,2

−〈α,∇tβ〉‖β‖1,2,ε

≤ 1

ε‖α‖L2 ,

‖∇tψ‖−1,2,ε = supb∈W 1,2

〈∇tψ, b〉‖b‖1,2,ε

= supb∈W 1,2

−〈ψ,∇tb〉‖b‖1,2,ε

≤ 1

ε‖ψ‖L2 .

Hence, we can conclude that

‖α + ψdt‖−1,2,ε + ‖dAα‖−1,2,ε + ‖d∗Aα‖−1,2,ε + ε‖∇tα‖−1,2,ε

+ ‖dAψ‖−1,2,ε + ε2‖∇tψ‖−1,2,ε ≤ c(‖α‖L2 + ε‖ψ‖L2)

We denote 1ε2d∗AdA + 1

ε2dAd

∗A −∇2

t by ∆Ξ, then

‖dAα‖1,2,ε ≤ε2‖∆ΞdAα‖−1,2,ε + ‖dAα‖0,2,ε

=ε2

∥∥∥∥ 1

ε2(d∗AdAdAα + dAd

∗AdAα)−∇2

tdAα

∥∥∥∥−1,2,ε

+ ‖dAα‖0,2,ε

≤ε2‖dA∆Ξα‖−1,2,ε + ‖α− πA(α)‖1,2,ε

≤cε2‖Dε1(α + ψ dt)‖0,2,ε + cε2‖πA(α)‖L2 + cε2‖∇tπA(α)‖L2

where in the third step we use the commutation formula (1.20) with the triangularinequality and the fact that the curvature on Σ vanishes and hence dAdA = d∗Ad

∗A = 0.

In the last step we used the last lemma and the triangular inequality. Analogously wehave

‖d∗Aα‖1,2,ε + ε‖∇t(α− πA(α))‖1,2,ε

≤cε2‖Dε1(α + ψdt)‖0,2,ε + cε2‖πA(α)‖L2 + cε2‖∇tπA(α)‖L2 ,

‖dAψdt‖1,2,ε + ε‖∇tψdt‖1,2,ε

≤cε2‖Dε(α + ψdt)‖0,2,ε + cε‖πA(α)‖L2 + cε2‖∇tπA(α)‖L2 ,

ε‖∇tα‖1,2,ε ≤ c(ε2‖Dε(α + ψdt)‖0,2,ε + ‖πA(α)‖L2

).

Finally, collecting these estimates and using the last two lemmas we conclude the proofof the theorem.


2.2 Proof of the theorem 15 for p ≥ 2

We first prove the next weaker theorem

Theorem 20. For 1 < p <∞ there exist two constants ε0 > 0 and c > 0 such that

‖ψ‖2,p,ε ≤ c(‖(d∗AdA − ε2∇t∇t)ψ‖Lp + ‖ψ‖1,p,ε

)(2.20)

‖α‖2,p,ε ≤ c(‖(dAd∗A + d∗AdA − ε2∇t∇t)α‖Lp + ‖α‖1,p,ε

)(2.21)

for every 1-form α ∈W2,p and every 0-form ψ ∈W2,p, 0 < ε < ε0.

Proof. We prove the theorem in four steps and in the first three we work in localcoordinates and hence we consider the following setting. We choose a metric g =gR2 ⊕ dt2 on U ×R ⊂ R2×R with U open and contained in a compact set, a constantconnection Ξc = Ac + Ψcdt ∈ Ω1(U × R, g) of the trivial bundle U × R× SO(3)→U × R which satisfies FAc = 0 and a positive constant c0. Furthermore we pick aconnection Ξ = A+ Ψdt ∈ Ω1(U × R, g) which satisfies

‖(A− A) + (Ψ−Ψ) dt‖∞,ε + ‖d∗A(A− A)‖L∞ ≤ c0,

‖dA(A− A) + dA(Ψ−Ψ) dt‖∞,ε ≤ c0,

ε‖∇t(A− A) +∇t(Ψ−Ψ) dt‖∞,ε ≤ c0.

(2.22)

Step 1. For 1 < p <∞ there exists a constant c, such that

‖ψ‖W 2,p ≤ c (‖d∗dψ‖Lp + ‖ψ‖W 1,p) (2.23)

‖α‖W 2,p ≤ c (‖(d∗d+ dd∗)α‖Lp + ‖α‖W 1,p) (2.24)

holds for every 0-form ψ ∈ W 2,pc (U×R, g) and every 1-form α ∈ W 2,p

c (U×R, T ∗(U×R)× g) with compact support in U × R.

Proof of step 1. The first step follows directly from the Calderon-Zygmund inequality,i.e.

‖u‖W 2,p ≤ c (‖∆gu‖Lp + ‖u‖W 1,p)

for every u ∈ W 2,pc (U ×R) with compact support in U ×R. We refer to the chapter 2

and 3 of [25] for the details.


‖ψ‖Ξc,2,p,ε ≤ c(∥∥d∗AcdAcψ − ε2∇Ψc

t ∇Ψct ψ

∥∥Lp

+ ‖ψ‖Ξc,1,p,ε

)(2.25)

‖α‖Ξc,2,p,ε ≤ c(∥∥(d∗AcdAc + dAcd

∗Ac − ε

2∇Ψct ∇Ψc

t

)α∥∥Lp

+ ‖α‖Ξc,1,p,ε

)(2.26)



2.2 Proof of the theorem 15 for p ≥ 2 29

Proof of step 2. First, since the norms ‖ · ‖W i,p and ‖ · ‖Ξc,i,p,1 are equivalent

‖ψ‖Ξc,2,p,1 ≤‖ψ‖W 2,p + c‖Ξc‖C1‖ψ‖W 1,p

≤c (‖(d∗d)ψ‖Lp + c‖ψ‖Ξc,1,p,1 + c‖Ξc‖L∞‖ψ‖Lp)≤c(∥∥d∗AcdAcψ −∇Ψc

t ∇Ψct ψ

∥∥Lp

+ (1 + ‖Ξc‖C1)‖ψ‖Ξc,1,p,1

)≤c(∥∥d∗AcdAcψ −∇Ψc

t ∇Ψct ψ

∥∥Lp

+ ‖ψ‖Ξc,1,p,1

)and analogously

‖α‖Ξc,2,p,1 ≤ c(∥∥(d∗AcdAc + dAcd

∗Ac −∇

Ψct ∇Ψc

t )α∥∥Lp

+ ‖α‖Ξc,1,p,1

).

Next, we define a 0-form ψ := ψ(x, εt), a 1-form α := α(x, εt) and the connectionA(x, t) + Ψ(x, t)dt = A(x, εt) + εΨ(x, εt)dt, then

‖ψ‖Ξc,2,p,ε =ε1p‖ψ‖A+Ψdt,2,p,1

≤cε1p

(∥∥∥d∗AdAψ −∇Ψt ∇Ψ

t ψ∥∥∥Lp

+ ‖ψ‖A+Ψdt,1,p,1

)=c(∥∥d∗AcdAcψ − ε2∇Ψc

t ∇Ψct ψ

∥∥Lp

+ ‖ψ‖Ξc,1,p,ε

)and analogously,

‖α‖Ξc,2,p,ε =ε1p‖α‖A+Ψdt,2,p,1

≤cε1p

(∥∥∥(d∗AdA + dAd∗A −∇

Ψt ∇Ψ

t )α∥∥∥Lp

+ ‖α‖A+Ψdt,1,p,1

)=c(∥∥(d∗AcdAc + dAcd

∗Ac − ε

2∇Ψct ∇Ψc

t )α∥∥Lp

+ ‖α‖Ξc,1,p,ε

).


‖ψ‖Ξ,2,p,ε ≤ c(∥∥∥d∗AdAψ − ε2∇Ψ

t ∇Ψt ψ∥∥∥Lp

+ ‖ψ‖Ξ,1,p,ε

)(2.27)

‖α‖Ξ,2,p,ε ≤ c(∥∥∥(d∗AdA + dAd

∗A− ε2∇Ψ

t ∇Ψt

)α∥∥∥Lp

+ ‖α‖Ξ,1,p,ε

)(2.28)



Proof of step 3. The third step follows from the second step and the assumption (2.22).

Step 4. We prove the theorem.

Proof of step 4. We choose a finite atlas Vi, ϕi : Vi → Σ× S1i∈I of our 3-manifoldΣ × S1. Furthermore, we fix a partition of the unity ρii∈I ⊂ C∞(Σ × S1, [0, 1]),∑

i∈I ρi(x) = 1 for every x ∈ Σ × S1 and supp(ρi) ⊂ ϕi(Vi) for any i ∈ I . Fur-thermore, we denote by Ξi = Ai + Ψidt ∈ Ω(Vi, g) the local representations of theconnectionA+Ψdt on Vi and by αi the local representations of α. We choose the atlas


in order that each Ξi satisfies the condition (2.22) for constant connections Ξci . Then

by the last step

‖(ρi ϕi)αi‖Ξi,2,p,ε ≤c(Ξi)∥∥(dAid∗Ai + dAidAi −∇

Ψit ∇Ψi

t

)((ρi ϕi)αi)

∥∥Lp(Ui)

+ c(Ξi)‖(ρi ϕi)αi‖Ξi,1,p,ε,

If we sum up all the estimates we obtain

‖α‖A+Ψdt,2,p,ε ≤c‖α‖1,p,ε +∑i∈I

‖(ρi ϕi)αi‖Ξi,2,p,ε

≤∑i∈I

c(Ξi)∥∥(dAid∗Ai + d∗AidAi − ε

2∇Ψit ∇Ψi

t

)((ρi ϕi)αi)

∥∥Lp(Ui)

+∑i∈I

c(Ξi)‖(ρi ϕi)αi‖Ξi,1,p,ε + c‖α‖1,p,ε

≤c(Ξ)(∥∥(dAd

∗A + d∗AdA − ε2∇t∇t)α

∥∥Lp

+ ‖α‖A+Ψdt,1,p,ε

).

In the same way we can prove (2.20).

Lemma 21. There are two positive constants c and ε0 such that the following holds.For any i-form ξ ∈ W 2,p, i = 0, 1 and 0 < ε < ε0∫

S1

‖ξ‖pL2(Σ) dt ≤ c

∫S1

‖ − ε2∇2t ξ + ∆Aξ‖pL2(Σ)dt+ c

∫S1

‖πA(ξ)‖pL2(Σ)dt. (2.29)

where ∆A = dAd∗A + d∗AdA.

Proof. In this proof we denote the norm ‖ · ‖L2(Σ) by ‖ · ‖. If we consider only theLaplace part of the operator, we obtain that∫

S1

‖ξ‖p−2〈ξ,−ε2∂2t ξ + ∆Aξ〉dt =

∫S1

‖ξ‖p−2(ε2‖∂tξ‖2 + ‖dAξ‖2 + ‖d∗Aξ‖2

)dt

+

∫S1

(p− 2)‖ξ‖p−4〈ξ, ∂tξ〉2dt

and thus∫S1

‖ξ‖p−2(ε2‖∂tξ‖2 + ‖dAξ‖2 + ‖d∗Aξ‖2

)dt

≤∫S1

‖ξ‖p−2〈ξ,−ε2∂2t ξ + ∆Aξ〉dt

≤∫S1

‖ξ‖p−1‖ε2∂sξ − ε2∂2t ξ + ∆Aξ‖ dt

≤(∫

S1

‖ξ‖pdt) p−1

p(∫

S1

‖ε2∂sξ − ε2∂2t ξ + ∆Aξ‖pdt

) 1p

(2.30)

2.2 Proof of the theorem 15 for p ≥ 2 31

where the second step follows from the Cauchy-Schwarz inequality and the fourthfrom the Holder inequality. Therefore, by lemma 8∫

S1

‖ξ‖pdt ≤∫S1

‖ξ‖p−2(‖dAξ‖2 + ‖d∗Aξ‖2 + ‖πA(ξ)‖2

)dt

and by (2.30) we have that

≤(∫

S1

‖ξ‖pdt) p−1

p(∫

S1

‖ − ε2∂2t ξ + ∆Aξ‖pdt

) 1p

+

∫S1

‖ξ‖p−1‖πA(ξ)‖ dt

in addition by the Holder inequality

≤(∫

S1

‖ξ‖pdt) p−1

p(∫

S1

‖ − ε2∂2t ξ + ∆Aξ‖pdt

) 1p

+

(∫S1

‖ξ‖pdt) p−1

p(∫

S1

‖πA(ξ)‖pdt) 1

p

;

thus, we can conclude that∫S1

‖ξ‖pdt ≤ c

∫S1

(‖ − ε2∂2

t ξ + ∆Aξ‖p + ‖πA(ξ)‖p)dt.

and hence we finished the proof of the lemma using that ‖Ψ‖L∞+‖∂tΨ‖L∞ is boundedby a constant.

Proof of theorem 15. By lemma 8, for any δ > 0 there is a c0 such that

‖α‖pLp ≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0

∫S1

‖α‖pL2dt

≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0c1

∫S1

‖πA(α)‖pL2dt

+ c0c1

∫S1

‖ − ε2∇2tα + ∆Aα‖pL2dt

≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0c1c2‖πA(α)‖pLp+ c0c1c2‖ − ε2∇2

tα + ∆Aα‖pLp≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0c1c2‖πA(α)‖pLp + c4ε

2p‖α‖pLp+ c0c1c2ε

2p‖Dε1(ξ)‖pLp + c4ε2p‖ψ‖pLp

where the second step follows form the lemma 21 and the third by the Holder’s in-

equality with c2 :=(∫

ΣdvolΣ

) p−2p . If we choose therefore δ and ε small enough we

can improve the estimate of the theorem 20 using the last estimate and we obtain (2.2),i.e.

‖ξ‖2,p,ε ≤ c(ε2‖Dε(ξ)‖0,p,ε + ‖πA(α)‖Lp

);


furthermore (2.3) can be proved by

‖(1− πA)ξ‖2,p,ε ≤cε2‖Dε((1− πA)ξ)‖0,p,ε

≤cε2 (‖Dε(ξ)‖0,p,ε + ‖−∇t∇tπA(α)− d ∗Xt(A)πA(α)‖Lp)

+ cε3

∥∥∥∥ 2

ε2∗ [πA(α) ∧ ∗ (∂tA− dAΨ)] dt

∥∥∥∥Lp

≤c(ε2‖Dε(ξ)‖0,p,ε + ε2 ‖∇t∇tπA(α)‖Lp + ε ‖πA(α)‖Lp

).

(2.4) follows from

‖(1− πA)α‖2,p,ε ≤cε2‖Dε((1− πA)α)‖0,p,ε

≤cε2‖Dε1((1− πA)α)‖Lp+ cε ‖2 ∗ [(1− πA)α ∧ ∗ (∂tA− dAΨ)]‖Lp

≤cε2‖Dε1(ξ)‖0,p,ε + cε ‖(1− πA)α‖Lp+ ε2 ‖−∇t∇tπA(α)− d ∗Xt(A)πA(α)‖Lp+ ε2 ‖2 [ψ, (∂tA− dAΨ)]‖Lp≤cε2 (‖Dε1(ξ)‖0,p,ε + ‖∇t∇tπA(α)‖Lp + ‖πA(α)‖Lp)

+ cε ‖(1− πA)α‖Lp + cε2‖ψ‖Lp ,

indeed, if we choose ε small enough and we use (2.3) to estimate cε2‖ψ‖Lp we con-clude

‖(1− πA)α‖2,p,ε ≤cε2(‖Dε1(ξ)‖Lp + ε2‖Dε2(ξ)‖Lp

)+ cε2 (‖∇t∇tπA(α)‖Lp + ‖πA(α)‖Lp) .

Quadratic estimates I 3In the next chapter we will prove the existence and the uniqueness of a map T ε,bbetween the perturbed geodesics and the perturbed Yang-Mills connection providedthat ε is small enough; in order to do this we need the following quadratic estimates.

Lemma 22. For any two constants p ≥ 2 and c0 > 0 there are two positive constantsc and ε0 such that for any two connections A+ Ψdt, A+ Ψdt ∈ A1,p(P × S1)∥∥(Dε(A+ Ψdt)−Dε(A+ Ψdt)

)(α, ψ)

∥∥0,p,ε

≤ c

ε2‖A− A+ (Ψ− Ψ) dt‖∞,ε‖α + ψdt‖1,p,ε

+c

ε2‖α + ψdt‖∞,ε‖A− A+ (Ψ− Ψ) dt‖1,p,ε

(3.1)

∥∥(Dε(A+ Ψdt)−Dε(A+ Ψdt))(α, ψ)

∥∥0,p,ε

≤ c

ε2‖α + ψdt‖∞,ε‖α + ψdt‖1,p,ε

+c

ε2(‖dAα‖L∞ + ‖d∗Aα‖L∞ + ε‖∇tα‖L∞) ‖α + ψdt‖0,p,ε

+c

ε2

(ε‖dAψ‖L∞ + ε2‖∇tψ‖L∞

)‖α + ψdt‖0,p,ε

(3.2)

holds for every α + ψ dt ∈ W 1,p and A + Ψ dt = A + Ψ dt + α + ψ dt with ‖α +ψ dt‖∞,ε ≤ c0 and any 0 < ε < ε0.

Proof. On the one side, the difference between the two first components can be writtenas(Dε1(A+ Ψdt)−Dε1(A+ Ψdt)

)(α, ψ)

=− 1

ε2∗[α ∧ ∗

(dA(A− A) +

1

2[(A− A) ∧ (A− A)]

)]− 1

ε2∗ [(A− A) ∧ ∗[(A− A) ∧ α]]

+1

ε2d∗A

[(A− A) ∧ α]− 1

ε2∗ [(A− A) ∧ ∗dAα]

− 2[ψ,(∇t(A− A)− dA(Ψ− Ψ) + [(Ψ− Ψ), (A− A)]

)]−[(Ψ− Ψ),

(∇tα + [(Ψ− Ψ), α]

)]−∇t[(Ψ− Ψ), α]

+1

ε2

[(A− A) ∧

(d∗Aα− ∗[(A− A) ∧ ∗α]

)]− 1

ε2dA ∗ [(A− A) ∧ ∗α] + d ∗Xt(A)α− d ∗Xt(A)α

(3.3)

33

34 3. Quadratic estimates I

and on the other side,(Dε2(A+ Ψdt)−Dε2(A+ Ψdt)

)(α, ψ)

=2

ε2∗[α ∧ ∗

(∇t(A− A)− dA(Ψ− Ψ)− [(A− A), (Ψ− Ψ)]

)]− 1

ε2∗[(A− A) ∧ ∗

([(A− A), ψ] + dAψ

)]+

1

ε2d∗A

[(A− A) ∧ ψ]−[(Ψ− Ψ),

([(Ψ− Ψ), ψ] +∇tψ

)]−∇t[(Ψ− Ψ), ψ].

(3.4)

The lemma follows estimating term by term the last two identities.

Next, we consider the expansions, for a connection A + Ψdt ∈ A2,p(P × S1) and a1-form α + ψdt ∈ W 2,p,

F ε1(A+ α,Ψ + ψ) = F ε1(A,Ψ) +Dε1(A,Ψ)(α, ψ) + C1(A,Ψ)(α, ψ)

F ε2(A+ α,Ψ + ψ) = F ε2(A,Ψ) +Dε2(A,Ψ)(α, ψ)dt+ C2(A,Ψ)(α, ψ)dt

and we prove the following estimates for the non linear terms C1(A,Ψ)(α, ψ) andC2(A,Ψ)(α, ψ).

Lemma 23. For any constants c0 > 0, p ≥ 2 and any reference connection A0 +Ψ0dt ∈ A2,p(P×S1), there are two positive constants c and ε0 such that forA+Ψdt ∈A2,p(P × S1)

‖C1(A,Ψ)(α, ψ) + C2(A,Ψ)(α, ψ)dt‖0,p,ε

≤ 1

ε2c ‖α + ψdt‖∞,ε‖α + ψdt‖1,p,ε

+1

ε2c ‖α + ψdt‖∞,ε‖A− A0 + (Ψ−Ψ0)dt‖1,p,ε,

(3.5)

‖πA0 (C1(A,Ψ)(α, ψ))‖Lp ≤c

ε2‖α + ψdt‖∞,ε‖(1− πA0)α + ψdt‖1,p,ε

+ c‖α‖L∞‖α‖Lp + ‖ψ‖L∞‖∇tπA0(α)‖Lp

+c

ε2‖α‖2

L∞ (‖α‖Lp + ‖A− A0‖Lp)

+c

ε2‖α‖L∞ (‖d∗A(A− A0)‖Lp + ‖(Ψ−Ψ0)dt‖1,p,ε)

+c

ε2‖A− A0‖2

L∞‖α‖Lp

+ c‖ψ‖2L∞‖A− A0‖L∞‖Ψ−Ψ0‖Lp

(3.6)

for every α + ψ dt ∈W1,p with norm ‖α + ψ dt‖∞,ε < c0 and every 0 < ε < ε0.

35

Proof. By definition, C1 and C2 are

C1(A,Ψ)(α, ψ) =Xt(A+ α)− ∗Xt(A)− d ∗X(A)α

+1

2ε2d∗A[α ∧ α] +

1

ε2∗ [α ∧ ∗(dAα + [α ∧ α])]

+∇t[ψ, α]− [ψ, [ψ, α]] +1

ε2[α, d∗A(A− A0) + d∗Aα]

+ [ψ, (∇tα− dAψ)] +1

ε2[α, ∗[α ∧ ∗(A− A0)]]

− 1

ε2dA ∗ [α ∧ ∗(A− A0)]

− [α ∧ (∇t(Ψ−Ψ0) +∇tψ + [ψ, ((Ψ−Ψ0) + ψ)])]

− dA[ψ, ((Ψ−Ψ0) + ψ)],

(3.7)

C2(A,Ψ)(α, ψ) =1

ε2∗ [α ∧ ∗(∇tα− dAψ − [α, ψ])− 1

ε2d∗A[ψ, α]

+1

ε2[ψ, (d∗A(A− A0 + α)− ∗[α ∧ ∗(A− A0)])]

+1

ε2∇t ∗ [α ∧ ∗(A− A0)]

− [ψ, (∇t(Ψ−Ψ0 + ψ) + [ψ, (Ψ−Ψ0)])]−∇t[ψ, (Ψ−Ψ0)]

(3.8)and if we estimate term by term, we have

‖C1(A,Ψ)(α, ψ) + C2(A,Ψ)(α, ψ)dt‖0,p,ε

≤ 1

ε2c ‖α + ψdt‖∞,ε‖α + ψdt‖1,p,ε

+1

ε2c ‖α + ψdt‖∞,ε‖A− A0 + (Ψ−Ψ0)dt‖1,p,ε.

Next, we consider

πA0C1(A,Ψ)(α, ψ) =πA0 (Xt(A+ α)− ∗Xt(A)− d ∗X(A)α)

+ πA0

(1

ε2∗ [α ∧ ∗(dAα + [α ∧ α])]

)− πA0

(1

2ε2∗ [(A− A0), ∗[α ∧ α]] + 2[ψ,∇tπA0(α)]

)+ πA0 ([∇tψ, α] + 2[ψ,∇t(1− πA0)α]− [ψ, [ψ, α]])

+ πA0

(1

ε2[α, d∗A(A− A0) + d∗Aα]

)+ πA0

(−[ψ, dAψ] +

1

ε2[α, ∗[α ∧ ∗(A− A0)]]

)− πA0

(1

ε2[(A− A0), ∗[α ∧ ∗(A− A0)]]

)+ πA0 (−[α ∧ (∇t(Ψ−Ψ0) +∇tψ)])

+ πA0 (−[α ∧ [ψ, ((Ψ−Ψ0) + ψ)]])

− πA0 ([(A− A0), [ψ, ((Ψ−Ψ0) + ψ)]]) ,

(3.9)

36 3. Quadratic estimates I

thus if we estimate all the summands we obtain (3.6).

The map T ε,b betweenthe critical connections 4

4.1 Definition and properties of T ε,b

In this chapter we will defined the map T ε,b which relates the perturbed closed geodesicsto the perturbed Yang-Mills connections and for this purpose we assume that the Jacobioperator is invertible for every geodesic. The definition will be based on the followingtwo theorems.

Theorem 24 (Existence). We choose a regular energy level b of EH and p ≥ 2. Thereare costants ε0, c > 0 such that the following holds. If Ξ0 = A0 + Ψ0dt ∈ CritbEH is

a perturbed closed geodesic and αε0(t) ∈ im(d∗A0(t) : Ω2(Σ, gP )→ Ω1(Σ, gP )

)is the

unique solution of

d∗A0dA0αε0 = ε2∇t(∂tA0 − dA0Ψ0) + ε2 ∗Xt(A

0), (4.1)

then, for any positive ε < ε0, there is a perturbed Yang-Mills connection Ξε ∈CritbYMε,H which satisfies

d∗εΞ0

(Ξε − Ξ0

)= 0,

∥∥Ξε − Ξ0∥∥

2,p,ε≤ cε2 (4.2)

and, for α + ψdt := Ξε − Ξ0,

‖(1− πA0)(α− αε0)‖2,p,ε + ε ‖ψdt‖2,p,ε ≤ cε4, (4.3)

‖πA0(α)‖2,p,1 + ε ‖πA0(α)‖L∞ ≤ cε2. (4.4)

Remark. As we already mentioned, the theorem 24 was proved by Hong in [9]. Inthe next section we show the proof for completeness reasons and because we need theconstruction of the map T ε,b in order to prove its surjectivity.

Remark. The operator d∗εΞ0 is defined using the L2-inner product as we explained inthe subsection 1.1.2 and thus, it does not depend on the choice of p.

Theorem 25 (Local uniqueness). For any perturbed geodesic Ξ0 ∈ CritbEH and anyc > 0 there are an ε0 > 0 and a δ > 0 such that the following holds for any positiveε < ε0. If Ξε, Ξε are two perturbed Yang-Mills connections that satisfy the condition

d∗εΞ0

(Ξε − Ξ0

)= d∗εΞ0

(Ξε − Ξ0

)= 0

and the estimates

ε∥∥Ξε − Ξ0

∥∥2,p,ε

+∥∥(1− πA0)(Ξε − Ξ0 − αε0)

∥∥1,p,ε≤ cε3

37

38 4. The map T ε,b between the critical connections

with αε0 defined uniquely as in (4.1) and∥∥Ξε − Ξ0∥∥

1,p,ε+∥∥Ξε − Ξ0

∥∥∞,ε ≤ δε, (4.5)

then Ξε = Ξε.

If a connection Ξε ∈ A(P × S1) satisfies∥∥∥Ξε − Ξ0

∥∥∥2,p,ε≤ δ′ε1+ 1

p , then it follows

from the Sobolev embedding theorem 14, that Ξε satisfies (4.5) with δ = (1 + cs)δ′,

where cs ist the constant of theorem 14. Therefore the inequality∥∥∥Ξε − Ξ0

∥∥∥2,p,ε≤ cε2

implies (4.5) whenever ε < ε1 and ε1 is sufficiently small, i.e. if

ε1 ≤ min

ε0,

(δ

2csc

) 1

1− 1p

where ε0 is given in theorem 25. Thus, if we choose in the theorem 25 ε0 satisfying

cε0 + cscε1− 1

p

0 < δ we have that, for each 0 < ε < ε0, in the ball Bcε2 (Ξ0, ‖ · ‖2,p,ε)there is a unique perturbed Yang-Mills connection Ξε which satisfies the conditiond∗εΞ0(Ξε − Ξ0) = 0.

δε

cε2Ξ0

existenceuniqueness

Figure 4.1: Existence and uniqueness.

Definition 26. For every regular value b > 0 of the energy EH there are three positiveconstants ε0, δ and c such that the assertions of the theorems 24 and 25 hold with

these constants. Shrink ε0 such that cε0 + ccsε1− 1

p

0 < δ, where cs is the constant ofSobolev theorem 14. Theorems 24 and 25 assert that, for every Ξ0 ∈ CritbEH and everyε with 0 < ε < ε0, there is a unique perturbed Yang-Mills connection Ξε ∈ CritbYMε,H

satisfying ∥∥Ξε − Ξ0∥∥

2,p,ε≤ cε2, d∗εΞ0(Ξε − Ξ0) = 0. (4.6)

We define the map T ε,b : CritbEH → CritbYMε,H by T ε,b(Ξ0) := Ξε where Ξε ∈CritbYMε,H is the unique Yang-Mills connection satisfying (4.6).

The map T ε,b is gauge equivariant because the construction of the perturbed Yang-Mills connection in the proof of theorem 24 is gauge equivariant, since the map F εand the operator Dε are so. Furthermore, since G0(P ) acts free on A(P ), the gauge

4.2 Proof of the existence theorem 39

group G0(P×S1) acts freely onA(P×S1) and on the set CritbEH and thus T ε,b definesa unique map

T ε,b : CritbEH/G0(P × S1)→ CritbYMε,b/G0(P × S1). (4.7)

In addition, there is a γ > 0 which bounds from below the distance between any twodifferent perturbed geodesics on Mg(P ). Therefore the map T ε,b is injective if wechoose ε < ε1 such that 2cε2

1 < γ and ε1 < ε0, where c and ε0 are the constants in thelast definition.

4.2 Proof of the existence theorem

Proof of theorem 24. For a perturbed geodesic Ξ0 ∈ CritbEH we construct in four stepsa perturbed Yang-Mills connection satisfying (4.2) and (4.3). We need therefore to finda solution Ξε ofF ε(Ξε) = 0 inBcε2 (Ξ0, ‖ · ‖2,p,ε) and in order to do this we use a New-ton’s iteration.

Step 1. In the first step we find a first approximation of a perturbed Yang-Mills con-nection using the next lemma.

Lemma 27. For any perturbed geodesic Ξ0 = A0 + Ψ0dt ∈ CritbEH there is a unique1-form αε0, α0(t) ∈ Ω1(Σ, gP ), which satisfies

d∗A0dA0αε0 = ε2∇t(∂tA0 − dA0Ψ0) + ε2 ∗Xt(A

0), αε0 ∈ im d∗A0 . (4.8)

In addition there is a constant c > 0 such that

‖αε0‖2,p,1 + ‖αε0‖L∞ + ‖dA0αε0‖L∞ + ‖∇tαε0‖L∞ ≤ cε2 (4.9)

for any varepsilon and for Ξε1 := Ξ0 + αε0 ∈ A(P × S1)

‖F ε1 (Ξε1)‖Lp ≤ cε2, ‖F ε2 (Ξε

1)‖Lp ≤ c. (4.10)

Proof. Ξ0 is a perturbed geodesic and therefore the 1-form ε2∇t(∂tA0−dA0Ψ0)+ε2 ∗

Xt(A0) is in the image of d∗A0 by lemma 11; the existence and the uniqueness of αε0

follow therefore from the bijectivity of the operator d∗A0dA0 : im d∗A → im d∗A. Sincethe perturbed geodesic is smooth, ‖∇i

t (∂tA0 − dA0Ψ0)‖L∞ is bounded for every i ≥ 0

and then we can conclude using the commutation formulas 1.20 and 1.21 that there isa constant c > 0 such that

‖αε0‖2,p,1 + ‖αε0‖L∞ + ‖dA0αε0‖L∞ + ‖∇tαε0‖L∞ ≤ cε2

for any ε. In addition, by the definition of αε0 and by d∗A0 (∂tA0 − dA0Ψ0):

F ε1 (Ξε1) =

1

ε2d∗A0+αε0

(dA0αε0 +

1

2[αε0 ∧ αε0]

)−∇t

(∂t(A

0 + αε0)− dA0+αε0Ψ0)− ∗Xt(A

0 + αε0)

=−∇t∇tαε0 +

1

2ε2d∗A0 [αε0 ∧ αε0]

+1

ε2∗[αε0 ∧ ∗

(dA0αε0 +

1

2[αε0 ∧ αε0]

)],


F ε2 (Ξε1) =d∗A0+αε0

(∂t(A

0 + αε0)− dA0+αε0Ψ0)

=1

ε2

(2 ∗[αε0 ∧ ∗

(∂tA

0 − dA0Ψ0)]

+ ∗ [αε0 ∧ ∗∇tαε0])dt

and thus we can estimate the norm of F ε using those of αε0:

‖F ε1 (Ξε1)‖Lp ≤

∥∥∥∥−∇t∇tαε0 +

1

2ε2d∗A0 [αε0 ∧ αε0]

∥∥∥∥Lp

+

∥∥∥∥ 1

ε2∗[αε0 ∧ ∗

(dA0αε0 +

1

2[αε0 ∧ αε0]

)]∥∥∥∥Lp

≤c(‖αε0‖2,p,1 +

1

ε2

(‖αε0‖L∞ + ‖αε0‖2

L∞

)‖αε0‖1,p,1

)≤ cε2,

‖F ε2 (Ξε1)‖Lp =

∥∥∥∥ 1

ε2

(2 ∗[αε0 ∧ ∗

(∂tA

0 − dA0Ψ0)]

+ ∗ [αε0 ∧ ∗∇tαε0])dt

∥∥∥∥Lp

≤ 1

ε2c ‖αε0‖L∞ (‖∇tα

ε0‖Lp + c) ≤ c.

Step 2. We start now a Newton’s iteration and then, in the fourth step, we prove that itconverges. The quadratic estimates of the chapter 2 allow us to generalize the estimatesof lemma 16: Indeed the ‖ ·‖2,p,ε-norm of a 1-form can be evaluated also usingDε(Ξε

1)instead of Dε(Ξ0) in the following way.

Lemma 28. For any perturbed geodesic Ξ0 = A0 + Ψ0dt and for Ξε1 defined as in

lemma 27 the following holds. There exist two constants c > 0 and ε0 > 0 such that

‖πA0(α)‖Lp + ‖∇tπA0(α)‖Lp + ‖∇2tπA0(α)‖Lp

≤cε ‖Dε(Ξε1)(α, ψ)‖0,p,ε + c ‖πA0Dε1(Ξε

1)(α, ψ)‖0,p,ε ,(4.11)

‖α− πA0(α) + ψ dt‖2,p,ε ≤cε2 ‖Dε(Ξε1)(α, ψ)‖0,p,ε + cε ‖πA0Dε1(Ξε

1)(α, ψ)‖0,p,ε ,

(4.12)

‖α− πA0(α)‖2,p,ε ≤cε2 ‖Dε1(Ξε1)(α, ψ)‖Lp + cε4 ‖Dε2(Ξε

1)(α, ψ)‖Lp , (4.13)

for every α + ψ dt ∈W2,p and any positive ε < ε0.

Next, we define Ξε2 = Ξε

1 + αε1 + ψε1dt such that

Dε(Ξε1)(αε1, ψ

ε1) = −F ε(Ξε

1)

and hence by lemmas 27 and 28 and the Sobolev theorem 14 (where for the harmonicpart we can use the Sobolev theorem for ε = 1) we have

‖αε1 − πA0(αε1)‖2,p,ε + ε1p‖αε1 − πA(αε1)‖∞,ε

≤c(ε2‖Dε1(Ξε

1)(αε1, ψε1)‖Lp + ε4‖Dε2(Ξε

1)(αε1, ψε1)‖Lp

)≤c(ε2‖F ε1(Ξε

1)‖Lp + ε4‖F ε2(Ξε1)‖Lp

)≤ cε4,

(4.14)


‖ψε1 dt‖2,p,ε + ε1p‖ψε1 dt‖∞,ε + ε‖πA0(αε1)‖2,p,1 + ε‖πA0(αε1)‖L∞

≤c(ε2‖Dε(Ξε

1)(αε1, ψε1)‖0,p,ε + ε‖πA0(Dε(Ξε

1)(αε1, ψε1))‖0,p,ε

)≤c(ε2‖F ε1(Ξε

1)‖Lp + ε3‖F ε2(Ξε1)‖Lp + ε‖πA0F ε1(Ξε

1)‖Lp)

≤cε3,

(4.15)

Therefore, since, for i = 1, 2,

F εi (Ξε2) = F εi (Ξε

1) +Dεi (Ξε1)(αε1, ψ

ε1) + Cε

i (Ξε1)(αε1, ψ

ε1) = Cε

i (Ξε1)(αε1, ψ

ε1)

by the quatratic estimates (3.5) and (3.6) we have

‖F ε1 (Ξε2)‖Lp = ‖Cε

1(Ξε1)(αε1, ψ

ε1)‖Lp ≤ c2ε

2, (4.16)

‖F ε2 (Ξε2)‖Lp = ‖Cε

2(Ξε1)(αε1, ψ

ε1)‖Lp ≤ c2ε, (4.17)

‖πA0F ε1 (Ξε2)‖0,p,ε = ‖πA0Cε

1(Ξε1)(αε1, ψ

ε1)‖Lp ≤ c2ε

3. (4.18)

Proof of lemma 28. On the one side by the quadratic estimate (3.2)

‖Dε(Ξε1)(α, ψ)−Dε(Ξ0)(α, ψ)‖0,p,ε

≤cε−2 (‖αε0‖L∞ + ‖dA0αε0‖L∞ + ε‖∇tαε0‖L∞) ‖α + ψ dt‖1,p,ε

≤c‖α + ψ dt‖1,p,ε.

(4.19)

where the last estimate follows from (4.9). On the other side, we remark that the ωdefined by (eq:thm:geod:dasdsgf) is exactly 1

ε2dA0αε0 and thus for the harmonic part

we obtain

πA0

(Dε1(Ξε

1)(α, ψ)−(Dε1(Ξ0)(α, ψ)− 1

ε2∗ [α ∧ ∗dA0αε0]

))=πA0

(− 1

ε2∗[α ∧ ∗1

2[αε0 ∧ αε0]

]− 1

ε2∗ [αε0 ∧ ∗[αε0 ∧ α]]− 2 [ψ,∇tα

ε0]

− 1

ε2∗ [αε0 ∧ ∗dA0α] +

1

ε2[αε0 ∧ (d∗A0α− ∗[αε0 ∧ ∗α])]

)and hence∣∣∣∣∣∣πA0

(Dε1(Ξε

1)(α, ψ)−Dε1(Ξ0)(α, ψ) +1

ε2∗ [α ∧ ∗dA0α0]

)∣∣∣∣∣∣0,p,ε

≤ c

ε2

(‖αε0‖2

L∞‖α‖Lp + (‖αε0‖L∞ + ε‖∇tαε0‖L∞) ‖(1− πA0)α + ψ dt‖1,p,ε

)≤cε2‖πA0(α)‖Lp + c‖(1− πA0)α + ψ dt‖1,p,ε.

(4.20)

By the lemma 16 we have∥∥ (1− πA0)α + ψdt∥∥

2,p,ε+ ε ‖πA0(α)‖2,p,1

≤cε2‖Dε(Ξ0)(α, ψ)‖0,p,ε + cε‖πA0

(Dε1(Ξ0)(α, ψ) + ∗[α ∧ ∗ω]

)‖Lp

≤cε2‖Dε(Ξε1)(α, ψ)‖0,p,ε + cε‖πA0Dε1(Ξε

1)(α, ψ)‖0,p,ε

+ cε‖α− πA0α + ψ dt‖1,p,ε + cε2‖πA0α‖1,p,ε.


where the second inequality follows from (4.19) and (4.20). Therefore (4.21) impliesthe first and the second estimate of the lemma choosing ε sufficiently small. The thirdestimates follows combining (2.9), (4.19), (4.20) with the first two inequality of thelemma:

∥∥ (1− πA0)α∥∥

2,p,ε≤cε2‖Dε(Ξ0)(α, ψ)‖Lp + cε4‖Dε(Ξ0)(α, ψ)‖Lp

+ cε2‖πA0

(Dε1(Ξ0)(α, ψ) + ∗[α ∧ ∗ω]

)‖Lp

≤cε2‖Dε1(Ξε1)(α, ψ)‖Lp + cε4‖Dε2(Ξε

1)(α, ψ)‖Lp+ cε2‖α + ψ dt‖1,p,ε

≤cε2‖Dε1(Ξε1)(α, ψ)‖Lp + cε4‖Dε2(Ξε

1)(α, ψ)‖Lp

(4.21)

Step 3. Analogously, we define Ξε3 = Ξε

2 + αε2 + ψε2dt such that

Dε(Ξε2)(αε2, ψ

ε2) = −F ε(Ξε

2)

and hence using the lemma 29

‖(1− πA0)αε2 + ψε2 dt‖2,p,ε + ε1p‖(1− πA0)αε2 + ψε2 dt‖∞,ε

+ ε‖πA0(αε2)‖2,p,1 + ε1‖πA0(αε2)‖L∞≤2c1

(ε2‖Dε(Ξε

2)(αε2, ψε2)‖0,p,ε + ε‖πA0Dε(Ξε

2)(αε2, ψε2)‖Lp

)≤2c1

(ε2‖F ε1(Ξε

2)‖Lp + ε3‖F ε2(Ξε2)‖Lp + ε‖πA0F ε1(Ξε

2)‖Lp)

≤6c1c2ε4,

(4.22)

where for the last estimate we used (4.16)-(4.18).

Lemma 29. For any Ξε2 defined as above there exist two constants c1 > 0 and ε0 > 0

such that

‖πA0(α)‖Lp + ‖∇tπA0(α)‖Lp + ‖∇2tπA0(α)‖Lp

≤c1ε ‖Dε(Ξε2)(α, ψ)‖0,p,ε + c1 ‖πA0Dε1(Ξε

2)(α, ψ)‖0,p,ε

(4.23)

‖α− πA0(α) + ψ dt‖2,p,ε ≤c1ε2 ‖Dε(Ξε

2)(α, ψ)‖0,p,ε + c1ε ‖πA0Dε1(Ξε2)(α, ψ)‖0,p,ε .

(4.24)

for every α + ψ dt ∈W2,p and any 0 < ε < ε0.

Proof. The lemma 29 follows from the lemma 28, the quadratic estimate (3.1) and theestimates (4.14) and (4.15).

Step 4. For k ≥ 3, we define inductively Ξεk = Ξε

k−1 + αεk−1 + ψεk−1dt such that

Dε(Ξε2)(αεk−1, ψ

εk−1) = −F ε(Ξε

k−1),

in this case we have the following estimates

‖(1− πA0)αεk + ψεkdt‖2,p,ε + ε1p‖(1− πA0)αεk + ψεkdt‖∞,ε ≤ 2−k48c1c2ε

4, (4.25)


‖πA0(αεk)‖2,p,1 + ‖πA0(αεk)‖L∞ ≤ 2−k48c1c2ε4, (4.26)

‖αεk + ψεkdt‖2,p,ε + ε1p‖αεk + ψεkdt‖∞,ε ≤ 2−k48c1c2ε

3, (4.27)

‖Ξεk − Ξε

2‖2,p,ε + ε1p‖Ξε

k − Ξε2‖∞,ε ≤ 92c1c2ε

3, (4.28)

‖F ε1(Ξεk) + F ε2(Ξε

k)‖0,p,ε ≤ 2−k48c2ε3, (4.29)

‖Ξεk − Ξ0‖2,p,ε + ε

1p‖Ξε

k − Ξ0‖∞,ε ≤ cε2, (4.30)

where c is a positive constant.

Proof. The estimate (4.28) follows directly from (4.27) and (4.30) from (4.9), (4.15)and (4.28). Therefore we only need to prove (4.25), (4.27) and (4.29). Since, fori = 1, 2,

F εi (Ξεk) =F εi (Ξε

k−1) +Dεi (Ξε2)(αεk−1 + ψεk−1dt)

+ Ci(Ξεk−1)(αεk−1 + ψεk−1dt)

+(Dεi (Ξε

k−1)−Dεi (Ξε2))

(αεk−1 + ψεk−1dt)

=Ci(Ξεk−1)(αεk−1 + ψεk−1dt)

+(Dεi (Ξε

k−1)−Dεi (Ξε2))

(αεk−1 + ψεk−1dt),

from the Sobolev theorem 14 and the lemmas 22 and 23 we get

‖F εi (Ξεk)‖0,p,ε ≤

c3

ε2+ 1p

‖αεk−1 + ψεk−1dt‖22,p,ε

+c3

ε2+ 1p

‖αεk−1 + ψεk−1dt‖2,p,ε‖Ξεk−1 − Ξε

2‖2,p,ε

and thus, we obtain

c1

(‖F ε1(Ξε

k)‖0,p,ε + ‖F ε2(Ξεk)‖0,p,ε

)≤2c1c3

ε2+ 1p

(2−k+148c1c2 + 92c1c2)ε3‖αεk−1 + ψεk−1dt‖1,p,ε

≤2−k28c1c2ε3

(4.31)

The second step holds if ε < ε0 and 208c21c2c3ε

1− 1p

0 ≤ 12. The estimates (4.31) implies

(4.29), (4.27) and (4.25) using the lemma 29

‖(1− πA0)αεk + ψεkdt‖2,p,ε + ε1p‖(1− πA0)αεk + ψεkdt‖∞,ε

+ ε‖πA0(αεk)‖2,p,1 + ε1‖πA0(αεk)‖L∞≤c1ε

2‖Dε(Ξε2)(αεk−1 + ψεk−1)‖0,p,ε

+ c1ε‖πA0Dε1(Ξε2)(αεk−1 + ψεk−1)‖Lp

≤c1

(ε2‖F ε(Ξε

k)‖0,p,ε + ε‖πA0F ε1(Ξεk)‖Lp

)≤ 2−k48c1c2ε

4

and thus we proved all the estimates (4.25)-(4.30).


By the last step, the sequence Ξk converges in Bcε2(Ξ0, ‖ · ‖2,p,ε), and hence con-tinuously, to a connection Ξε = Aε + Ψεdt which satifies (4.2) and F ε(Ξε) = 0; inparticular, d∗εΞ0 (Ξε − Ξ0) = 0. In addition by the estimates (4.14), (4.15), (4.22), (4.25)and (4.26) we have (4.3) and (4.4), i.e.

‖(1− πA0)(α− αε0)‖2,p,ε + ε ‖ψdt‖2,p,ε ≤ cε4,

‖πA0(α)‖2,p,1 + ε ‖πA0(α)‖L∞ ≤ cε2

for Ξε − Ξ0 = α + ψdt. One can also easily remark that

2∣∣∣YMε,H(Ξε)− EH(A0)

∣∣∣=

∣∣∣∣∫ 1

0

(1

ε2‖FAε‖2 + ‖∂tAε − dAεΨε‖2 − ‖∂tA0 − dA0Ψ0‖2

)dt

∣∣∣∣≤∣∣∣ ∫ 1

0

( 1

ε2

∥∥∥∥dA0(Aε − A0) +1

2[(Aε − A0) ∧ (Aε − A0)]

∥∥∥∥2

+∥∥∇t(A

ε − A0)− dA0(Ψε −Ψ0)− [(Aε − A0), (Ψε −Ψ0)]∥∥2)dt

≤ c

ε2

∥∥Ξε − Ξ0∥∥2

1,2,ε+

c

ε2

∥∥Ξε − Ξ0∥∥4

0,4,ε≤ cε2

and hence since the energy level b of EH is regular, there is an ε0 such that for anyε < ε0 we have that Ξε ∈ CritbYMε,H .

4.3 Proof of the local uniqueness theorem

Proof of theorem 25. Since Ξ0 is a geodesic, by lemma 27 we can define a connectionΞε

1 = Ξ0 + αε0 such that ‖αε0‖2,p,1 + ‖dA0αε0‖L∞ + ε‖∇tα

ε0‖L∞ + ‖αε0‖L∞ ≤ cε2 and

‖F ε1(Ξε1)‖Lp ≤ cε2, ‖F ε2(Ξε

1)‖Lp ≤ c. (4.32)

Therefore we have, for Ξε − Ξε1 =: αε + ψε dt and cε < δ,

‖Ξε − Ξε1‖1,p,ε + ‖Ξε − Ξε

1‖∞,ε ≤ 2δε (4.33)

and for i = 1, 2, since Ξε is a Yang-Mills connection which satisfies d∗εΞ0

(Ξε − Ξ0

)=

0, and thus F ε(Ξε) = 0,

Dεi (Ξε1)(Ξε − Ξε

1) = −Cεi (Ξ

ε1)(Ξε − Ξε

1)−F εi (Ξε1). (4.34)

By lemma 28 we get

‖(1− πA0)αε + ψε dt‖2,p,ε + ε‖πA0(αε)‖Lp + ε‖∇tπA0(αε)‖Lp≤ c(ε2‖Dε(Ξε

1)(Ξε − Ξε1)‖0,p,ε + ε‖πA0(Dε(Ξε

1)(Ξε − Ξε1))‖0,p,ε

)≤cε2‖Cε(Ξε

1(Ξε − Ξε1)‖0,p,ε + cε2‖F ε(Ξε

1)‖0,p,ε

+ cε‖πA0(Cε1(Ξε

1)(Ξε − Ξε1))‖0,p,ε + cε‖πA0(F ε1(Ξε

1))‖0,p,ε

≤cε3 + cδ‖(1− πA0)αε + ψε dt‖1,p,ε

+ cδ (ε‖πA0(αε)‖Lp + ε‖πA0(αε)‖Lp)

4.4 Local uniquess modulo gauge 45

where in the second step we use (4.34) and the third step follows from lemma 23 andthe estimate of the curvatures (4.32). Thus we proved the estimates ‖Ξε − Ξε

1‖2,p,ε ≤cε2 and hence ‖Ξε−Ξε‖2,p,ε ≤ cε2. Since Ξε satisfies F ε(Ξε) = 0 by the assumptions,we can write

Dεi (Ξε1)(Ξε − Ξε) = (Dεi (Ξε) + (Dεi (Ξε

1)−Dεi (Ξε))) (Ξε − Ξε)

=− Cεi (Ξ

ε)(Ξε − Ξε) + (Dεi (Ξε1)−Dεi (Ξε)) (Ξε − Ξε)

and by the quadratic estimates of the chapter 3

ε2‖Cε(Ξε1)(Ξε − Ξε

1)‖0,p,ε + cε‖πA0(Cε1(Ξε

1)(Ξε − Ξε1))‖0,p,ε

≤cε1− 1p‖(Ξε − Ξε)− πA0(Ξε − Ξε)‖1,p,ε

+ cε2− 1p‖∇tπA0(Ξε − Ξε)‖0,p,ε + cε1− 1

p‖πA0(Ξε − Ξε)‖0,p,ε,

ε2‖(Dεi (Ξε1)−Dεi (Ξε))(Ξε − Ξε)‖0,p,ε + cε‖πA0((Dεi (Ξε

1)−Dεi (Ξε)) (Ξε − Ξε))‖0,p,ε

≤cε1− 1p‖(Ξε − Ξε)− πA0(Ξε − Ξε)‖1,p,ε

+ cε1− 1p‖∇tπA0(Ξε − Ξε) ∧ dt‖0,p,ε + cε2− 1

p‖πA0(Ξε − Ξε)‖0,p,ε,

we obtain by the lemma 28

‖(1− πA0)(Ξε − Ξε)‖2,p,ε + ε‖πA0(Ξε − Ξε)‖Lp+ ε‖∇tπA0(Ξε − Ξε)‖2,p,1

≤cε2‖Dε(Ξ1)(Ξε − Ξε)‖0,p,ε + cε‖πA0Dε1(Ξ1)(Ξε − Ξε)‖0,p,ε

≤cε1− 1p‖Ξε − Ξε − πA0(Ξε − Ξε)‖2,p,ε

+ cε2− 1p(‖πA0(Ξε − Ξε)‖Lp + ‖∇tπA0(Ξε − Ξε)‖Lp

)and thus, ‖Ξε − Ξε‖2,p,ε = 0 and hence Ξε = Ξε in for ε small enough.

4.4 Local uniquess modulo gauge

The following theorem states a uniqueness property. The result is interesting, but itwill not be used in the next chapters and in particular it will not enter in the proof ofthe surjectivity of T ε,b on the contrary to what one might expect.

Theorem 30 (Uniqueness). We choose p > 3. For every perturbed geodesic Ξ0 ∈CritbEH there are constants ε0, δ1 > 0 such that the following holds. If 0 < ε < ε0 andΞε ∈ CritbYMε,H is a perturbed Yang-Mills connection satisfying∥∥∥Ξε − Ξ0

∥∥∥1,p,ε≤ δ1ε

1+1/p, (4.35)

then there is a g ∈ G2,p0 (P × S1) such that g∗Ξε = T ε,b(Ξ0).


Theorem 31. Assume that q ≥ p > 2 and q > 3. Let Ξ0 = A0 +Ψ0dt ∈ A1,p(P ×S1)be a connection flat on the fibers, i.e. FA0 = 0. Then for every c0 > 0 there exist δ0 >0, c > 0 such that the following holds for 0 < ε ≤ 1. If Ξ = A+ Ψdt ∈ A1,p(P × S1)satisfies∥∥∥d∗A0(A− A0)− ε2∇Ψ0

t (Ψ−Ψ0)∥∥∥Lp≤ c0ε

1/p,∥∥Ξ− Ξ0

∥∥0,q,ε≤ δ0ε

1/q, (4.36)

then there exists a gauge transformation g ∈ G2,p0 such that d∗εΞ0(g∗Ξ− Ξ0) = 0 and

‖g∗Ξ− Ξ‖1,p,ε ≤ cε2(1 + ε−1/p‖Ξ− Ξ0‖1,p,ε

) ∥∥d∗εΞ0(Ξ− Ξ0)∥∥Lp. (4.37)

Proof. The proof is the same as that of proposition 6.2 in [7]. In fact the theorem31 is the 3-dimensional version of the proposition 6.2 in [7]1 which works with 4-dimensional connections. Between this two statements there are a few changes thatare a consequence of the differences in the Sobolev properties (theorem 14 above andlemma 4.1 in [7]). Therefore here we can work with q > 3 instead of q > 4 becausewe have a 3-dimensional manifold and we do not need the condition qp/(q − p) > 4;furthermore, we can replace ε2/p, ε−2/p, ε2/q by ε1/p, ε−1/p, ε1/q because in the proof ofthe Sobolev theorem 14 we rescale a 1-dimensional domain instead of a 2-dimensionalone. In addition, we remark that the gauge transformation g is an element of G2,p

0 (P )and this follows from the proof of the theorem; in fact, the gauge transformationg ∈ G2,p(P × S1) is a limit of a sequence gii∈N ⊂ G2,p(P × S1) defined bygi = exp(η0) exp(η1)... exp(ηi) where ηi ∈ W 2,p(Σ × S1, gP ) are 0-forms. There-fore, the sequence gii∈N lies in the unit component of the gauge group and hence inG2,p

0 (P × S1).

Remark. In the proof of the last theorem we can not use the local slice theorem di-rectly, because although the operator d∗εΞ0dΞ0 is Fredholm and invertible on the com-plement of its kernel, the norm of its inverse depends on ε and hence we do not obtainan estimate independent on the metric and thus not independent on ε.

Proof of the uniqueness theorem 30. Let Ξε = Aε + Ψεdt be a perturbed Yang-Millsconnection which satisfies (4.35) with Ξ0 = A0 + Ψ0dt; then∥∥∥d∗A0(Aε − A0)− ε2∇Ψ0

t (Ψε −Ψ0)∥∥∥Lp

≤∥∥d∗A0(Aε − A0)

∥∥Lp

+ ε2∥∥∥∇Ψ0


≤ 2∥∥∥Ξε − Ξ0

∥∥∥1,p,ε≤ 2δ1ε

1+1/p

(4.38)

and therefore the first condition of the assumption (4.36) of theorem 31 is satisfied for εsufficiently small; the second one follows if we choose δ1ε < δ0 and q = p. Thus thereexists, by theorem 31, a gauge transformation g ∈ G2,p

0 such that d∗Ξ0(g∗Ξε − Ξ0) = 0

1The 0-form d∗εΞ0(Ξ−Ξ0) in the cited proposition is defined by d∗A0(A−A0)− ε2∇Ψ0

t (Ψ−Ψ0)−ε2∇Φ0

t (Φ− Φ0) and the norms are defined in chapter 4 of the paper.

4.4 Local uniquess modulo gauge 47

and ∥∥∥g∗Ξε − Ξε∥∥∥

1,p,ε≤cε2

(1 + ε−1/p

∥∥∥Ξε − Ξ0∥∥∥

1,p,ε

)∥∥∥d∗εΞ0(Ξε − Ξ0)∥∥∥Lp

≤2c∥∥∥d∗A0(Aε − A0)− ε2∇Ψ0


≤4cδ1ε1+1/p.

(4.39)

Then∥∥∥g∗Ξε − Ξ0∥∥∥

1,p,ε≤∥∥∥g∗Ξε − Ξε

∥∥∥1,p,ε

+∥∥∥Ξε − Ξ0

∥∥∥1,p,ε≤ (4cδ1 + δ1)ε1+1/p (4.40)

and by the Sobolev embedding theorem 14 we have also that∥∥∥g∗Ξε − Ξ0∥∥∥∞,ε≤ csε

−1/p∥∥∥g∗Ξε − Ξ0

∥∥∥1,p,ε≤ cs(4c+ 1)δ1ε, (4.41)

where cs is the constant in theorem 14. Finally, we can apply theorem 25 with δ1 <δ/((cs + 1)(4c + 1)) for Ξε = g∗Ξε and Ξε = T ε,b(Ξ0) and we can conclude thatg∗Ξε = T ε,b(Ξ0).

A priori estimates forthe perturbed

Yang-Mills connections 5In this chapter we explain some a priori estimates that we will need to prove the sur-jectivity of the map T ε,b.

5.1 L2(Σ)-estimates for the curvature term FA

Theorem 32. We choose p ≥ 2 and two constants b, c1 > 0. Then there are twopositive constants ε0, c such that the following holds. For any perturbed Yang-Millsconnection A+ Ψdt ∈ CritbYMε,H , with 0 < ε < ε0, which satisfies

supt∈S1

‖∂tA− dAΨ‖L4(Σ) ≤ c1, (5.1)

we have the estimates1

‖FA‖3,2,ε ≤ cε2, (5.2)

supt∈S1

(‖FA‖L2(Σ) + ‖FA‖L∞(Σ) + ‖d∗AFA‖L2(Σ)

+ ‖dAd∗AFA‖L2(Σ) + ε‖∇tFA‖L2(Σ) + ε2‖∇t∇tFA‖L2(Σ)

)≤ cε2−1/p.

(5.3)

First we prove the next theorem

Theorem 33. We choose δ, b > 0, then there is a positive constant ε0 such that thefollowing holds. For any perturbed Yang-Mills connection A+Ψdt ∈ CritbYMε,H , with0 < ε < ε0,

supt∈S1

‖FA‖L∞(Σ) ≤ δ.

Proof. The theorem follows from the perturbed Yang-Mills equation and the Sobolevtheorem 14 in the following way. If we derive the identity

1

ε2d∗AFA −∇tBt − ∗Xt(A) = 0

1The operator ∇t is defined using Ψ.

49

50 5. A priori estimates for the perturbed Yang-Mills connections

by dA and ∇t we obtain

0 =1

ε2dAd

∗AFA − dA∇tBt − dA ∗Xt(A)

=1

ε2dAd

∗AFA −∇t∇tFA + [Bt ∧Bt]− dA ∗Xt(A)

0 =1

ε2∇td

∗AFA −∇t∇tBt −∇t ∗Xt(A)

=1

ε2d∗AdABt −∇t∇tBt −

1

ε2∗ [Bt, ∗FA]−∇t ∗Xt(A)

and the L2-norm of the Laplace part of the last two identities is

ε4

∥∥∥∥ 1

ε2dAd

∗AFA −∇t∇tFA

∥∥∥∥2

L2

= ‖dAd∗AFA‖2L2 + ε4 ‖∇t∇tFA‖2

L2

+ ε2 ‖∇td∗AFA‖

2L2 + ε2〈[Bt ∧ d∗AFA],∇tFA〉

+ ε2〈∇td∗AFA, ∗[Bt, ∗FA]〉,

ε4

∥∥∥∥ 1

ε2d∗AdABt −∇t∇tBt

∥∥∥∥2

L2

= ‖d∗AdABt‖2L2 + ε4 ‖∇t∇tBt‖2

L2

+ ε2 ‖∇tdABt‖2L2 + ε2〈− ∗ [Bt, ∗dABt],∇tBt〉

− ε2〈∇tdABt, [Bt, Bt]〉.

Therefore we can estimate the ‖·‖2,2,ε-norm of FA and ofBt using the Holder inequal-ity and the Sobolev theorem 14:

1

2‖FA‖2

2,2,ε ≤ε4

∥∥∥∥ 1

ε2dAd

∗AFA −∇t∇tFA

∥∥∥∥2

L2

+ ‖FA‖2L2

+ cε‖Bt‖L2‖d∗AFA‖L4‖∇tFA ∧ dt‖0,4,ε

+ δε2 ‖∇td∗AFA‖

2L2 + cε2‖Bt‖2

L2‖FA‖2L∞

≤ε4 ‖[Bt ∧Bt]− dA ∗Xt(A)‖2L2 + ‖FA‖2

L2

+ cε12‖FA‖2

2,2,ε + δε2 ‖∇td∗AFA‖

2L2

≤cε3‖Bt‖L2‖Bt‖2,2,ε + ε4‖dA ∗Xt(A)‖2L2 + ‖FA‖2

L2

+ cε12‖FA‖2

2,2,ε + δε2 ‖∇td∗AFA‖

2L2

≤cε3‖Bt‖2,2,ε + ‖FA‖2L2 + cε

12‖FA‖2

2,2,ε + δε2 ‖∇td∗AFA‖

2L2 ,

5.1 L2(Σ)-estimates for the curvature term FA 51

1

2‖Bt‖2

2,2,ε ≤ε4

∥∥∥∥ 1


∥∥∥∥2

L2

+ ‖Bt‖2L2

+ cε‖Bt‖L2‖dABt‖L4‖∇tBt ∧ dt‖0,4,ε

+ δε2 ‖∇tdABt‖2L2 + cε2‖Bt‖2

L2‖Bt‖2L∞

≤ε4

∥∥∥∥ 1

ε2∗ [Bt, ∗FA] +∇t ∗Xt(A)

∥∥∥∥2

L2

+ ‖Bt‖2L2

+ cε12‖Bt‖2

2,2,ε + δε2 ‖∇tdABt‖2L2

≤c‖FA‖2L2‖Bt‖2

L∞ + cε4‖Bt‖2L2 + cε4 + ‖Bt‖2

L2

+ cε12‖Bt‖2

2,2,ε + δε2 ‖∇tdABt‖2L2

≤2‖Bt‖2L2 + cε4 + cε

12‖Bt‖2

2,2,ε + δε2 ‖∇tdABt‖2L2 .

Hence we can conclude that

‖Bt‖22,2,ε ≤ 4‖Bt‖2

L2 + cε4 ≤ c,

‖FA‖22,2,ε ≤ c‖FA‖2

L2 + ε3‖Bt‖2L2 + cε7 ≤ cε2

and thus, by the Sobolev theorem 14, ‖FA‖L∞ ≤ cε−12‖FA‖2

2,2,ε ≤ cε12 .

In order to prove the theorem 32 we need the following lemma.

Lemma 34. We choose R, r > 0, u : BR+r ⊂ R → R a C2 function, f, g : BR+r ⊂R→ R such that

f ≤ g + ∂2t u, u ≥ 0, f ≥ 0, g ≥ 0,

then ∫BR

f dt ≤∫BR+r

g dt+4

r2

∫BR+r\BR

u dt. (5.4)

Furthermore, if g = cu for a positive constant c, then

1

2supBR

u ≤(c+

4

r

)∫BR+r

u dt. (5.5)

Proof. For BR ⊂ R2 and the Laplace operator instead of ∂2t the first estimate was

proved by Gaio and Salamon in [8] and the second one by Dostoglou and Salamon inthe lemma 7.3 of [7]. These two proofs apply also for our case.

Proof of theorem 32. In this proof we write Bt instead of ∂tA − dAΨ and we denoteby ‖ · ‖ and by 〈·, ·〉 respectively the L2-norm and the L2-product on Σ. In orderto prove the theorem 32 we will apply the last lemma where we choose u to be theL2-norms on Σ of FA, ∇tFA, d∗AFA and ∇t∇tFA; since the perturbed Yang-Mills aresmooth provided that we choose ε sufficiently small, as we discussed in the section1.5, the regularity assumption of the lemma 34 is satisfied. In addition we recall thatthe Bianchi identity tell us that

dABt = ∇tFA (5.6)


and by the assumptions of the theorem∫ 1

0

(1

ε2‖FA‖2 + ‖Bt‖2

)dt ≤ b, sup

t∈S1

‖Bt‖L4(Σ) ≤ c1. (5.7)

Furthermore by the theorem 33 we can assume that supt∈S1 ‖FA‖2 ≤ δ where δ satis-fies the assumptions of the lemmas 6 and 7 for p = 2, which allows us to estimate any2-form in the following way

‖β‖Lq(Σ) ≤ c‖d∗Aβ‖,∀β ∈ Ω2(Σ, gP ), 2 ≤ q <∞. (5.8)

Step 1. We prove the estimate (5.2).

Proof of step 1. If we derive ‖FA‖2 we obtain

∂2t ‖FA‖2 =2‖∇tFA‖2 + 2〈∇t∇tFA, FA〉 = 2‖∇tFA‖2 + 2〈∇tdABt, FA〉

=2‖∇tFA‖2 + 2〈dA∇tBt, FA〉+ 2〈[Bt ∧Bt], FA〉=2‖∇tFA‖2 + 2〈∇tBt, d

∗AFA〉+ 2〈[Bt ∧Bt], FA〉

=2‖∇tFA‖2 +2

ε2‖d∗AFA‖2 − 2〈∗Xt(A), d∗AFA〉+ 2〈[Bt ∧Bt], FA〉

≥2‖∇tFA‖2 +2

ε2‖d∗AFA‖2 − 2|〈∗Xt(A), d∗AFA〉| − ‖Bt‖2

L4(Σ)‖FA‖

≥2‖∇tFA‖2 +2

ε2‖d∗AFA‖2 − c‖FA‖ − c‖d∗AFA‖

(5.9)

where the second equality follows from the Bianchi identity (5.6), the third from thecommutation formula (1.20), the fifth from the perturbed Yang-Mills equation (1.26)and the last one from (5.7). Thus, (5.8) and (5.9) imply that

‖FA‖2 ≤ c‖d∗AFA‖2 + cε2‖∇tFA‖2 ≤ c∂2t (ε

2‖FA‖2) +cε4

δ0

+ cδ0‖FA‖2 (5.10)

and hence for δ0 sufficiently small

‖FA‖2 + ‖d∗AFA‖2 + ε2‖∇tFA‖2 ≤ c∂2t (ε

2‖FA‖2) + cε4; (5.11)

applying the lemma 34 for (5.11)∫ 1

0

(‖FA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)dt ≤ cε4 + cε2

∫ 1

0

‖FA‖2dt ≤ cε4 (5.12)

by (5.7). Analogously to (5.9) one can show that

∂2t

(ε4‖∇tFA‖2 + ε2‖d∗AFA‖2

)≥ε4‖∇t∇tFA‖2 + ε2‖d∗A∇tFA‖2 + ‖dAd∗AFA‖2 − cε4,

(5.13)

∂2t

(ε6‖∇t∇tFA‖2 + ε4‖∇tFA‖2 + ε2‖d∗AFA‖2

)≥ε6‖∇t∇t∇tFA‖2 + ε4‖d∗A∇t∇tFA‖2

+ ε4‖∇t∇tFA‖2 + ε2‖d∗A∇tFA‖2 + ‖dAd∗AFA‖2 − cε4;

(5.14)

5.1 L2(Σ)-estimates for the curvature term FA 53

we will prove these estimates in the section 5.2. Hence by the lemma 34∫ 1

0

(ε4‖∇t∇tFA‖2 + ε2‖∇td

∗AFA‖2 + ‖dAd∗AFA‖

)dt

≤c∫ 1

0

(ε2‖∇tFA‖2 + ε2‖d∗AFA‖2 + ε2‖FA‖2 + cε4

)dt ≤ cε4,

(5.15)

∫ 1

0

(ε6‖∇t∇t∇tFA‖2 + ε4‖∇t∇td

∗AFA‖2

)dt ≤ cε4. (5.16)

and thus, ‖FA‖3,2,ε ≤ cε2 by (5.12), (5.15) and (5.16) and therefore we proved (5.2).

Step 2.∫ 1

0

(‖FA‖2p

L2(Σ) + ε2p‖∇tFA‖2pL2(Σ) + ε4p‖∇t∇tFA‖2p

L2(Σ)

)dt ≤ cε4p.

Proof of step 2. Using the estimates (5.13), (5.14) combined with the lemma 6 weobtain (

ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2)

≤cε4 + cε2∂2t

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)and since for f(t) = (ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2)

∂2t f(t)p =

p

2f(t)p−1∂2

t f(t) +p(p− 1)

4f(t)p−2(∂tf(t))2 ≥ p

2f(t)p−1∂2

t f(t)2,

we have

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)p≤cε4

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ε2‖d∗AFA‖2

)p−1

+ cε2∂2t

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)p.

Then, we apply the inequality ab ≤ ap

p+ bq

qwith q = p

p−1for the first term on the right

side of the inequality for a = cε4 and b = f(t)p−1 and hence

1

p

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)p≤cε4p + cε2∂2

t

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)p.

(5.17)

Finally using the previous lemma 34∫ 1

0

(‖d∗AFA‖

2pL2(Σ) + ε2p‖∇tFA‖2p

L2(Σ) + ε4p‖∇t∇tFA‖2pL2(Σ)

)dt

≤c2ε4p + ε2

∫ 1

0

(‖d∗AFA‖

2pL2(Σ) + ε2p‖∇tFA‖2p

L2(Σ) + ε4p‖∇t∇tFA‖2pL2(Σ)

)dt

and hence we conclude the proof of the third step choosing ε sufficiently small.

Step 3. For any p ≥ 2, the estimate (5.3) holds.


Proof of step 3. The estimate (5.17) yields to

0 ≤cε2(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2 + ε4− 2

p

)p+ ε2∂2

t

(ε4‖∇t∇tFA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2 + ε4− 2

p

)pand thus by the lemma 34

supt∈S1

(ε2‖d∗AFA‖2p + ε2+2p‖∇FA‖2p + ε2+4p‖∇t∇tFA‖2p

)≤cε4p + ε2

∫ 1

0

(‖d∗AFA‖2p + ε2p‖∇FA‖2p + ε4p‖∇t∇tFA‖2p

)dt ≤ cε4p.

By the perturbed Yang-Mills equation we can also estimate ‖dAd∗AFA‖ in the followingway:

‖dAd∗AFA‖ ≤ε2‖dA∇tBt‖+ cε2

≤ε2‖∇tdABt‖+ ε2‖[Bt ∧Bt]‖+ cε2

≤ε2‖∇t∇tFA‖+ 4ε2‖Bt‖4L4(Σ) + cε2

where the second inequality follows from (5.8) and the commutation formula (1.20)and the third from the Bianchi identity (5.6) and the Holder inequality. By the last twoestimates and by the lemma 6 we can conclude that

supt∈S1

(‖FA‖+ ‖FA‖L∞(Σ) + ‖d∗AFA‖

+ ‖dAd∗AFA‖+ ε‖∇tFA‖+ ε2‖∇t∇tFA‖)≤ cε2− 1

p .

With the fourth step we finished also the proof of the theorem 32.

5.2 ComputationsUnder the assumption of the theorem 32 we can prove the following estimates. Inaddition, we write Bt instead of ∂tA − dAΨ and we denote by ‖ · ‖ and by 〈·, ·〉respectively the L2-norm and the L2-product on Σ.

Claim 35. There are two constants c, ε0 > 0 such that

∂2t

(‖d∗AFA‖2 + ε2‖∇tFA‖2

)≥‖∇t∇tFA‖2 + ‖∇td

∗AFA‖2 +

1

ε2‖dAd∗AFA‖2 − cε2

holds for any positive ε < ε0.

Proof. On the one hand, if we compute

∂2t ‖∇tFA‖2 =2‖∇t∇tFA‖2 + 2〈∇t∇t∇tFA,∇tFA〉

5.2 Computations 55

we can apply the Bianchi identity (5.6)

=2‖∇t∇tFA‖2 + 2〈∇t∇tdAB,∇tFA〉

and the commutation formula (1.20)

=2‖∇t∇tFA‖2 + 2〈∇tdA∇tB,∇tFA〉 − 2〈∇t[B ∧B],∇tFA〉

furthermore, if we consider the Yang-Mills equation (1.26)

=2‖∇t∇tFA‖2 +2

ε2〈∇tdAd

∗AFA,∇tFA〉

− 2〈∇t[B ∧B],∇tFA〉 − 〈dA ∗Xt(A),∇t∇tFA〉

and the commutation formulas (1.20) and (1.21), we can conclude

=2‖∇t∇tFA‖2 +2

ε2‖∇td

∗AFA‖2

− 2

ε2〈∇td

∗AFA, ∗[B ∧ ∗FA]〉 − 2

ε2〈[B ∧ d∗AFA],∇tFA〉

− 2〈∇t[B ∧B],∇tFA〉 − 〈dA ∗Xt(A),∇t∇tFA〉;

thus, by the last identity and since ‖ω‖L4(Σ) ≤ ‖dAω‖ for every 2-form ω by the lemma6,

∂2t ‖∇tFA‖2 ≥2‖∇t∇tFA‖2 +

2

ε2‖∇td

∗AFA‖2

− c

ε2‖∇td

∗AFA‖ · ‖B‖L4(Σ)‖d∗AFA‖

− c

ε2‖B‖L4(Σ)‖d∗AFA‖ · ‖d∗A∇tFA‖

− c‖B‖L4(Σ)‖∇tB‖ · ‖d∗A∇tFA‖ − ‖FA‖ · ‖∇t∇tFA‖

and hence

∂2t ‖∇tFA‖2 ≥ ‖∇t∇tFA‖2 +

1

ε2‖∇td

∗AFA‖2 − c

ε2‖d∗AFA‖2. (5.18)

On the other hand,

∂2t ‖d∗AFA‖2 =2‖∇td

∗AFA‖2 + 2〈∇t∇td

∗AFA, d

∗AFA〉

by the commutation formula (1.21) and the Bianchi identity (5.6):

=2‖∇td∗AFA‖2 + 2〈∇td

∗AdAB, d

∗AFA〉+ 2〈∗∇t[B ∧ ∗FA], d∗AFA〉

=2‖∇td∗AFA‖2 + 2〈d∗A∇tdAB, d

∗AFA〉

+ 2〈[B ∧ dAB], d∗AFA〉+ 2〈∗∇t[B ∧ ∗FA], d∗AFA〉


and if we apply the commutation formula (1.20)

=2‖∇td∗AFA‖2 + 2〈d∗AdA∇tB, d

∗AFA〉 − 2〈d∗A[B ∧B], d∗AFA〉

+ 2〈[B ∧ dAB], d∗AFA〉+ 2〈∗∇t[B ∧ ∗FA], d∗AFA〉

and the Yang-Mills equation (1.26), we can conclude

=2‖∇td∗AFA‖2 +

2

ε2‖dAd∗AFA‖2 − 2〈[B ∧B], dAd

∗AFA〉

+ 2〈[B ∧ dAB], d∗AFA〉+ 2〈∗[B ∧ ∗FA],∇td∗AFA〉

− 2〈d∗AdA ∗Xt(A), d∗AFA〉;

thus

∂2t ‖d∗AFA‖2 ≥2‖∇td

∗AFA‖2 +

2

ε2‖dAd∗AFA‖2 − c‖B‖2

L4(Σ)‖dAd∗AFA‖

+ c‖B‖L4(Σ)‖∇tFA‖ · ‖dAd∗AFA‖+ 2〈∗[B ∧ ∗FA],∇td∗AFA〉

−c‖FA‖ · ‖dAd∗AFA‖

and hence

∂2t ‖d∗AFA‖2 ≥‖∇td

∗AFA‖2 +

1

ε2‖dAd∗AFA‖2 − cε2. (5.19)

Finally, combining (5.18) and (5.19) we obtain

∂2t

(‖d∗AFA‖2 + ε2‖∇tFA‖2

)≥2‖∇td

∗AFA‖2 +

2

ε2‖dAd∗AFA‖2 − cε2.


∂2t ‖∇t∇tFA‖2 ≥‖∇t∇t∇tFA‖2 +

1

ε2‖d∗A∇t∇tFA‖2

− c

ε2‖∇t∇tFA‖2 − c

ε2‖d∗A∇tFA‖2 − c

ε6‖dAd∗AFA‖2


Proof. If we consider

∂2t ‖∇t∇tFA‖2 = 2‖∇t∇t∇tFA‖2 + 2〈∇t∇t∇t∇tFA,∇t∇tFA〉

by the Bianchi identity (5.6) we have

=2‖∇t∇t∇tFA‖2 + 2〈∇t∇t∇tdAB,∇t∇tFA〉

and by the commutation formula (1.20):

=2‖∇t∇t∇tFA‖2 + 2〈∇t∇tdA∇tB,∇t∇tFA〉− 2〈∇t∇t[B ∧B],∇t∇tFA〉

5.3 L2(Σ)-estimates for the curvature term ∂tA− dAΨ 57

in addition, using the Yang-Mills equation, we obtain

=2‖∇t∇t∇tFA‖2 +2

ε2〈∇t∇tdAd

∗AFA,∇t∇tFA〉

− 2〈∇t∇t[B ∧B],∇t∇tFA〉 −2

ε2〈∇t∇tdA ∗Xt(A),∇t∇tFA〉

and by the commutation formulas (1.20) and (1.21) we can conclude that

=2‖∇t∇t∇tFA‖2 +2

ε2‖∇t∇td

∗AFA‖+

2

ε2〈∇t∇td

∗AFA, ∗[B ∧ ∗∇tFA]〉

+2

ε2〈∇t∇td

∗AFA, ∗∇t[B ∧ ∗FA]〉+

2

ε2〈[B ∧∇td

∗AFA],∇t∇tFA〉

− 4

ε2〈[B ∧ d∗AFA],∇t∇t∇tFA〉 −

2

ε2〈∇t∇tdA ∗Xt(A),∇t∇tFA〉;

thus

∂2t ‖∇t∇tFA‖2 ≥2‖∇t∇t∇tFA‖2 +

2

ε2‖∇t∇td

∗AFA‖

− c

ε2‖∇t∇td

∗AFA‖ · ‖B‖L4(Σ)‖d∗A∇tFA‖

− c

ε4‖∇t∇td

∗AFA‖ · ‖FA‖ · ‖dAd∗AFA‖

− c

ε2‖B‖L4‖∇td

∗AFA‖ · ‖d∗A∇t∇tFA‖

− c

ε2‖B‖ · ‖dAd∗AFA‖ · ‖∇t∇t∇tFA‖

− c


ε2‖∇t∇tFA‖ · ‖d∗A∇tFA‖ · ‖Bt‖L4

− c

ε2‖d∗A∇t∇tFA‖ · ‖FA‖

and hence by the Cauchy-Schwarz inequality and the commutation formula (1.21)

∂2t ‖∇t∇tFA‖2 ≥‖∇t∇t∇tFA‖2 +

1

ε2‖d∗A∇t∇tFA‖2

− c


ε2‖d∗A∇tFA‖2 − c

ε6‖dAd∗AFA‖2.

5.3 L2(Σ)-estimates for the curvature term ∂tA− dAΨ

Theorem 37. We choose two constants c1, c2 > 0, an open interval Ω ⊂ R and acompact set K ⊂ Ω. Then there are two positive constants δ, c such that the followingholds. For any perturbed Yang-Mills connection A+ Ψdt ∈ Crit∞YMε,H which satisfies

supt∈Ω‖FA‖L2(Σ) ≤ δ, sup

t∈Ω‖∂tA− dAΨ‖L4(Σ) ≤ c1, (5.20)


we have the estimates, for Bt = ∂tA− dAΨ,

supt∈K

ε2‖Bt‖2L2(Σ) ≤ c

∫Ω

(ε2‖Bt‖2

L2(Σ) + ‖FA‖2L2(Σ) + ε2cXt(A)

)dt, (5.21)

supt∈K‖dABt‖2

L2(Σ) ≤ c

∫Ω

(‖dABt‖2

L2(Σ) +1

ε2‖FA‖2

L2(Σ) + ‖Bt‖2L2(Σ)

)dt (5.22)

where √cXt(A) is a constant which bounds the L∞-norm of Xt(A). The constants cand δ depend on Ω and on K, but only on their length and on the distance betweentheir boundaries. Furthermore, if 0 < ε < c2, then

supt∈S1

‖d∗AdABt‖2L2(Σ) ≤ c

∫S1

(ε2‖Bt‖2

L2(Σ) + ‖FA‖2L2(Σ) + ε2cXt(A)

)dt. (5.23)

Remark. The estimates (5.21) and (5.22) hold for any ε and this will play a funda-mental role in the next section where we will have a sequence of perturbed Yang-Millsconnections in Crit∞YMεi,H with εi →∞.

Proof. During this proof we denote by ‖ · ‖ and by 〈·, ·〉 respectively the L2-norm andthe inner product over Σ. We choose δ small enough to apply the lemma 6 and hence‖FA‖ ≤ c‖d∗AFA‖ holds for a constant c.

Step 1. There is a constant c > 0 such that

supt∈K

ε2‖Bt‖2 ≤ c

∫Ω

(ε2‖Bt‖2 + ‖FA‖2 + ε2cXt

)dt.

∫K

‖dABt‖2dt ≤ c

∫Ω

(‖FA‖2 +

1

ε2‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt.

Proof of step 1. In order to prove the first step we compute ∂2t ‖Bt‖2 and then we apply

the lemma 34. By the perturbed Yang-Mills equation (1.26), we have tha

1

2∂2t ‖Bt‖2 =‖∇tBt‖2 + 〈∇t∇tBt, Bt〉

=‖∇tBt‖2 +1

ε2〈∇td

∗AFA, Bt〉 − 〈∇t ∗Xt(A), Bt〉

=‖∇tBt‖2 +1

ε2〈d∗A∇tFA, Bt〉+

1

ε2〈∗[Bt, ∗FA], Bt〉

− 〈d ∗Xt(A)Bt + Xt(A), Bt〉

=‖∇tBt‖2 +1

ε2‖dABt‖2 +

1

ε2〈∗[Bt, ∗FA], Bt〉

− 〈d ∗Xt(A)Bt + Xt(A), Bt〉.

where third step follows from the commutation formula (1.20) and the fourth from theBianchi identity (5.6). Thus, using the Holder, the Cauchy-Schwarz inequality and the


Sobolev estimate ‖Bt‖L4(Σ) ≤ c (‖Bt‖+ ‖dABt‖), one can easily see that

∂2t ‖Bt‖2 ≥‖∇tBt‖2 +

1

ε2‖dABt‖2 − c

ε2‖Bt‖L4(‖Bt‖+ ‖dABt‖)‖FA‖

− c‖Bt‖2 − c‖Xt(A)‖ · ‖Bt‖

≥‖∇tBt‖2 +1

ε2‖dABt‖2 − c

ε4‖FA‖2

− c

ε2‖FA‖2 − c‖Bt‖2 − c‖Xt(A)‖2.

(5.24)

Hence using the lemma 34 we can conclude the second estimate of the first step:∫S1

(ε2‖∇tBt‖2 + ‖dABt‖2

)dt

≤c∫S1

(1

ε2‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A) + c‖FA‖2

)dt.

Since ‖FA‖L2(Σ) ≤ δ and ‖FA‖ ≤ c‖d∗AFA‖, by the theorems 6 and 7 there is aA1 ∈ A0(P ) such that ‖A− A1‖L2 ≤ c‖FA‖L2 and thus we can write

dA ∗Xt(A) = dA1 ∗Xt(A1) + [(A− A1) ∧ ∗Xt(A1)] (5.25)

where dA1 ∗Xt(A1) = 0. Therefore, by the fifth line of the computation (5.9)

1

2∂2t ‖FA‖2 ≥ 1

4ε2‖d∗AFA‖2 +

1

4‖∇tFA‖2 − cε2‖Bt‖2 − c‖FA‖2 (5.26)

and with (5.24) it follows that for a constant c0 big enough

1

2∂2t

(c0‖FA‖2 + ε2‖Bt‖2 + ε2cXt

)≥ −c

(c0‖FA‖2 + ε2‖Bt‖2 + ε2cXt

).

Finally by lemma 34, we can conclude that

supt∈K

ε2‖Bt‖2 ≤ c

∫Ω

(ε2‖Bt‖2 + ‖FA‖2 + ε2cXt

)dt.

Step 2. There is a positive constant c > 0 such that

supt∈K‖d∗AdABt‖2 ≤ c

∫Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A) + ‖d∗AdABt‖2

)dt.

Proof of step 2. Analogously to the previous steps we need to compute 12∂2t ‖d∗AdABt‖2:

1

2∂2t ||d∗AdABt||2 = ||∇td

∗AdABt||2 + 〈∇t∇td

∗AdABt, d

∗AdABt〉

by the commutation formula (1.20) and the Yang-Mills equation (1.26) we have

=||∇td∗AdABt||2 +

1

ε2〈∇td

∗AdAd

∗AFA, d

∗AdABt〉

− 〈∇td∗AdA ∗Xt(A), d∗AdABt〉

+ 〈∇t (− ∗ [Bt∧, ∗dABt] + d∗A[Bt ∧Bt]) , d∗AdABt〉


and applying one more time t(1.20)


1

ε2〈d∗AdAd∗A∇tFA, d

∗AdABt〉

+1

ε2〈− ∗ [Bt ∧ ∗dAd∗AFA] + d∗A[Bt ∧ d∗AFA], d∗AdABt〉

− 1

ε2〈dA ∗ [Bt, ∗FA], dAd

∗AdABt〉 − 〈dA ∗ ∇tXt(A), dAd

∗AdABt〉

− 〈d∗A[Bt ∧ ∗Xt(A)]− ∗[Bt ∧ ∗dA ∗Xt(A)], d∗AdABt〉+ 〈∇t (− ∗ [Bt∧, ∗dABt] + d∗A[Bt ∧Bt]) , d

∗AdABt〉

finally, by the Bianchi identity (5.6) and the perturbed Yang-Mills equation (1.26) wecan conclude that


1

ε2||dAd∗AdABt||2

− 〈[∗Bt ∧ ∗dA(∇tBt + ∗Xt(A))], d∗AdABt〉+1

ε2〈d∗A[Bt ∧ d∗AFA], d∗AdABt〉

− 1


∗AdABt〉 − 〈dA ∗ ∇tXt(A), dAd

∗AdABt〉


∗AdABt〉;

The last computation implies

∂2t ‖d∗AdABt‖2 ≥ −cε2‖Bt‖2 − c 1

ε2‖d∗AFA‖2 − cε2‖∇tBt‖2 − c‖d∗AdABt‖2.

Therefore combining (5.24), (5.26) and (5.3)

∂2t

(‖d∗AdABt‖2 + c0‖FA‖2 + c0ε

2‖Bt‖)≥− cε2‖Bt‖2 − c‖FA‖2

− c‖d∗AdABt‖2 − ε2cXt

and hence we conclude by the lemma 34 that

supt∈K‖d∗AdABt‖2 ≤c

∫Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A) + ‖d∗AdABt‖2

)dt.

Step 3. There is a constant c > 0 such that

supt∈K‖dABt‖ ≤ c

∫Ω

(‖dABt‖2 +

1

ε2‖FA‖2 + ‖Bt‖2

)dt

and if 0 < ε < c2, then∫S1

‖d∗AdABt‖2dt ≤ cε2

∫S1

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt.


Proof of step 3. Like in the previous steps we will prove this one using the lemma 34and therefore we need to compute 1

2∂2t ||dABt||2. We consider

1

2∂2t ||dABt||2 =||∇tdABt||2 + 〈∇t∇tdABt, dABt〉

using the commutation formula (1.20) and the Yang-Mills flow equation (1.26), wehave

=||∇tdABt||2 +1

ε2〈∇tdAd

∗AFA, dABt〉

− 〈∇tdA ∗Xt(A), dABt〉+ 〈∇t[Bt ∧Bt], dABt〉

by the commutation formula (1.20)

=||∇tdABt||2 +1

ε2〈dAd∗A∇tFA, dABt〉

+1

ε2〈[Bt ∧ d∗AFA], dABt〉 −

1

ε2〈∗[Bt, ∗FA], d∗AdABt〉

− 〈∇tdA ∗Xt(A), dABt〉+ 〈∇t[Bt ∧Bt], dABt〉

next, the Bianchi identity (5.6) yields to

=||∇tdABt||2 +1

ε2||d∗AdABt||2 +

1

ε2〈[Bt ∧ d∗AFA], dABt〉

− 1

ε2〈∗[Bt, ∗FA], d∗AdABt〉 − 〈∇tdA ∗Xt(A), dABt〉

+ 〈2[∇tBt ∧Bt], dABt〉

and thus

1

2∂2t ‖dABt‖2 ≥1

2‖∇tdABt‖2 +

1

2ε2‖d∗AdABt‖2 − c

ε2‖d∗AFA‖2

− cε2‖Bt‖2 − cε2‖Xt(A)‖2L∞ − cε2‖∇tBt‖2.

(5.27)

and

1

2∂2t ‖dABt‖2 ≥1

2‖∇tdABt‖2 +

1

2ε2‖d∗AdABt‖2 − c

ε4‖d∗AFA‖2

− c‖dABt‖2 − c‖Bt‖2 − c

ε2‖FA‖2.

(5.28)

Therefore, (5.28) combined with (5.24) yields to

∂2t

(‖dABt‖2 + c0

1

ε2‖FA‖2 + c0‖Bt‖2

)≥− c‖Bt‖2 − c‖Xt(A)‖L∞ −

c

ε2‖FA‖2 − c‖dABt‖2

(5.29)

where we use that∂2t ‖FA‖2 ≥ −cε2‖Bt‖2 − cε2‖dABt‖2


by the fifth line of (5.9). The lemma 34 applyed the last estimate give us

supt∈K‖dABt‖ ≤ c

∫Ω

(‖dABt‖2 +

1

ε2‖FA‖2 + ‖Bt‖2

)dt.

The estimate (5.27) combined with (5.24) yields to

∂2t

(‖dABt‖2 + c0‖FA‖2 + c0ε

2‖Bt‖2)

≥‖∇tdABt‖2 +1

ε2‖d∗AdABt‖2 − cε2‖Bt‖2 − cε2‖Xt(A)‖L∞ − c‖FA‖2

(5.30)

for a constant c0 big enough. Hence, if 0 < ε < c2, by lemma 34 we have∫S1

(ε2‖∇tdABt‖2 + ‖d∗AdABt‖2

)dt

≤cε2

∫S1

(‖dABt‖2 + ‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt

≤cε2

∫S1

(‖d∗AdABt‖2 + ‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt

≤cε2

∫S1

(ε2‖dABt‖2 + ‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt

≤cε2

∫S1

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt

(5.31)

where the second estimate follows from the lemma 6, the third inequality follows fromthe first one and the first step implies the last estimate.

The estimate (5.23) follows combining the second and the third step; hence, we fin-ished the proof of the theorem 37.

5.4 L∞-boundTheorem 38. We choose a constant b > 0. Then there are ε0, c > 0 such that for everypositive ε < ε0 the following holds. If Ξε := Aε + Ψεdt ∈ CritbYMε,H is a perturbedYang-Mills connection, then

‖∂tAε − dAεΨε‖L∞(Σ) ≤ c. (5.32)

Proof. If we prove that ‖∂tAε − dAεΨε‖L4(Σ) is uniformly bounded by a constant, thenby the theorem 37 and the Sobolev estimate it follows that

‖∂tAε − dAεΨε‖L∞(Σ) ≤ ‖∂tAε − dAεΨε‖L4(Σ) + ‖d∗AεdAε(∂tAε − dAεΨε)‖L2(Σ) ≤ c

and hence (5.32) is satisfied for ε sufficiently small. We prove the theorem by an in-direct argument assuming that there is a sequence Ξεν = Aεν + Ψενdtν∈N, εν → 0,of perturbed Yang-Mills connections that satisfies YMεν ,H(Ξεν ) ≤ b and mν :=supt∈S1 ‖∂tAεν − dAενΨεν‖L4(Σ) → ∞. In addition we assume that there is a con-vergent sequence tν → t∞ in S1 such that∥∥∂tAεν (tν)− dAεν (tν)Ψ

εν (tν)∥∥L4(Σ)

= mν . (5.33)

5.4 L∞-bound 63

We need to check three cases that depend from the behavior of the sequence ενmν .

Case 1: limν→∞ ενmν = 0. We define a new sequence of connections Ξεν := Aεν +Ψενdt by Aεν (t) := Aεν (tν + t/mν), and Ψεν (t) := 1

mνΨεν (tν + t/mν). This sequence

satisfies the perturbed Yang-Mills equations

1

ε2νm

2ν

d∗AενFAεν = ∇t

(∂tA

εν − dAεν Ψεν)

+1

m2ν

∗Xtν+ tmν

(Aεν ),

d∗Aεν(∂tA


= 0.

In addition, we have the following estimates for the norms for εν := ενmν

supt∈[−mν2 ,mν

2 ]

∥∥∂tAεν − dAεν Ψεν∥∥L4(Σ)

=∥∥∂tAεν (0)− dAεν (0)Ψ

εν (0)∥∥L4(Σ)

= 1,

(5.34)1

ε2ν

‖FAεν ‖2L2 =

∫ mν2

−mν2

1

ε2ν

‖FAεν ‖2L2(Σ) dt

=

∫ 12

− 12

1

mνε2ν

‖FAεν ‖2L2(Σ) dt ≤

b2

mν

,

(5.35)

∥∥∂tAεν − dAεν Ψεν∥∥2

L2 =

∫ −mν2

−mν2


L2(Σ)dt

=

∫ − 12

− 12

1

m2ν

‖∂tAεν − dAενΨεν‖2L2(Σ) mνdt ≤

b2

mν

.

(5.36)

We denote ∂tAεν −dAεν Ψεν by Bνt and we remark that the L∞-norm of 1

m2νXtν+ t

mν(A)

can be estimate by cm3ν

where c is a positive constant; thus, by the Sobolev estimate andthe theorem 37 we can conclude that

supt∈[−mν2 ,mν

2 ]

∥∥Bνt

∥∥2

L4(Σ)≤c sup

t∈[−mν2 ,mν2 ]

(‖Bν

t ‖2L2(Σ) + ‖d∗Aεν dAεν B

νt ‖2

L2(Σ)

)≤c∫ mν

2

−mν2

(‖Bν

t ‖2L2(Σ) +

1

ε2νm

2ν

‖FAεν ‖2L2(Σ) +

1

m3ν

)dt

≤ c

mν

(1 +

1

mν

+1

m2ν

)which converges to 0 in contradiction to (5.34).

Case 2: limν→∞ ενmν = c1 > 0. This time we choose a different rescaling to defineΞεν := Aεν + Ψενdt, i.e.

Aεν (t) := Aεν (tν + ενt), Ψεν (t) := ενΨεν (tν + ενt)

which satisfies the perturbed Yang-Mills equations

d∗AενFAεν = ∇t

(∂tA


+ ε2ν ∗Xtν+ενt(A

εν ),


d∗Aεν(∂tA


= 0

and

supt∈[− 1

2εν, 12εν

]


=∥∥∂tAεν (0)− dAεν (0)Ψ

εν (0)∥∥L4(Σ)

≤ 2c1

(5.37)for ν sufficiently big. Furthermore, we have the estimates

‖FAεν ‖2L2 =

∫ 12εν

− 12εν

‖FAεν ‖2L2(Σ) dt

=

∫ 12

− 12

1

εν‖FAεν ‖2

L2(Σ) dt ≤ bεν ,

(5.38)


L2 =

∫ 12εν

− 12εν


L2(Σ)dt

=

∫ 12

− 12

ε2ν ‖∂tAεν − dAενΨεν‖2

L2(Σ)

1

ενdt ≤ bεν .

(5.39)

If we denote ∂tAεν−dAεν Ψεν by Bνt and we consider cε3

ν as the bound for the L∞-normof ε2

νXtν+ενt(A), then, by the Sobolev estimate and the theorem 37 we can concludethat

sup∥∥Bν

t

∥∥2

L4(Σ)≤c sup

(‖Bν

t ‖2L2(Σ) + ‖d∗Aεν dAεν B

νt ‖2

L2(Σ)

)≤c∫S1

(‖Bν

t ‖2L2(Σ) + ‖FAεν ‖2

L2(Σ) + ε3ν

)dt

≤cεν(1 + εν + ε2

ν

)which converges to 0 in contradiction to (5.37).

Case 3: limν→∞ ενmν = ∞. First, we define Ξεν := Aεν + Ψενdt as in the first case,i.e.

Aεν (t) := Aεν(tν +

t

mν

), Ψεν (t) :=

1

mν

Ψεν

(tν +

t

mν

).

The new sequence satisfies then

d∗AενFAεν = ε2νm

2ν∇t

(∂tA


+ ε2ν ∗Xtν+t/mν (A

εν ), (5.40)

d∗Aεν(∂tA


= 0.

In addition, we obtain the following estimates for any compact set K ⊂ R that

supt∈[− 1

2mν, 12mν

]


=∥∥∂tAεν (0)− dAεν (0)Ψ

εν (0)∥∥L4(Σ)

= 1,

(5.41)

‖FAεν ‖2L2(Σ×K) =

∫K

‖FAεν ‖2L2(Σ) dt ≤ mν

∫K

‖FAεν ‖2L2(Σ) dt ≤ cε2

νmν ,

5.4 L∞-bound 65


L2(K)=

∫K


L2(Σ)dt

≤∫K

1

m2ν

‖∂tAεν − dAενΨεν‖2L2(Σ)mνdt ≤

b

mν

.

(5.42)

Analogously as in the first two cases we denote ∂tAεν−dAεν Ψεν by Bνt and we consider

1m3ν

as the bound for the L∞-norm of 1m2νXtν+ t

mν(A), then, by the Sobolev estimate and

the theorem 37 we can conclude that, for a compact set K and an open set Ω with0 ∈ K ⊂ Ω,

supt∈K

∥∥Bνt

∥∥2

L4(Σ)≤c sup

t∈K

(‖Bν

t ‖2L2(Σ) + ‖dAεν Bν

t ‖2L2(Σ)

)≤c∫

Ω

(‖Bν

t ‖2L2(Σ) +

1

ε2νm

2ν

‖FAεν ‖2L2(Σ)

)dt

+ c

∫Ω

(ε2ν

mν

+ ‖dAεν Bνt ‖2

L2(Σ)

)dt

≤ c

mν

+c

mν

∫S1

‖dAεν (∂tAεν − dAενΨεν )‖2

L2(Σ) dt

≤ c

mν

+cε

12ν

mν

where the last step follows from the next claim. Then the L4-norm of Bνt converges to

0 by the last estimate in contradiction to (5.41).

Claim 39. For any perturbed Yang-Mills connection Ξ = A+ Ψdt

‖dABt‖L2 ≤ cε14

where we denote ∂tA− dAΨ by Bt.

Proof. If we consider the identity

ε2

∥∥∥∥ 1

ε2d∗AFA −∇tBt − ∗Xt(A)

∥∥∥∥2

L2

+ ‖∇tFA − dABt‖2L2

=1

ε2‖d∗AFA‖

2L2 + ε2 ‖∇tBt‖2

L2 + ε2 ‖Xt(A)‖2L2

+ ‖∇tFA‖2L2 + ‖dABt‖2

L2 − 2ε2

⟨∗Xt(A),

1

ε2d∗AFA −∇tBt

⟩− 2 〈d∗AFA,∇tBt〉 − 2 〈∇tFA, dABt〉 ,

(5.43)

we can remark that first line vanishes by the perturbed Yang-Mills equation (1.26) andby the Bianchi identity∇tFA = dABt; in addition, the last line can be written as

−2 〈d∗AFA,∇tBt〉 − 2 〈∇tFA, dABt〉 = 2 〈FA, [Bt ∧Bt]〉


by the commutation formula (1.20). The identity (5.43) yields therefore to

‖dABt‖2L2 + ε2‖∇tBt‖2

L2

≤2 |〈dA ∗Xt(A), FA〉|+ ε2|〈∗∇tXt(A), Bt〉|+ c‖FA‖L2 · ‖Bt‖2L4

≤c‖FA‖2L2 + ε2(1 + ‖Bt‖2

L2) + cε−12‖FA‖L2 · ‖Bt‖2

1,2,ε

≤cε2(1 + ‖Bt‖2L2) + cε

12

(‖Bt‖2

L2 + ‖dABt‖2L2 + ε2‖∇tBt‖2

L2

)≤cε

12 + cε

12

(‖dABt‖2

L2 + ε2‖∇tBt‖2L2

)where we use the Holder inequality and the Sobolev estimate in the second estimateand the assumption 1

ε2‖FA‖2

L2 + ‖Bt‖2L2 ≤ 2b in the last two estimates. Thus choosing

ε small enough the claim holds.

Since we have found a contradiction for all the tree cases, we can conclude thatsupt∈S1 ‖∂tA − dAΨ‖L4 is uniformly bounded for ε sufficiently small and thus theproof of the theorem 38 is finished.

Surjectivity of T b,ε 6In the fifth chapter we defined the injective map T ε,b in a unique way, in this one weshow that it is also surjective provided that ε is chosen sufficiently small.

Theorem 40. Let b > 0 be a regular value of EH . Then there is a constant ε0 > 0such that


is bijective for 0 < ε < ε0.

Proof. The indirect proof will be divided in five steps. First, we assume that there is adecreasing sequence εν , ν → ∞, converging to 0 and a sequence of perturbed Yang-Mills connections Ξν = Aν + Ψνdt ∈ CritbYMεν ,H that are not in the image of T εν ,b.By the theorems 32 and 38 the sequence satisfies

‖FAν‖L∞(Σ) ≤ cε2− 1p , ‖∂tAν − dAνΨν‖L∞(Σ) ≤ c, (6.1)

sups∈S1

(‖FAν‖L2(Σ) + ‖d∗AνFAν‖L2(Σ) + ‖dAνd∗AνFAν‖L2(Σ)

+ εν∥∥∇Ψν

t FAν∥∥L2(Σ)

+ ε2ν

∥∥∇Ψν

t ∇Ψν

t FAν∥∥L2(Σ)

)≤ cε2−1/p

ν .(6.2)

In the estimate (6.2) the constant c depends on p ≥ 2 which can be taken as big as wewant. In order to conclude the proof we will need to choose p > 6 as we will see inthe proof of the fifth step.

In step 1, for each Ξν we will define a connection Ξν = Aν + Ψνdt near Ξν , flaton the fibers, which satisfies, for a constant c > 0,

∥∥πAν (F0(Ξν))∥∥

Lp≤ cε

1−1/pν .

Next, in the second step, we will find a representative Ξ0 of a perturbed geodesic forwhich ‖Ξν − Ξ0‖1,p,1 + ‖Ξν − Ξ0‖L∞ ≤ cε

1−1/pν for a subsequence of Ξνν∈N (step

3). Then, in step 5, we will improve this last estimate in order to apply the localuniqueness theorem 25 which requires that the norms are bounded by δε for δ and εsufficiently small; in this way we will have a contradiction, because a subsequence ofΞνν∈N will turn out to be in the image of T εν ,b.

Step 1. There are two positive constants c and ν0 such that the following holds. Forevery Ξν , ν > ν0, there is a connection Ξν = Aν + Ψνdt which satisfies

i) FAν = 0, ii) d∗Aν

(∂tAν − dAν Ψν) = 0,

iii)∥∥Ξν − Ξν

∥∥Ξν ,1,p,εν

≤ cε2− 1

pν , iv)

∥∥πAν (F0(Ξν))∥∥

Lp≤ cε

1−1/pν .

67

68 6. Surjectivity of T b,ε

Proof of step 1. Since ‖FAν‖L∞(Σ) ≤ cε2− 1p , by lemma 7 there is a positive constant

c such that for any Aν there is a unique 0-form γν which satisfies FAν+∗dAν γν = 0,‖dAνγν‖L∞(Σ) ≤ c ‖FAν‖L∞(Σ). We denote Ξν := Aν + Ψνdt where Aν := Aν +∗dAνγν , αν := ∗dAνγν and Ψν := Ψν +ψν is the unique 0-form such that d∗

Aν(∂tA

ν −dAν Ψ

ν) = 0; Ψν is well defined because d∗AdA : Ω0(Σ, gP ) → Ω0(Σ, gp) is bijectivefor any flat connection A. Hence, αν satisfies the estimate

‖αν‖L∞(Σ) = ‖dAνγν‖L∞(Σ) ≤ c ‖FAν‖L∞(Σ) ≤ cε2− 1

pν . (6.3)

Since the Ξν is a perturbed Yang-Mills connection, i.e.

1

ε2ν

d∗AνFAν = ∇Ψν

t (∂tAν − dAνΨν) + ∗X(Aν), (6.4)

we have that the connections Ξν satisfy

∇Ψν

t (∂tAν − dAν Ψν) + ∗X(Aν)

= ∇Ψν

t (∂tAν − dAνΨν) + ∗X(Aν)

+ [ψν , (∂tAν − dAνΨν)] +∇Ψν

t (∇Ψν

t αν − dAνψν)

=1

ε2ν

d∗AνFAν −1

ε2ν

∗ [αν ∧ ∗FAν ] + 2[ψν , (∂tAν − dAνΨν)]

+∇Ψν

t ∇Ψν

t αν − dAν∇Ψν

t ψν − [ψν , dAνψν ]

where in the last equality we used (6.4) and the commutation formula (1.20); thus,

πAν(F0(Aν , Ψν)

)=− πAν

(∇Ψν

t (∂tAν − dAν Ψν) + ∗X(Aν)

)=πAν

(∗ 1

ε2ν

[αν ∧ ∗FAν ]− 2[ψν , (∂tAν − dAνΨν)]

)− πAν

([ψν , dAνψ

ν ] +∇Ψν

t ∇Ψν

t αν + [αν ,∇Ψν

t φν ]).

Therefore, by (6.3) and the next lemma,∥∥∥πAν (F0(Aν , Ψν)) ∥∥∥

Lp

≤∥∥∥∥ 1

ε2ν

πAν (∗ [αν ∧ ∗FAν ])∥∥∥∥Lp

+ ‖πAν (2[ψν , (∂tAν − dAνΨν)])‖Lp

+∥∥∥πAν (∇Ψν

t ∇Ψν

t αν + [αν ,∇Ψν

t φν ]− [ψν , dAνψν ])∥∥∥

Lp

≤ cε2− 2

pν +

∥∥∥πAν (∇Ψν

t ∇Ψν

t αν)∥∥∥

Lp

where ∥∥∥πAν (∇Ψν

t ∇Ψν

t αν)∥∥∥

Lp

≤∥∥πAν (∇Ψν

t [ψν , αν ] + ∗∇Ψν

t [(∂tAν − dAνΨν), γν ]

)∥∥Lp

+∥∥πAν (dAν∇Ψν

t ∇Ψν

t γν + [(∂tAν − dAνΨν),∇Ψν

t γν ])∥∥

Lp

≤cε1−1/pν

69

follows from lemma 41 and hence∥∥∥πAν (F0(Aν ,Ψν)) ∥∥∥

0,p,ε≤ cε1−1/p

ν . (6.5)

The estimate∥∥Ξν − Ξν

∥∥Ξν ,1,p,εν

≤ cε2− 1

pν follows from the lemma 41.

Lemma 41. There are constants c > 0, ε0 > 0 such that

‖ψν‖L∞(Σ) + ‖dAνψν‖Lp(Σ) ≤ cε2−1/pν ,∥∥∇Ψν

t αν∥∥Lp(Σ)

+∥∥∇Ψν

t γν∥∥L∞(Σ)

≤ cε1−1/pν ,∥∥∇Ψν

t ψν∥∥L∞(Σ)

+ ε∥∥∇Ψν

t ∇Ψν


≤ cε1−1/pν

for any 0 < εν < ε0.

Proof of lemma 41. Since the Yang-Millas connections Ξν satisfy

d∗Aν (∂tAν − dAνΨν) = 0,

from the definition of ψν we have

0 =d∗Aν(∂tA

ν − dAν Ψν)

=− ∗ [αν ∧ ∗ (∂tAν − dAνΨν)] + d∗Aν∇

Ψν

t αν − d∗AνdAνψν

(6.6)

where

d∗Aν∇Ψν

t αν =− ∗[(∂tAν − dAνΨν) ∧ ∗αν ]− ∗[αν ∧ ∗∇Ψν

t αν ] +∇Ψν

t d∗Aναν ,

∇Ψν

t d∗Aναν =∇Ψν

t d∗Aνd∗Aν ∗ γν = ∇Ψν

t ∗ [FAν ∧ γν ]= ∗ [∇Ψν

t FAν ∧ γν ] + ∗[FAν ∧∇Ψν

t γν ].

Since we know that ‖αν‖L∞(Σ) + ‖γν‖L∞(Σ) ≤ cε2− 1

pν , the proof of the first two in-

equalities of the lemma is completed by showing that there exists a constant c suchthat

‖∇Ψν

t αν‖Lp(Σ) + ‖∇Ψν

t γν‖Lp(Σ) ≤ cε1−1/pν (6.7)

and estimating the norms of ψν and of dAνψν using (6.6):

‖ψν‖L∞(Σ) ≤c‖dAνψν‖Lp(Σ) ≤ c‖d∗AνdAνψν‖L2(Σ)

=‖ − ∗ [αν ∧ ∗ (∂tAν − dAνΨν)] + d∗Aν∇

Ψν

t αν‖L2(Σ)

≤c‖αν‖L2(Σ) + ‖αν‖L∞(Σ)‖∇Ψν

t αν‖L2(Σ)

+ ‖γν‖L∞(Σ)‖∇Ψν

t FAν‖L2(Σ) + ‖FAν‖L∞(Σ)‖∇Ψν

t γν‖L2(Σ)

≤cε2− 1

pν .

In order to show (6.7) we derive

FAν + dAν ∗ dAνγν +1

2[dAνγ

ν ∧ dAνγν ] = 0


by ∇Ψν

t and we obtain

dAν ∗ dAν∇Ψν

t γν =−∇Ψν

t FAν − [dAν∇Ψν

t γν ∧ dAνγν ]− [[(∂tA

ν − dAνΨν) ∧ γν ] ∧ dAνγν ]− [(∂tA

ν − dAνΨν) ∧ ∗dAνγν ](6.8)

and hence, by (6.1),

‖dAν ∗ dAν∇Ψν

t γν‖L2(Σ)

≤c‖∇Ψν

t FAν‖L2(Σ) + c‖αν‖L∞(Σ)‖dAν ∗ dAν∇Ψν

t γν‖L2(Σ)

+ c ‖∂tAν − dAνΨν‖L∞(Σ) ‖αν‖L∞(Σ)

(1 + ‖γν‖L2(Σ)

)≤c(‖∇Ψν

t FAν‖L2(Σ) + ‖αν‖L∞(Σ)

)+ cε

2−

1p‖dAν ∗ dAν∇Ψν

t γν‖L2(Σ).

Choosing ε sufficiently small, we have by (6.2),

‖dAν ∗ dAν∇Ψν

t γν‖L2(Σ) ≤ cε1− 1

pν

which yields to

‖∇Ψν

t γν‖L∞(Σ) ≤c‖dAν ∗ dAν∇Ψν

t γν‖L2(Σ)

≤c(‖∇Ψν

t FAν‖L2(Σ) + ‖αν‖L∞(Σ)

)≤ cε1−1/p

ν

by lemma 6 and

‖∇Ψν

t αν‖Lp(Σ) = ‖∇Ψν

t dAνγν‖Lp(Σ)

≤‖dAν∇Ψν

t γν‖Lp(Σ) + ‖[(∂tAν − dAνΨν), γν ]‖Lp(Σ)

≤c‖dA ∗ dAν∇Ψν

t γν‖L2(Σ) + c‖γν‖Lp(Σ) ≤ c2ε1−1/pν .

Analogously, one can obtain the third inequality of the lemma; the starting point is toderive (6.6) and (6.8) by ∇Ψν

t and to use the estimate∥∥∇Ψν

t ∇Ψν

t FAν∥∥L2(Σ)

≤ c2ε−1/p

in order to show∥∥∇Ψν

t ψν∥∥L∞(Σ)

+ εν∥∥∇Ψν

t ∇Ψν


≤ cε1−1/pν .

In the following, by the Nash embedding theorem, we considerMg(P ) to be a com-pact submanifold of Rn.

Step 2. The sequenceuν :=

[Aν]

ν∈N has a subsequence, still denoted by uν , whichconverges to a perturbed geodesic u0 respect to the norm ‖ · ‖W 1,p or more precisely∥∥uν − u0

∥∥W 1,p ≤ cε1−1/p

ν

for a constant c > 0.

71

Proof of step 2. Since FAν = 0, the vector ∂tAν lies on the tangent space TAνA0(P )and hence in the kernel of dAν ; thus dAν (∂tAν − dAν Ψ

ν) = 0. Every [Aν ] is there-fore a curve in the moduli space Mg(P ) with velocity ∂tA

ν − dAν Ψν ; moreover

it approximates a geodesic in the sense of inequality (6.5). Therefore uνν∈N is abounded Palais-Smale sequence and hence, using next lemma, it has a strong conver-gent subsequence that converge in the norm ‖ · ‖W 1,p to a perturbed geodesic u0 and

‖uν − u0‖W 1,p ≤ cε1− 1

pν .

Lemma 42. Let p ≥ 2 and M be a compact embedded manifold. We choose theenergy

E(u) =1

2

∫ 1

0

(|∇u|2 +Ht(u)

)dt

for any u ∈ W 1,p(S1,M) where Ht : M → R is a smooth Hamiltonian. For everybounded sequence uνν∈N ⊂ W 1,p(S1,M) which satisfies

‖dE(uν)‖Lp → 0

there is a critical curve u∞ ∈ W 1,p(S1,M) such that for a subsequence uινν∈N ⊂uνν∈N we have

1. ‖uιν − u0‖W 1,p → 0 (k →∞);

2. The LE(uιν )ν∈N, where LE denote the linearisation of dE, converges in Lp

to the Jacobi operator of u0;

3. If the Jacobi operator of u∞ is invertible, then there is a constant c > 0 suchthat ‖uιν − u0‖W 1,p ≤ c‖dE(uν)‖Lp .

Proof of lemma 42. 1. The energy functional E satisfies the Palais-Smale conditionfor the norm ‖·‖W 1,2: We refer the reader to [19], theorem 4.4, for the proof in the caseM is a surface and Ht = 0, but the proof applied also for the general case. Therefore,uνν∈N has a subsequence, still denoted by uνν∈N, which converges to a perturbedgeodesic u0 in W 1,2(S1,M). It remains to prove that the sequence converges to u0

also in ‖ · ‖W 1,p , in fact

‖uν − u0‖W 1,p ≤ supv∈W 1,q ,‖v‖W1,q=1

∫ 1

0

(〈∇(uν − u0),∇v〉+ 〈uν − u0, v〉

)dt

= supv∈W 1,q ,‖v‖W1,q=1

(−∫ 1

0

〈dH(uν)− dHt(u0), v〉dt

+

∫ 1

0

〈∆(uν) + dHt(uν), v〉dt+

∫ 1

0

〈uν − u0, v〉dt)

converges to 0.

2. The Lp convergence of dE(uν) implies the convergence of∇tuν because

||∇tuν −∇tu

0||Lp ≤ ||dE(uν)− dE(u0)||Lp + ||dH(uν)− dH(u0)||Lp


goes to 0 for ν →∞. We denote by R the Riemann tensor of the manifoldM and byΠ the projection on its tangent space. Then the linearisation of dE respect to the loopsuν is (cf. appendix B in [22])

LE(uν)X(uν) = −∇uν∇uνX(uν)−R(X(uν), uν)uν −∇X(uν)∇Ht(uν)

for any vector field X onM and the first term can be written as

∇uν∇uνX(uν) =∇uν (Π(uν)dX(uν)uν)

= Π(uν) (dΠ(uν)uν) (dX(uν)uν)

+ Π(uν)d2X(uν)uν uν)

+ Π(uν)dX(uν)∇uν uν .

Thus, for a constant c > 0,∥∥LE(uν)− LE(u0)∥∥Lp≤c(∥∥uν − u0

∥∥Lp

+∥∥uν − u0

∥∥Lp

)+ c∥∥∇uν u

ν −∇u0u0∥∥Lp.

The sequence LE(uν)k∈N converges therefore to the Jacobi operator of u0 in Lp.

3. The third conclusion of the theorem can be proved using the following theorem (seeProposition A.3.4. in [12]). In our case we chose

f : W 2,p((uν)∗TM) → L2((uν)∗TM)x 7→ f(x) := gx(F0(expuν (x))

where gx : Lp(expuν (x)∗TM) → Lp((uν)∗TM) is the parallel transport along t 7→expuν ((1− t)x).

Theorem 43. Let X and Y be Banach spaces, U ⊂ X be an open set, and f : U → Ybe a continuously differentiable map. Let x0 ∈ U be such that D := df(x0) : X → Yis surjective and has a (bounded linear) right inverse Q : Y → X . Choose positiveconstants δ and c such that ‖Q‖ ≤ c, Bδ(x0;X) ⊂ U , and

‖x− x0‖ < δ ⇒ ‖df(x)−D‖ ≤ 1

2c.

Suppose that x1 ∈ X satisfies

‖f(x1)‖ < δ

4c, ‖x1 − x0‖ <

δ

8.

Then there exists a unique x ∈ X such that

f(x) = 0, x− x1 ∈ imQ, ‖x− x0‖ ≤ δ.

Moreover, ‖x− x1‖ ≤ 2c‖f(x1)‖.

73

Step 3.There is a lift Ξ0 of the closed geodesic u0 and a sequence gν ⊂ G2,p0 (P × S1)

such that∥∥g∗νΞν − Ξ0∥∥

1,p,1+∥∥g∗νΞν − Ξ0

∥∥L∞≤ cε1−1/p

ν , ‖dA0(g∗νAν − A0)‖Lp ≤ cε2−2/p

ν .(6.9)

and d∗A0(g∗νAν − A0)‖L2 = 0.

For expositional reasons we will still denote by Ξν the sequence g∗νΞν .

Proof of step 3. We choose now a representative Ξ0 = A0 + Ψ0dt of the geodesic u0.Since the sequence of curves on the moduli space converges to a geodesic [Ξ0] inW 1,p,i.e. ∥∥[Ξν

]− [Ξ0]

∥∥W 1,p(S1,M)

≤ cε1−1/pν , (6.10)

by the Sobolev embedding theorem we have that∥∥[Ξν]− [Ξ0]

∥∥L∞≤ cε1−1/p

ν .

Therefore there is a sequence gν ⊂ G2,p0 (P × S1) such that

d∗A0

(g∗νA

ν − A0)

= 0 (6.11)

and in order to symplify the exposition we still denote the sequence g∗νΞν by Ξν . The

condition (6.11) means that we choose the closest connection in the orbit of Aν to A0

respect to the L2(Σ)-norm. The existence of gν is assured by the lemma 5 and by thelocal slice theorem (see theorem 8.1 in [25]). Therefore

∥∥Ξν − Ξ0∥∥L∞≤ cε

1−1/pν and

thus by the first step ∥∥Ξν − Ξ0∥∥L∞≤ cε1−1/p

ν . (6.12)

SincedA0

(Aν − A0

)= FAν −

1

2

[(Aν − A0

)∧(Aν − A0

)],

we have the estimate∥∥dA0

(Aν − A0

)∥∥Lp≤ ‖FAν‖Lp + c‖Aν − A0‖L∞‖Aν − A0‖Lp ≤ cε2−2/p

ν . (6.13)

Next, we remark, using∇t := ∂t + [Ψ0, ·], that

0 = ∇td∗A0

(Aν − A0

)= d∗A0∇t

(Aν − A0

)+ ∗

[(∂tA

0 − dA0Ψ0)∧ ∗(Aν − A0

)],

thus,

d∗A0dA0

(Ψν −Ψ0

)= d∗A0

(∂tA

0 − dA0Ψ0)− d∗Aν (∂tA

ν − dAνΨν)

+ d∗A0∇t

(Aν − A0

)− d∗A0

[(Aν − A0

)∧(Ψν −Ψ0

)]− ∗

[(Aν − A0) ∧ ∗ (∂tA

ν − dAνΨν)]

=− ∗[(Aν − A0

)∧ ∗ (∂tA

ν − dAνΨν)]

− ∗[(Aν − A0

)∧ ∗(∂tA

0 − dA0Ψ0)]

− ∗[∗(Aν − A0) ∧ dA0(Ψν −Ψ0)

](6.14)


allows us to compute the estimate using (6.1)∥∥dA0

(Ψν −Ψ0

)∥∥Lp

+∥∥Ψν −Ψ0

∥∥Lp≤ c

∥∥dA0 ∗ dA0

(Ψν −Ψ0

)∥∥Lp

≤c‖Aν − A0‖Lp + ‖Aν − A0‖L∞‖dA0(Aν − A0)‖Lp ≤ cε1−1/pν .

(6.15)

Furthermore, since, by (6.10),∥∥(∂tAν − dAνΨν)−

(∂tA

0 − dA0Ψ0)∥∥

Lp≤ cε1−1/p

ν ,∥∥∇t

(Aν − A0

)∥∥Lp≤ cε1−1/p

ν . (6.16)

On the other side, we have

d∗A0dA0∇t

(Ψν −Ψ0

)=∇td

∗A0dA0

(Ψν −Ψ0

)− ∗

[(∂tA

0 − dA0Ψ0)∧ ∗dA0

(Ψν −Ψ0

)]+ ∗

[dA0 ∗

(∂tA

0 − dA0Ψ0)∧(Ψν −Ψ0

)]− ∗

[∗(∂tA

0 − dA0Ψ0)∧ dA0

(Ψν −Ψ0

)]and deriving (6.14) by∇t we obtain∥∥∇t

(Ψν −Ψ0

)∥∥Lp≤∥∥d∗A0dA0∇t

(Ψν −Ψ0

)∥∥Lp

≤c‖dA0∇t(Ψν −Ψ0)‖Lp + c

∥∥∇t

(Aν − A0

)∥∥Lp

+∥∥∇t

(Aν − A0

)∥∥L2p

∥∥dA0

(Ψν −Ψ0

)∥∥L2p

+ c‖Aν − A0‖L∞(

1 +1

ε2‖dAnud∗AνFAν‖L2(Σ)

)+ ‖Aν − A0‖L∞

∥∥d∗A0dA0∇t

(Ψν −Ψ0

)∥∥Lp

(6.17)

where in the second estimate we use that, by the perturbed Yang-Mills equations,

‖∇Ψν

t (∂tAν − dAνΨν)‖Lp ≤ c+ c

1

ε2‖d∗AνFAν‖Lp(Σ) ≤ c+ c

1

ε2‖dAnud∗AνFAν‖L2(Σ) ;

thus, ∥∥∇t

(Ψν −Ψ0

)∥∥Lp≤ cε1−1/p

ν . (6.18)

Finally by the estimates (6.12), (6.13), (6.11), (6.15), (6.16) and (6.18) we have∥∥Ξν − Ξ0∥∥

1,p,1+ ε1/p

ν

∥∥Ξν − Ξ0∥∥L∞≤ cε1−1/p

ν (6.19)

wich proves the third step.

Step 4. Let p > 3. There is sequence gνν∈N of gauge transformations gν ∈G2,p

0 (P × S1) such thatd∗ενΞ0 (g∗νΞ

ν − Ξ0) = 0, (6.20)∥∥d∗A0(g∗νAν − A0)

∥∥Lp≤ cε3−1/p

ν ,∥∥dA0(g∗νA

ν − A0)∥∥Lp≤ cε2−2/p

ν , (6.21)

and ∥∥g∗νΞν − Ξ0∥∥

1,p,1+ ε1/p

ν

∥∥g∗νΞν − Ξ0∥∥L∞≤ cε1−1/p

ν . (6.22)

75

Proof of step 4. By the last step the perturbed Yang-Mills connections Ξν , that satisfythe estimate (6.19) and in addition

ε2ν

∥∥d∗εΞ0(Ξν − Ξ0)∥∥Lp≤∥∥d∗A0(Aν − A0)

∥∥Lp

+ ε2ν

∥∥∇t(Ψν −Ψ0)

∥∥Lp≤ cε3−1/p

ν ,∥∥Ξν − Ξ0∥∥

0,p,ε≤∥∥Ξν − Ξ0

∥∥1,p,ε≤ cε1−1/p

ν ≤ δ0ε1/pν

for all 0 < εν ≤ ε0, cε1−2/p0 ≤ δ0 and the δ0 given in theorem 31; hence Ξν , Ξ0 satisfy

the assumption (4.36) of theorem 31 with q = p. Therefore by this last theorem wecan find a sequence gν ∈ G2,p

0 (P × S1) such that

d∗ενΞ0 (g∗νΞ

ν − Ξ0) = 0

and‖g∗νΞν − Ξν‖1,p,ε ≤ cε2

∥∥d∗εΞ0(Ξν − Ξ0)∥∥Lp≤ cε3−1/p

ν (6.23)

and therefore, by the Sobolev theorem 14,

ε1/pν

∥∥g∗νΞν − Ξ0∥∥∞,ε ≤c

∥∥g∗νΞν − Ξ0∥∥

1,p,ε

≤c(‖g∗νΞν − Ξν‖1,p,ε +

∥∥Ξν − Ξ0∥∥

1,p,ε

)≤cε1−1/p

ν .

The estimates (6.19), (6.23) and the triangular inequality yield also to∥∥d∗A0(g∗νAν − A0)

∥∥Lp≤∥∥d∗A0(Aν − A0)

∥∥Lp

+ ‖d∗A0(g∗νAν − Aν)‖Lp ≤ cε3−1/p

ν ,

∥∥dA0(g∗νAν − A0)

∥∥Lp≤∥∥dA0(Aν − A0)

∥∥Lp

+ ‖dA0(g∗νAν − Aν)‖Lp ≤ cε2−2/p

ν ,

∥∥g∗νΞν − Ξ0∥∥

1,p,1≤∥∥Ξν − Ξ0

∥∥1,p,1

+1

ε2‖g∗νΞν − Ξν‖1,p,ε ≤ cε1−1/p

ν .

Thus, we concluded the proof of the fourth step.

We still denote the new sequence g∗νΞν by Ξν in order to semplify the notation.

Step 5. There are three positive constants δ1, ε0, c such that for any positive εν < ε0

‖πA0(Aν − A0)‖L2 + ‖πA0(Aν − A0)‖L∞ ≤ cε1+δ1ν . (6.24)

End of the proof: Since our sequence satisfies the assumptions of the uniquenesstheorem 25 because by the fourth step d∗εΞ0(Ξν −Ξ0) = 0 and by the fourth and the laststep

‖Ξν − Ξ0‖1,2,ε + ‖Ξν − Ξ0‖∞,ε ≤ δεν ,

for ν big enough Ξν = T εν ,b(Ξ0) which is a contradiction.


δε

uniqueness

Ξ0

cε2− 1p

cε1− 1p

situation step 4

πA0(Ξ)

(1− πA0)Ξ

Figure 6.1: Uniqueness (circle) and the result of step 4 (ellipse).

Proof of step 5. In order to estimate the norms of πA0(Aν − A0) we use the estimate(2.11), i.e.

‖πA(Aν − A0)‖L2 + ‖∇tπA(Aν − A0)‖L2

≤c‖πA(Dεν1 (Ξ0)(Aν − A0,Ψν −Ψ0) + ∗[(Aν − A0) ∧ ∗ω]

)‖L2

+ c‖Aν − A0 − πA(Aν − A0)‖L2

+ c‖∇t(Aν − A0 − πA(Aν − A0))‖L2

+ cε2‖∇t(Ψν −Ψ0)‖L2 + ε2‖Ψν −Ψ0‖L2

+ cε2ν‖Dεν2 (Ξ0)(Aν − A0,Ψν −Ψ0)‖L2

which, by (4.20), can be written as

‖πA(Aν − A0)‖L2 + ‖∇tπA(Aν − A0)‖L2

≤c‖πA(Dεν1 (Ξ0 + ε2

να0)(Aν − A0,Ψν −Ψ0))‖L2

+ c‖(Aν − A0)− πA(Aν − A0)‖1,2,εν + cε3− 1

pν

+ c‖∇t((Aν − A0)− πA0(Aν − A0))‖L2

+ ε2ν‖Dεν2 (Ξ0 + ε2

να0)(Aν − A0,Ψν −Ψ0)‖L2

(6.25)

where α0 ∈ im d∗A0 is defined in lemma 27 choosing ε = 1 and satisfies

‖α0‖2,2,1 + ‖α0‖L∞ ≤ c; (6.26)

we denote Ξ1,ν = Ξ0 + ε2να0 = A1,ν + Ψ0dt and we recall also that, always by lemma

27,‖F ε1(Ξ1,ν)‖L2 ≤ cε2, ‖F ε2(Ξ1,ν)‖L2 ≤ c.

In the following, we will work with the difference Ξν − Ξ1,ν = αν + ψνdt + φνdswhich by step 4 and (6.26) satisfies

‖Ξν − Ξ1,ν‖1,2,1 + ε1/pν ‖Ξν − Ξ1,ν‖L∞ ≤ cε1−1/p

ν . (6.27)

Furthermore we consider the decomposition

Aν −A1,ν = (Aν −Aν1) + (Aν1 −A0) + (A0−A1,ν) = αν + αν − ε2να0 = αν (6.28)

77

where αν = Aν − Aν1 is the 1-form defined in the first step and αν := Aν1 − A0.

The idea of the proof is to use the situation described in the picture 6.2 and in orderto compute the norms of Aν − A0 we use the properties of the orthogonal splittingH1A0 ⊕ im dA0 ⊕ im d∗A0 combined with the facts that αν ∈ im d∗Aν1 and that the norm

of Πim d∗A0

(αν) can be estimate using the identity dA0αν = −12[αν ∧ αν ] which can be

deduced from the flat curvatures FAν1 and FA0 .

FA = 0

αν(t)

Aν1(t) A0(t)

A1,ν(t)Aν(t)

αν(t)

ε2να0(t)αν(t)

Figure 6.2: The splitting of the fifth step.

Claim 44. ‖αν − πA0(αν)‖1,2,εν ≤ cε2−2/pν .

Proof of claim 44. By the triangular inequality and d∗A0α0 = 0 we obtain

‖dA0(Aν − A1,ν)‖L2 ≤ ε2ν‖dA0α0‖L2 + ‖dA0(Aν − A0)‖L2 ≤ cε2−3/p

ν , (6.29)

‖d∗A0αν‖L2 ≤ ‖d∗A0(Aν − A0)‖L2 + ε2‖d∗A0α0‖L2 ≤ cε3−1/pν . (6.30)

and therefore by (6.27), (6.29) and (6.30)

‖αν − πA0(αν)‖1,2,εν ≤ cε2−2/pν . (6.31)

Claim 45. ε2ν‖Dεν2 (Ξ1,ν)(αν , ψν)‖L2 ≤ cε2−3/p.

Proof of claim 45. The estimate follows from

Dεν2 (Ξ1,ν)(αν , ψν) = −Cεν2 (Ξ1,ν)(αν , ψν)−F εν2 (Ξν,1),

where ‖F εν2 (Ξν,1)‖L2 ≤ c and the quadratic estimates of the lemma 23.

Claim 46.

‖πA0(Dεν1 (Ξ1,ν)(αν , ψν)‖L2 ≤cε2−3/p +1

ε2‖πA0([αν ∧ ∗dA0(αν − αν)])‖L2

+ ε1/2−2/pν ‖πA0αν‖L2 .

(6.32)


Proof of the claim 46. By ‖F εν1 (Ξν,1)‖L2 ≤ cε2ν and by the identity

Dεν1 (Ξ1,ν)(αν , ψν) = −Cεν1 (Ξ1,ν)(αν , ψν)−F εν1 (Ξν,1),

we have

‖πA0(Dεν1 (Ξ1,ν)(αν , ψν)‖L2 ≤‖F εν1 (Ξ1,ν)‖L2 + ‖πA0(Cεν1 (Ξ1,ν)(αν , ψν))‖L2

≤cε2−3/p +1

ε2‖πA0([αν ∧ ∗(dA0αν +

1

2[αν ∧ αν ])])‖L2

≤cε2−3/p +1

ε2‖πA0([αν ∧ ∗dA0(αν − αν)])‖L2

+ ε1/2−2/pν ‖πA0αν‖L2

(6.33)

where for the second inequality we estimate every term of πA0(Cεν1 (Ξ1,ν)(αν , ψν))

using the formula (3.9) and for the third one we we applied

0 = FA0+αν = dA0αν +1

2[αν ∧ αν ], ‖αν‖L2(Σ) + ‖dAναν‖L2(Σ) ≤ cε2−1/p

and the decomposition of αν .

Claim 47. ‖∇t(αν − πA0(αν))‖L2 ≤ cε

2−3/pν .

Proof of claim 47. We denote by Πim dA0 and Πim d∗A0

respectively the projections onthe linear spaces im dA0 and im d∗A0 using the orthogonal splitting (1.3). For αν −πA0αν = dA0 γν + dA0ων , where γ is a 0-form and ω a 2-form, we then have that

‖∇t(αν − πA0(αν))‖L2 ≤c‖αν − πA0(αν)‖L2 +

∥∥Πim dA0 (∇tdA0 γν)∥∥L2

+∥∥∥Πim d∗

A0(∇td

∗A0ων)

∥∥∥L2

≤cε2−3/pν

where the last estimate follows from the next two:∥∥∥Πim d∗A0

(∇td∗A0ων)

∥∥∥L2≤‖dA0∇td

∗A0ων‖L2

≤‖∇tdA0αν‖L2 + ‖∂tA0 − dA0Ψ0‖L∞‖αν − πA0αν‖L2

≤‖∇t[αν ∧ αν ]‖L2 + c‖αν − πA0αν‖L2

≤c‖αν‖L∞‖∇tαν‖L2 + c‖αν − πA0αν‖L2 ≤ cε2−3/p

ν ,∥∥Πim dA0 (∇tdA0 γν)∥∥L2 ≤

∥∥Πim dA0

(∇t

(Πim dA0 (αν)

))∥∥L2

+∥∥Πim dA0

(∇t

(Πim dA0 (αν)

))∥∥L2

≤c∥∥d∗A0∇t

(Πim dA0 (αν)

)∥∥L2

+∥∥Πim dA0

(∇t

(Πim dA0 (∗[αν , γν ])

))∥∥L2

≤‖∇td∗A0αν‖L2 + ‖∂tA0 − dA0Ψ0‖L∞

∥∥Πim dA0 (αν)∥∥

+ c ‖[αν , γν ]‖L2 + ‖∇t [αν , γν ]‖L2

≤cε2−2/pν .

79

Claim 48. εν‖∇t(αν − πA0αν − αν)‖0,2,εν + ‖dA0(αν − αν)‖0,2,εν ≤ cε

3−6/pν .

Therefore, using (6.25) and (6.26), we can estimate the norm of the harmonic part by

‖πA(αε)‖L2 + ‖∇tπA(αε)‖L2

≤‖πA0(Dεν1 (Ξ1,ν)(αν , ψν)‖L2 + ‖αν − πA0(αν)‖1,2,ε

+ ‖∇t(αν − πA0(αν))‖L2 + ε2

ν‖Dεν2 (Ξ1,ν)(αν , ψν)‖L2

by the claims 44-46

≤cε2−3/pν +

1

ε2ν

‖πA0([αν ∧ ∗dA0(αν − αν)])‖L2

+ ‖∇t(αν − πA0(αν))‖L2 + ε1/2−2/p

ν ‖πA0αν‖L2

≤cε2−5/pν +

1

ε2ν

‖αν‖L∞‖dA0(αν − αν)‖L2

+ ‖∇t(αν − πA0αν − αν)‖L2

+ ‖∇t(αν − πA0αν)‖L2 + ε1/2−2/p

ν ‖πA0αν‖L2

and because of the claims 47, 48 we can conclude

≤cε2−6/pν + ε1/2−2/p

ν ‖πA0αν‖L2 .

We finish therefore the proof of the fifth step by choosing p > 6.

Proof of claim 48. We choose an operator as follows.

Qεν (Ξ0)(αν , ψν

):=Dεν (Ξ0)

(αν , ψν

)+

1

2εν2d∗A0 [αν ∧ αν ]

+ ∗ 1

ε2ν

[αν ∧ (dA0αν +

1

2[αν ∧ αν ])

]Since dA0αν + 1

2[αν ∧ αν ] = 0,

d∗A0dA0αν +1

2d∗A0 [αν ∧ αν ] = d∗A0dA0(αν − αν) (6.34)

and ‖d∗Aν αν‖L2 ≤ cε3−1/pν by

‖d∗Aν αν‖L2 ≤‖d∗Aν (Aν − A0)‖L2 + ‖d∗Aν (Aν − A0 − αν)‖L2

≤cε3−1/pν + ‖d∗Aν ∗ dAνγν‖L2

≤cε3−1/pν + 2‖FAν‖L∞‖γν‖L2 ≤ cε3−1/p

ν

(6.35)

and hence‖d∗A0αν‖L2 ≤ ‖d∗Aν αν‖L2 + c‖αν‖L4‖αν‖L4 ≤ cε3−2/p

ν (6.36)

〈dA0d∗A0αν , αν − αν〉 ≥ ‖d∗A0(αν − αν)‖2L2 − c‖d∗A0α‖L2‖d∗A0(αν − αν)‖L2 . (6.37)


Then

ε2ν〈Qεν

1 (Ξ0)(αν , ψν

), αν − πA0αν − αν〉 ≥ ‖d∗A0(αν − αν)‖2

L2

+ ‖dA0(αε − αε)‖2L2 +

ε2ν

2‖∇t(α

ν − πA0αν − αν)‖2L2

− cε3−2/pν ‖d∗A0(αν − αν)‖L2

− cε2ν‖αν‖L2‖αε − πA0αν − αν‖L2 − cε2

ν‖αν − πA0αν − αν‖0,2,εν‖ψν‖0,2,εν

− cε2ν |〈∇tπA0(αν),∇t(α

ν − πA0αν − αν)〉|

− cε2|〈∇t(αν − πA0(αν)),∇t(α

ν − αν − πA0αν)〉|− ‖αν‖2

L∞‖dA0(αν − αν)‖L2‖αν − πA0αν − αν‖L2 .

(6.38)

We can conclude therefore that

εν‖∇t(αν − πA0αν − αν)‖0,2,εν + ‖dA0(αν − αν)‖0,2,εν

≤ε2ν‖Qεν (Ξ0)

(αν , ψν

)‖0,2,εν + cε3−2/p

ν

+ cε2ν‖αν‖0,2,εν + cε2

ν‖ψν‖0,2,εν

+ cε2ν‖∇tπA0(αν)‖0,2,εν

+ cε2ν‖πA0(αν)‖0,2,εν ≤ cε3−6/p

ν

(6.39)

where the last step follows because

‖Qεν1 (Ξ1,ν)(αν , φν)−Qεν

1 (Ξ0)(αν , φν)‖L2 ≤ c‖αν‖1,2,ε + c‖πA0(αν)‖L∞‖πA0(αν)‖L2

and

Qεν1 (Ξ1,ν)(Ξν − Ξ1) =−F εν1 (Ξ1,ν)− C1(Ξ1,ν)(Ξν − Ξ1)

+1

2ε2ν

d∗A0 [αν ∧ αν)] + ∗ 1

ε2ν

[αν , ∗[αν ∧ αν ]]

whose norm can be bounded by cε1−6/pν by the triangular and the Holder inequalities.

Bijection between thecritical connections 7

The main theorem of this part states the bijectivity of the map T ε,b which followsdirectly from its definition 26 and the theorem 40, which prove its surjectivity, and inaddition it shows that T b,ε maps perturbed closed geodesics of Morse index k in toperturbed Yang-Mills of the same Morse index.

Theorem 49. We assume that Jacobi operator is invertible for every perturbed geodesicand we choose a regular value b of the energy EH and p ≥ 2. Then there are two pos-itive constants ε0 and c such that the following holds. For every ε ∈ (0, ε0) there is aunique gauge equivariant map


satisfying, for Ξ0 ∈ CritbEH ,

d∗εΞ0

(T ε,b

(Ξ0)− Ξ0

)= 0,

∥∥T ε,b (Ξ0)− Ξ0

∥∥Ξ0,2,p,ε

≤ cε2.

Furthermore, this map is bijective and indexEH (Ξ0) = indexYMε,H (T ε,b(Ξ0)).

Theorem 50. We choose a regular value b > 0 of the energy EH and an ε0 > 0 as indefinition 26, then there is a constant c > 0 such that for every Ξ0 = A0 + Ψ0dt ∈CritbEH the following holds. Let Ξε = Aε + Ψεdt := T ε,b(Ξ0), 0 < ε < ε0, then

1. ε2〈α+ψdt,Dε(Ξε)(α+ψdt)〉 ≥ c‖α+ψdt‖21,2,ε for any 1-form α(t)+ψ(t)dt ∈

dA0Ω0(Σ, gP )⊕ d∗A0Ω2(Σ, gP )⊕ Ω0(Σ, gP ) dt;

2. indexEH (Ξ0) = indexYMε,H (Ξε).

Proof. As we have already mentioned, Weber in [23] proved that the Morse index ofa perturbed geodesic is finite and for a generic Hamiltonian Ht its nullity is zero. Weare therefore interested in the behavior of the operator Dε(Ξε) respect to D0(Ξ0) andin order to investigate that we consider the two parts of the orthogonal splitting of the1-forms

Ω1(Σ, gP ) =(dA0Ω0(Σ, gP )⊕ d∗A0Ω2(Σ, gP )⊕ Ω0(Σ, gP ) dt

)⊕H1

A0(Σ, gP ).

We also recall that

‖Ξε − Ξ0‖2,p,ε + ε1p‖Ξε − Ξ0‖∞,ε ≤ cε2,

∥∥dA0(Aε − A0 − αε0)∥∥L2 ≤ cε4

81

82 7. Bijection between the critical connections

by the theorem 24 and the Sobolev estimate (1.35) provided that ε is sufficiently smallwhere αε0 is defined in the theorem 24. Using the estimate (2.17) we obtain for aα + ψdt ∈ dA0Ω0(S1,M, gP )⊕ d∗A0Ω2(S1,M, gP )⊕ Ω0(S1,M, gP ) dt:

ε2〈α + ψdt,Dε(Ξε)(α + ψdt)〉=ε2〈α + ψdt,Dε(Ξ0)(α + ψdt)〉

+ ε2〈α + ψdt,(Dε(Ξε)−Dε(Ξ0)

)(α + ψdt)〉

≥c‖α + ψdt‖21,2,ε + ε2〈α + ψdt,

(Dε(Ξε)−Dε(Ξ0)

)(α + ψdt)〉

≥c‖α + ψdt‖21,2,ε − cε−

12‖Ξε − Ξ0‖1,2,ε‖α + ψdt‖1,2,ε‖α + ψdt‖0,2,ε

≥c‖α + ψdt‖21,2,ε − cε3/2‖α + ψdt‖2

1,2,ε

≥c‖α + ψdt‖21,2,ε

(7.1)

where the third step follows by the quadratic estimates of the lemma 22 from theSobolev estimate of lemma 14 and the last one holds for ε small enough. We choosenow α(t) ∈ H1

A0(Σ, gP ) and then we pick ψ(t) ∈ Ω0(Σ, gP ), such that

d∗A0dA0ψ = −2 ∗ [α ∧ ∗(∂tA0 − dA0Ψ0)]

Then

〈α + ψdt,Dε(Ξε)(α + ψdt)〉 = 〈D0(Ξ0)(α), α〉+ ε2‖∇tψ‖2L2 +Q

Where

Q :=1

ε2

⟨∗[α ∧ ∗

(dA0(Aε − A0 − α0) +

1

2

[(Aε − A0) ∧ (Aε − A0)

])], α

⟩+

1

ε2

∥∥[(Aε − A0) ∧ α]∥∥2

L2 +1

ε2

∥∥[(Aε − A0) ∧ ∗α]∣∣2L2

+ ε2‖[(Ψε −Ψ0), ψ]‖2L2 − 〈d ∗Xt(A

ε)α− d ∗Xt(A0)α, α〉

−⟨2[ψ,(∇t(A

ε − A0)− dA0(Ψε −Ψ0)−[(Aε − A0) ∧ (Ψε −Ψ0)

])], α⟩

−⟨2 ∗[α ∧ ∗

(∇t(A

ε − A0)− dA0(Ψε −Ψ0))], ψ⟩

+⟨2 ∗[α ∧ ∗

[(Aε − A0) ∧ (Ψε −Ψ0)

]], ψ⟩

+ ‖[(Aε − A0), ψ]‖2L2 + ‖[(Ψε −Ψ0) ∧ α]‖2

L2

and hence|Q| ≤ c1ε

1/2(‖α‖2

L2 + ‖∇tα‖2L2

)(7.2)

for a positive constant c1; in order to compute (7.2) we need also to use

‖ψ‖L2 ≤ ‖α‖L2 , ‖α‖L4 ≤ ‖α‖L2 + ‖∇tα‖L2

where the first estimate follows from the definition of ψ and the second from theSobolev inequality. Therefore there is a constant c > 0 such that if α is an element ofthe negative eigenspace of D0(Ξ0), then

〈α + ψdt,Dε(Ξε)(α + ψdt)〉≤ − c (‖α‖L2 + ‖∇tα‖L2)2 + c1ε

1/2(‖α‖2

L2 + ‖∇tα‖2L2

);

(7.3)

83

and if α is in the positive eigenspace for D0(Ξ0), then

〈α + ψdt,Dε(Ξε)(α + ψdt)〉≥c (‖α‖L2 + ‖∇tα‖L2)2 − c1ε

1/2(‖α‖2

L2 + ‖∇tα‖2L2

).

(7.4)

Thus, by (7.1), (7.3) and (7.4) the dimensions of the negative eigenspaces of D0(Ξ0)and Dε(Ξε) are equal provided that ε is small enough and hence we can conclude thatthe Morse indices are equal.

Part II

Isomorphism between the Morsehomologies

85

Flow equations andMorse homologies 8

8.1 Geodesic flow

Every continuously differentiable map [Ξ] : S1 × R → Mg(P ) can be seen as aconnection Ξ = A + Ψdt + Φds ∈ A(P × S1 × R) which satisfies the followingconditions

FA = 0, ∂tA− dAΨ ∈ H1A, ∂sA− dAΦ ∈ H1

A. (8.1)

In fact, for any [Ξ] we can choose a lift A : S1×R→ A0(P ); the second and the thirdcondition of (8.1) yield to unique 0-forms Ψ(t, s),Φ(t, s) ∈ Ω0(P, gP ). One can alsoconsider Φ to have an exponential convergence as |s| → ∞ (cf. [5]). The connection Ξis clearly not uniquely defined, but for every two connections Ξ1 and Ξ2 with the aboveproperties there is a map u ∈ G0(P × S1 ×R) such that u∗Ξ1 = Ξ2, the existence andthe uniqueness of u follow from the definition ofMg(P ) and from the equivariance ofthe conditions (8.1). The gauge group G0 (P × S1 × R) is defined as the set of smoothmaps g : S1 × R→ G0(P ).

Furthermore, [Ξ] is a heat flow between the perturbed geodesics Ξ± ∈ CritbEH , b ∈ R,if it satisfies the flow equation for the functional EH , i.e.

∂sA− dAΦ− πA(∇t (∂tA− dAΨ) + ∗Xt(A)

)= 0,

lims→±∞

Ξ(s) = Ξ±(8.2)

where ∇t := ∂t + [Ψ, ·] and the perturbation term Xt will be discussed in the nextsection. Since

d∗A (∇t (∂tA− dAΨ) + ∗Xt(A)) =∇td∗A (∂tA− dAΨ) + ∗dAXt(A)

+ ∗ [(∂tA− dAΨ) ∧ ∗ (∂tA− dAΨ)] = 0,

we can write the first line of (8.2) as

∂sA− dAΦ−∇t (∂tA− dAΨ)− ∗Xt(A) + d∗Aω = 0,

dAd∗Aω = − [(∂tA− dAΨ) ∧ (∂tA− dAΨ)] + dA ∗Xt(A)

(8.3)

where ω(t, s) ∈ Ω2(Σ, gP ) is uniquely defined by the second condition which is ob-tained deriving the first one by dA using the commutation formula (1.20) and (8.1).

87

88 8. Flow equations and Morse homologies

The linearised operator for a heat flow Ξ is

D0(Ξ)(πA(α)) :=πA(∇sπA(α)− 2[ψ0, (∂tA− dAΨ)]

−∇t∇tπA(α)− d ∗Xt(A)πA(α) + ∗[α ∧ ∗ω]) (8.4)

for any α : S1 × R → Ω1(Σ, gP ) and where ∇s := ∂s + [Φ, ·] and ψ0 is defineduniquely by d∗AdAψ0 = −2 ∗ [πA(α) ∧ ∗(∂tA− dAΨ)].

8.2 PerturbationIn order to achieve the transversality we have to perturb the equations and for this pur-pose we choose an abstract perturbation on L(Mg(P )).

First, we choose a perturbation V : L(Mg(P ))→ R on the loop space ofM(P ). Weassume that V satisfies the following condition (see condition (V4), chapter 2 of [17]or condition (V3), chapter 1 of [24]). For any two integers k > 0 and l ≥ 0 there is aconstant c = c(k, l) such that∣∣∇l

t∇ks V(A)

∣∣ ≤ c∑kj ,lj

(Πj,lj>0

∣∣∣∇ljt ∇kj

s A∣∣∣)Πj,lj=0

(∣∣∇kjs A∣∣+∥∥∇kj

s A∥∥Lpj

)(8.5)

for every smooth map A : R → L(A0(P )) : s 7→ A(·, s) and every (t, s) ∈ S1 × R;here pj ≥ 1 and

∑lj=0

1pj

= 1; the sum runs over all partitions k1 + · · ·+ km = k andl1+· · ·+lm ≤ l such that kj+lj ≥ 1 for all j. For k = 0 the same inequality holds withan additional summand c on the right. In addition, ∇kj

s A and ∇ljt A should be inter-

preted as ∇kj−1s (∂sA− dAΦ) and ∇lj−1

t (∂tA− dAΨ). Ψ(t, s),Φ(t, s) ∈ Ω0(Σ, gP )are uniquely defined by d∗A (∂tA− dAΨ) = 0 and d∗A (∂sA− dAΦ) = 0.

Using the results of Weber (cf. [24], theorem 1.13) we can consider the following.For any regular value b of the energy functional EH there is a Banach manifold Ob ofperturbations V that satisfy the above condition (8.5) and such that EH + V have thesame critical loops as EH . Moreover there is a residual subset Obreg ⊂ Ob such thatthe perturbed functional EH + V is Morse-Smale below the energy level b if V ∈ Obreg.From now on all the computations are done using a generic perturbation of this kind.Next, we define ∗Xt(A) ⊂ Ω1(Σ, gP ), A ∈ L(A0(P )), t ∈ S1, by∫ 1

0

〈∗Xt(A), ∂sA(t, 0)〉dt :=d

ds

∣∣∣s=0

(V(A(s)) +

∫ 1

0

Ht(A(s))dt

)(8.6)

for every smooth variation A(t, s) of A(t).

Furthermore, using the results of Weber (cf. [24], theorems 1.7, 1.8), for any constantb there are positive constants c, ρ, c0, c1, c2, . . . such that the following holds. If theconnection Ξ = A+ Ψdt+ Φds satisfies (8.2) and EH(A(·, s)) ≤ b, then for every s

‖∂tA− dAΨ‖L∞ + ‖∇t(∂tA− dAΨ)‖L∞ ≤ c, (8.7)

8.3 Yang-Mills flow 89

‖∂sA− dAΦ‖L∞ + ‖∇t(∂sA− dAΦ)‖L∞ + ‖∇s(∂sA− dAΦ)‖L∞ ≤ c, (8.8)

‖∂sA− dAΦ‖Ck(S1×[T,∞)) ≤ cke−ρT , (8.9)

‖∂sA− dAΦ‖Ck(S1×(−∞,−T ]) ≤ cke−ρT , (8.10)

for every T ≥ 1. Moreover [A] converges to a perturbed geodesic in C2(S1) ass→ ±∞.

Next, we need to choose a gauge invariant extension V : L(A(P )/G0(P )) → R,V(A) = V(A) for A ∈ L(A0(P )), such that V satisfies (8.5) for any smooth mapA : R → L(A(P )) with ‖FA(s)‖L2(Σ) ≤ δ0 for every s ∈ R, where δ0 is chosensuch that the lemmas 6 and 7 hold for p = 2 and q = 4. Another possibility is toextend V in the following way. We choose a smooth map ρ : [0,∞) → [0, 1] withthe property that ρ(x) = 0 if x ≥ δ0 and ρ(x) = 1 if x ≤ 2δ0

3. Next, we define

V : L(A(P )/G0(P ))→ R by

V(A) = ρ

(supt∈S1

‖FA(t, s)‖L2

)V (A+ ∗dAη(A)) (8.11)

where η(A) is the unique 0-form, by lemma 7, which satisfies

FA+∗dAη(A) = 0, ‖dAη(A)‖L4(Σ) ≤ c‖FA‖L2(Σ).

We define Xt(A) in the same way as in (8.6).

8.3 Yang-Mills flow

The Yang-Mills flow equations forA(s)+Ψ(s)dt ∈ A(P×S1) and for an ε-independentmetric are

∂sA− dAΦ + d∗AFA −∇t (∂tA− dAΨ)− ∗Xt(A) = 0,

∂sΨ−∇tΦ− d∗A (∂tA− dAΨ) = 0(8.12)

where Φ(t, s) ∈ Ω0(Σ, gP ) in order to make the equations gauge invariant. We canconsider the s-dependent connection A(s) + Ψ(s)dt together with the 0-form Φ as aconnection Ξ := A+ Ψdt+ Φds on the 4-manifold Σ×S1×R. In our case we shrinkthe metric on Σ by ε2 and therefore the adjoint of the exterior derivative dA contributewith a factor 1

ε2to the flow equation and if we consider the flow lines between two

perturbed Yang-Mills connections Ξ± ∈ CritbEH we have the equation

∂sA− dAΦ +1

ε2d∗AFA −∇t (∂tA− dAΨ)− ∗Xt(A) = 0,

∂sΨ−∇tΦ−1

ε2d∗A (∂tA− dAΨ) = 0, lim

s→±∞Ξ = Ξ±.

(8.13)

Another viewpoint to see the last equation is to consider (8.12) for the connection

A(t, s)+Ψ(t, s)dt+Φ(t, s)ds = A(εt, ε2s)+εΨ(εt, ε2s)dt+ε2Φ(εt, ε2s)ds, (8.14)

which is equivalent to (8.13) for (t, s) ∈[0, 1

ε

]× R.


8.4 Norms II

We choose a reference connection Ξ := A + Ψdt + Φds ∈ A(Σ × S1 × R); letξ := α + ψdt + φds ∈ Ω1(Σ × S1 × R, gP ), α(t, s), ψ(t, s), φ(t, s) ∈ Ωj(Σ, gP ),j = 0, 1, then

‖α + ψdt+ φds‖∞,ε := ‖α‖L∞ + ε‖ψ‖L∞ + ε2‖φ‖L∞ ,

‖α + ψdt+ φds‖p0,p,ε :=

∫S1×R

(‖α‖pLp(Σ) + εp‖ψ‖pLp(Σ) + ε2p‖φ‖pLp(Σ)

)dt ds,

‖α + ψdt+ φds‖p1,p,ε := ‖α + ψdt+ φds‖p0,p,ε

+

∫S1×R

(‖d∗Aα‖

pLp(Σ) + ‖dAα‖pLp(Σ) + εp‖∇tα‖pLp(Σ) + ε2p‖∇sα‖pLp(Σ)

)dt ds

+

∫S1×R

(εp‖dAψ‖pLp(Σ) + ε2p‖∇tψ‖pLp(Σ) + ε3p‖∇sψ‖pLp(Σ)

)dt ds

+

∫S1×R

(ε2p‖dAφ‖pLp(Σ) + ε3p‖∇tφ‖pLp(Σ) + ε4p‖∇sφ‖pLp(Σ)

)dt ds,

‖α + ψdt+ φds‖p1,2;p,ε := ‖α + ψdt+ φds‖p1,p,ε

+

∫S1×R

(‖d∗AdAα‖

pLp(Σ) + εp‖∇tdAα‖pLp(Σ)

)dt ds

+

∫S1×R

(ε‖d∗A(dAψ −∇tα)‖pLp(Σ) + ε2‖∇t(dAψ −∇tα)‖pLp(Σ)

)dt ds

(8.15)

where∇t := ∂t+[Ψ, ·] and∇s := ∂s+[Φ, ·]. The ε-dependent norms are created usingthe following simple rule that is given from the linearisationDε of the Yang-Mills flowequations. For every ∇t and every 0-form ψ, which descends from a 1-form in the t-direction, we put an ε in front of the norm; for every ∇s and every 0-form φ, comingfrom a 1-form in the s-direction, we multiply by ε2. The definition (8.15) contains,in the first line, all the 0-order Lp-norms and the Lp-norms of all the first derivatives;in the last two lines we can find the Lp-norms of some second derivatives. These canbe interpreted in the following way. We split α + ψdt in two orthogonal componentsαi + ψidt ∈ im dΞ and αk + ψkdt ∈ ker d∗εΞ ; on the one side, if α+ ψdt ∈ ker d∗εA+Ψdt,then

ε‖d∗AdAψ − ε2∇t∇tψ‖Lp =ε‖d∗AdAψ −∇td∗Aα‖Lp

≤ε‖d∗A(dAψ −∇tα)‖Lp + ε‖[(∂tA− dAΨ) ∧ ∗α]‖Lp ,

‖dAd∗Aα− ε2∇t∇tα‖Lp =ε2‖dA∇tψ −∇t∇tα‖Lp≤ε2‖∇t(dAψ −∇tα)‖Lp + ε2‖[(∂tA− dAΨ), ψ]‖Lp

and thus ε‖d∗AdAψ‖Lp , ε3‖∇t∇tψ‖Lp , ‖d∗AdAα‖Lp and ε2‖∇t∇tα‖Lp can be estimatesby ‖α + ψdt + φds‖1,2;p,ε as we will discuss in the next chapter. On the other side, ifα + ψdt = dAγ +∇tγdt, γ ∈ Ω0(Σ× S1 × R, gP ), then

d∗AdAα = d∗A[FA, γ], d∗A(dAψ −∇tα) = −d∗A[(∂tA− dAΨ), γ],

∇tdAα = ∇t[FA, γ], ∇t(dAψ −∇tα) = −∇t[(∂tA− dAΨ), γ];

8.5 Morse homologies 91

therefore, under some extra conditions on the curvature FA−(∂tA−dAΨ)dt, for exam-ple that FA = 0 and ∂tA− dAΨ is smooth, the last two lines of (8.15) can be estimatewith the first two if α+ ψdt ∈ im dΞ. Thus, ‖ · ‖1,2;p,ε considers the Lp-norm of ξ andof its derivatives, but the Lp-norm of the second derivatives in the Σ × S1-directionsonly for the ker d∗εA+Ψdt-part of ξ. This orthogonal splitting plays a fundamental role inthe proof of the linear estimates of the next chapter.

Next, we can define the Sobolev spaces

W 1,2;p := W 1,2;p(Σ× S1 × R, T ∗(Σ× S1 × R)⊗ gP×S1×R

),

W 1,p := W 1,p(Σ× S1 × R, T ∗(Σ× S1 × R)⊗ gP×S1×R

)as the completion, respect to the norm ‖ · ‖1,2;p,1 and ‖ · ‖1,p,1, of the 1-forms

Ω1(Σ× S1 × R, gP×S1×R

)with compact support; we denote by W 1,2;p(Ξ−,Ξ+) the space of all connections Ξthat satisfy Ξ − Ξ0 ∈ W 1,2;p for a smooth connection Ξ0 ∈ A(P × S1 × R) and thelimit conditions lims→±∞ Ξ = Ξ±. Furthermore, we denote by G2,p

0 (P × S1 × R) thecompletion of G0 (P × S1 × R) with respect to the Sobolev W 1,p-norm on 1-forms,i.e. g ∈ G2,p

0 (P × S1 × R) if g−1dΣ×S1×Rg ∈ W 1,p and by G1,2;p0 (P × S1 × R)

the completion respect to the norm ‖ · ‖1,2;p,ε on the 1-forms. In addition, we de-note by G1,2;2

0 (P × S1 × R) the gauge group such that an element g is locally inG1,2;2

0 (P × S1 × R), i.e. we allow also elements that do not vanish at ±∞. We con-clude this chapter proving the following Sobolev estimates.

Theorem 51 (Sobolev estimate II). We choose 1 ≤ p, q <∞. Then there is a constantcS such that for any ξ ∈ W 1,p, 0 < ε ≤ 1:

1. If −4q≤ 1− 4

p, then

‖ξ‖0,q,ε ≤ cSε3q− 3p‖ξ‖1,p,ε. (8.16)

2. If 0 < 1− 4p, then

‖ξ‖∞,ε ≤ cSε− 3p‖ξ‖1,p,ε. (8.17)

Proof. Analogously as for the lemma 4.1 in [7] and for the theorem 14, we can defineξ = α + ψdt + φds by α(t, s) = α(εt, ε2s), ψ(t, s) = εψ(εt, ε2s) and φ(t, s) =

ε2φ(εt, ε2s). Thus, ‖ξ‖1,p,ε = ε3p‖ξ‖W 1,p for 0 ≤ t ≤ ε−1, ε ∈ (0, 1]. Therefore the

theorem follows from the standard Sobolev’s inequality.

8.5 Morse homologiesIn this section we want to define the Morse homologies defined using the heat flowand the Yang-Mills L2-flow. We start with the Morse homology of the loop space onMg(P ). First, we introduce the moduli spaces

M0(Ξ−,Ξ+) :=Ξ ∈ W 1,2;2(Ξ−,Ξ+); Ξ satisfies (8.1), (8.2),M0(Ξ−,Ξ+) :=M0(Ξ−,Ξ+)/PG∞


where

PG∞ := g ∈ G1,2;20

(P × S1 × R

);∃S > 0 such that for |s| ≥ S, g(s) = 1.

Now, we choose a regular value b of the energy functional EH . In order to definethe Morse homology of the loop space LbMg(P ) in our Morse-Bott setting, where acritical loop is an equivalence class [A+ Ψdt] of perturbed geodesics with A+ Ψdt ∈CritbEH , we need to count the flow lines between the critical loops with Morse indexdifference 1 in the following way. We define

[CritbEH

]:= CritbEH/G0(P ×S1) and we

consider the space of flow lines between two loops γ± ∈[CritbEH

]:

FL0(γ−, γ+) = Ξ ∈ W 1,2;2(Ξ−,Ξ+); Ξ satisfies (8.1), (8.2), [A±] = γ±

and thus the moduli space

M0(γ−, γ+) := FL0(γ−, γ+)/G1,2;20 (P × S1 × R);

then we organise the critical loops of [CritbEH ] in a chain complex where

CEH ,bk := ⊕γ∈[Critb

EH],index

EH(γ)=kZ2γ

and the boundary operator ∂EHk : CEH ,bk → CEH ,b

k−1 by

∂EH

k γ− :=∑

γ+∈CEH,b

k−1

(]Z2

(M0(γ−, γ+)/R

))γ+.

If the functional EH satisfies the transversality condition, then ∂EHk+1∂EH

k = 0 and inthis case we can define the Morse homolgy

HM∗(LbMg(P ),Z2

):= ker ∂E

H

∗ /im ∂EH

∗+1. (8.18)

As we have already mentioned, by the work of Weber (cf. [24]), for a generic per-turbation, the transversality condition is satisfied and thus the Morse homology of theloop space HM∗

(LbMg(P ),Z2

)is well defined.

Remark. For any two perturbed geodesics γ± ∈ [CritbEH ] and any two representativesΞ± we can identify the moduli spacesM0(γ−, γ+) and M0(Ξ−,Ξ+), in particular wehave

]Z2

(M0(γ−, γ+)/R

)= ]Z2

(M0(Ξ−,Ξ+)/R

).

Next, we define the Morse homology for the Yang-Mills case. First, we denote byMε(Ξ−,Ξ+) and by Mε(Ξ−,Ξ+) the moduli spaces

Mε(Ξ−,Ξ+) :=Ξ ∈ W 1,2;2(P × S1 × R); Ξ satisfies (8.13),Mε(Ξ−,Ξ+) :=Mε(Ξ−,Ξ+)/PG∞.

Also in this case we can define a Morse homology for

Aε,b(P × S1

)/G0

(P × S1

);

8.6 Outline 93

in order to do that we consider the chain complex CYMε,H ,b

k := ⊕θ∈[CritbYMε,H ]Z2θ,

where[CritbYMε,H ] := CritbYMε,H/G0(Σ× S1),

with the boundary operator ∂YMε,H

k : CYMε,H ,b

k → CYMε,H ,b

k−1 defined by

∂YMε,H

k θ− :=∑

θ+∈CYMε,H,b

k−1

]Z2 (Mε(θ−, θ+)/R) θ+

whereMε(θ−, θ+) is the moduli space

Mε(θ−, θ+) := FLε(θ−, θ+)/G1,2;20 (P × S1 × R),

FLε(θ−, θ+) = Ξ ∈ W 1,2;2(Ξ−,Ξ+); Ξ satisfies (8.13), [Ξ±] = θ±.The functional YMε,H will inherits the transversality property of EH provided that εis small enough, in this case ∂YM

ε,H

k+1 ∂YMε,H

k = 0 and thus we can define the Morsehomology

HM∗(Aε,b

(P × S1

)/G0

(P × S1

),Z2

):= ker ∂YM

ε,H

∗ /im ∂YMε,H

∗+1 . (8.19)

Remark. Also in this case, for any two orbits of perturbed Yang-Mills connectionsθ± ∈ [CritbYMε,H ] and any two representatives Ξ± ∈ CritbYMε,H we can identify themoduli spacesMε(θ−, θ+) and Mε(Ξ−,Ξ+); in particular we have

]Z2 (Mε(θ−, θ+)/R) = ]Z2

(Mε(Ξ−,Ξ+)/R

).

8.6 OutlineThe aim of this part is to show that the two Morse homologies (8.18) and (8.19) areisomorph and we give the proof in the chapter 18. In order to do this we need to showthat there is a bijective map

Rb,ε :M0(Ξ−,Ξ+)→Mε(T b,ε(Ξ−), T b,ε(Ξ+)

)for each regular value b of EH , every pair Ξ−,Ξ+ ∈ CritbEH with index difference 1and for ε sufficiently small; for this purpose, we will proceed in the following way. Inchapter 9 we will prove some linear estimates using the linear operator, for a 1-formξ = α + ψdt+ φds ∈ W 1,2;p

Dε(Ξ)(ξ) := Dε1(Ξ)(ξ) +Dε2(Ξ)(ξ)dt+Dε3(Ξ)(ξ)ds (8.20)

where the first two terms are the linearization of (8.13), i.e.

Dε1(Ξ)(ξ) :=∇sα− dAφ+1

ε2d∗AdAα +

1

ε2∗ [α ∧ ∗FA]

−∇t∇tα + dA∇tψ − 2[ψ, (∂tA− dAΨ)]− d ∗Xt(A)α,

Dε2(Ξ)(ξ) :=∇sψ −∇tφ+2

ε2∗ [α ∧ ∗(∂tA− dAΨ)]

− 1

ε2∇td

∗Aα +

1

ε2d∗AdAψ.

(8.21)


and the third one is, for a fixed reference connection Ξ0 = A0 + Ψ0dt+ Φ0ds,

Dε3(Ξ)(ξ) := ∇Φ0

s φ−1

ε4d∗A0α +

1

ε2∇Ψ0

t ψ. (8.22)

The linear operator Dε(Ξ) can also be seen as the linearisation of the map

F ε(Ξ) := F ε1(Ξ) + F ε2(Ξ)dt+ F ε3(Ξ)ds

where

F ε1(Ξ) :=∂sA− dAΦ +1

ε2d∗AFA −∇t (∂tA− dAΨ) ,

F ε2(Ξ) :=∂sΨ−∇tΦ−1

ε2d∗A (∂tA− dAΨ) ,

F ε3(Ξ)(ξ + φds) :=∇Φ0

s (Φ− Φ2)− 1

ε4d∗A0(A− A2) +

1

ε2∇Ψ0

t (Ψ−Ψ2)

(8.23)

and A2 + Ψ2dt+ Φ2ds := Kε2(A0 + Ψ0dt+ Ψ0ds); we will discuss the map Kε2 in thechapter 11.

After computing some quadratic estimates in chapter 10, we will prove the existenceand the local uniqueness of the map Rb,ε (chapter 12). In the following chapters 13,15 and 14 we will prove some a priori estimates that we will use in the chapter 17 inorder to prove the surjectivity of Rb,ε. The section 16 is devoted to show a Coulombgauge condition theorem.

Linear estimates for theYang-Mills flow operator 9

As we already mentioned, by the Weber’s transversality (cf. [24], theorem 1.13), wecan assume that the energy functional EH is Morse-Smale. In this chapter we willprove a linear estimate, theorem 59, for the operator Dε(Ξ) for a perturbed geodesicflow Ξ = A + Φdt + Ψds. The main idea is to divide the linear operator respect tothe orthogonal splitting ker d∗εA+Ψdt ⊕ im dA+Ψdt and to use different linear estimateson the two parts. In order to investigate this we need to decompose, in a unique way,every 1-form ξ = α + ψdt as (αk + ψkdt) + (αi + ψidt) where 1

ε2d∗Aαk −∇tψk = 0

and αi + ψidt = dAγ +∇tγdt for a 0-form γ. Formally, first, we solve the equation

1

ε2d∗AdAγ −∇t∇tγ =

1

ε2d∗Aα−∇tψ (9.1)

which has a unique solution γ whose existence and uniqueness can be proved as in thelemma 6.4 of [7]; then we define

αi + ψidt := dAγ +∇tγdt, αk + ψkdt := (α + ψdt)− (αi + ψidt) (9.2)

and since the splitting is orthogonal,

‖dAγ +∇tγdt‖0,p,ε ≤ ‖α + ψdt‖0,p,ε, ‖αk + ψkdt‖0,p,ε ≤ ‖α + ψdt‖0,p,ε. (9.3)

By definition and using the commutation formulas (1.20), (1.21) we have also that

dA∇tψk =1

ε2dAd

∗Aαk,

dA∇tψi =∇t∇tαi − 2[(∂tA− dAΨ), ψi]− [∇t(∂tA− dAΨ), γ],

∇td∗Aαi =d∗A∇tdAγ + ∗[αi ∧ ∗(∂tA− dAΨ)]

=d∗AdAψi + ∗[αi ∧ ∗(∂tA− dAΨ)] + d∗A[(∂tA− dAΨ), γ]

=d∗AdAψi + 2 ∗ [αi ∧ ∗(∂tA− dAΨ)]− ∗[dA ∗ (∂tA− dAΨ), γ].

(9.4)

Now, we can write the components of the linear operator using this splitting. On theone hand, the first component of Dε(Ξ)(ξ + φds), defined by (8.21), is, using theidentities (9.4),

Dε1(Ξ)(ξ + φds) =∇sαk +1

ε2d∗AdAαk +

1

ε2dAd

∗Aαk −∇t∇tαk

+∇sαi − dAφ− [∇t(∂tA− dAΨ), γ]

− 2[ψk, (∂tA− dAΨ)]− d ∗Xt(A)α;

95

96 9. Linear estimates for the Yang-Mills flow operator

in the other hand, using (9.4), the second component becomes

Dε2(Ξ)(ξ + φds) =∇sψk −∇t∇tψk +1

ε2d∗AdAψk +

2

ε2∗ [αk ∧ ∗(∂tA− dAΨ)]

+∇sψi −∇tφ.

The third component is the easiest to investigate, because it depends only on αi +ψidtand on φ:

Dε3(Ξ)(ξ + φds) = ∇sφ−1

ε4d∗Aα +

1

ε2∇tψ = ∇sφ−

1

ε4d∗Aαi +

1

ε2∇tψi.

Next, the idea is to considerDε(Ξ)(ξ+φds) as the sum of the following three operators

Dε,1(Ξ)(ξ + φds) :=∇sαk +1

ε2d∗AdAαk +

1

ε2dAd

∗Aαk − [ψk, (∂tA− dAΨ)]

−∇t∇tαk +

(∇sψk −∇t∇tψk +

1

ε2d∗AdAψk

)dt

+1

ε2∗ [αk ∧ ∗(∂tA− dAΨ)] dt,

Dε,2(Ξ)(ξ + φds) :=∇sαi − dAφ+ (∇sψi −∇tφ) dt

+

(∇sφ−

1

ε4d∗Aα +

1

ε2∇tψ

)ds,

Restε(Ξ)(ξ + φds) :=− [∇t(∂tA− dAΨ), γ]− [ψk, (∂tA− dAΨ)]

− d ∗Xt(A)α +1

ε2∗ [αk ∧ ∗(∂tA− dAΨ)] dt

and to project them in to the two parts of the orthogonal splitting im dA+Ψdt⊕ker d∗εA+Ψdt.The result is that the important part ofDε,1(Ξ) lies in ker d∗εA+Ψdt and that ofDε,2(Ξ) inim dA+Ψdt as is showed in the next lemma; in other words, we interchange the operatorDε(Ξ) with the projection in the two parts of the splitting. We recall that, by (8.7) and(8.7), we can assume that

‖∂tA− dAΨ‖L∞ + ‖∇t (∂tA− dAΨ)‖L∞ + ‖∂sA− dAΦ‖L∞ ≤ c0 (9.5)

for a positive constant c0.

Lemma 52. We choose b, p > 0. For any geodesic flow Ξ = A + Ψdt + Φds ∈M0(Ξ−,Ξ+), Ξ± ∈ CritbEH , there exists a positive constant c such that∥∥Πim dA+Ψdt

Dε,1(Ξ)(ξ + φds)∥∥

0,p,ε≤ c (‖αk‖Lp + ‖ψk‖Lp + ‖∇tαk‖Lp) ,∥∥(1− Πim dA+Ψdt

)(∇sαi − dAφ+ (∇sψi −∇tφ) dt)

∥∥0,p,ε≤ c‖αi‖Lp ,

ε2 ‖Restε(Ξ)(ξ + φds)‖0,p,ε ≤ cε ‖ξ‖0,p,ε

for all ξ+φds ∈ W 1,2;p and using the splitting ξ =: (αk +ψkdt) + (αi+ψidt) definedby (9.1) and (9.2). We denote by Πim dA+Ψdt

the projection in to the linear subspaceim dA+Ψdt.

97

Proof. First, we remark that

〈∇sαk +1

ε2d∗AdAαk, dAω〉+ ε2〈∇sψk,∇tω〉

= 〈∇s(d∗Aαk − ε2∇tψk), ω〉+

1

ε2∗ [FA ∧ ∗dAαk]

+ 〈∗[(∂sA− dAΦ) ∧ ∗αk] + ε2[(∂sΨ− ∂tΦ + [Ψ,Φ]), ψk], ω〉= 〈∗[(∂sA− dAΦ) ∧ ∗αk] + ε2[(∂sΨ− ∂tΦ + [Ψ,Φ]), ψk], ω〉,

where we used the commutation formulas (1.21) and

[∇s,∇t]ω = [(∂sΨ− ∂tΦ + [Ψ,Φ]), ω] (9.6)

for any 0-form ω ∈ Ω0(Σ× S1 × R, gP ).

〈 1

ε2dAd

∗Aαk − dA∇tψk, dAω〉 = 0,

〈 1

ε2∇td

∗Aαk −∇t∇tψk,∇tω〉 = 0,

〈−∇t∇tαk, dAω〉+ 〈−d∗A∇tαk,∇tω〉 = − ∗ [(∂tA− dAΨ) ∧ ∗∇tαk], ω〉,〈∇tdAψk, dAω〉+ 〈d∗AdAψk,∇tω〉

= 〈∗[(∂tA− dAΨ) ∧ ∗dAψk], ω〉 = 〈− ∗ [∗(∂tA− dAΨ) ∧ ψk], dAω〉

where for the last step we used that d∗A (∂tA− dAΨ) = 0. Next, we choose q such that1p

+ 1q

= 1. Then∥∥∥Πim dA+ΨdtDε,1(Ξ)(ξ + φds)

∥∥∥0,p,ε

≤ supω∈Ω0(Σ×S1×R)

〈Dε,1(Ξ)(ξ + φds), dAω +∇tωdt〉‖dAω +∇tωdt‖q

≤c(‖αk‖Lp + ε2‖ψk‖Lp + ‖∇tαk‖Lp

).

The last estimate follows directly from the next identities, from the Holder’s inequality,from ‖ω‖Lq ≤ c‖dAω‖Lq and from the lemma 53. The second estimate of the lemmafollows from the identity

∇sαi − dAφ+ (∇sψi −∇tφ) dt = ∇sdAγ − dAφ+ (∇s∇tγ −∇tφ) dt

=dA+Ψdt (∇sγ − φ) + [(∂sA− dAΦ), γ]− [(∂tΦ− ∂sΨ− [Φ,Ψ]), γ] dt,

from the a priori estimate (9.5) and from ‖γ‖Lp ≤ c‖dAγ‖Lp = c‖αi‖Lp . The thirdestimate follows directly from the definition of Restε and from the L∞-bound (9.5) forthe curvature terms ∂tA− dAΨ,∇t(∂tA− dAΨ).

Lemma 53. We choose a regular value b of EH . There is a positive constant c suchthat for any perturbed geodesic flow A+ Ψdt+ Φds ∈M0(Ξ−,Ξ+), Ξ± ∈ CritbEH ,

d∗AdA (∂sΨ−∇tΦ) = 2 ∗ [Bs ∧ ∗Bt], (9.7)

‖∂sΨ−∇tΦ‖L∞ ≤ c (9.8)

hold.


Proof. We define Bt = ∂tA− dAΨ und Bs = ∂sA− dAΦ then d∗ABt = d∗ABs = 0 andtherefore

∇sd∗ABt = ∗ [Bs ∧ ∗Bt] + d∗A∇sBt = 0

∇td∗ABs = ∗ [Bt ∧ ∗Bs] + d∗A∇tBs

=− ∗[Bs ∧ ∗Bt] + d∗A∇tBs = 0

yields to d∗A∇tBs − d∗A∇sBt = 2 ∗ [Bs ∧ ∗Bt] where

d∗A∇tBs − d∗A∇sBt = d∗AdA (∇sΨ−∇tΦ− [Φ,Ψ]) .

Finally, we can finish the proof of the lemma, i.e.

d∗AdA (∂sΨ−∇tΦ) = d∗AdA (∇sΨ−∇tΦ− [Φ,Ψ]) = 2 ∗ [Bs ∧ ∗Bt].

Furthermore, for a positive constant c

‖∂sΨ−∇tΦ‖L∞ ≤ 8‖Bs‖L∞‖Bt‖L∞ ≤ c

by (9.5).

Theorem 54. We choose a regular value b ofEH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , any 1-form ξ = α + ψdt+ φds ∈ W 1,2;p and for 0 < ε < ε0

‖ξ‖1,2;p,ε ≤ cε2 ‖Dε(Ξ)(ξ)‖0,p,ε + c‖πA(α)‖Lp , (9.9)

‖(1− πA)(ξ)‖1,2;p,ε ≤cε2 ‖Dε(Ξ)(ξ)‖0,p,ε + cε‖πA(α)‖Lp+ cε2‖∇sπA(α)‖Lp + cε2‖∇t∇tπA(α)‖Lp ,

(9.10)

‖(1− πA)α‖1,2;p,ε ≤cε2 (‖Dε(Ξ)ξ‖0,p,ε + ‖∇sπA(α)‖Lp + ‖πA(α)‖Lp)+ cε2 (‖∇tπA(α)‖Lp + ‖∇t∇tπA(α)‖Lp) .

(9.11)

In order to prove the last statement we need the next two theorems that will be provenin the sections 9.1, 9.2.

Theorem 55. We choose a regular value b ofEH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , any 1-form1 α ∈ W 2,2,1;p, any 0-form ψ ∈ W 2,2,1;p and for 0 < ε < ε0

‖α‖Lp + ‖dAα‖Lp + ‖d∗Aα‖Lp + ‖d∗AdAα‖Lp + ‖dAd∗Aα‖Lp+ ε‖∇tα‖Lp + ε2‖∇t∇tα‖Lp + ε‖dA∇tα‖Lp + ε‖∇tdAα‖Lp+ ε‖d∗A∇tα‖Lp + ε‖∇td

∗Aα‖Lp + ε2‖∇sα‖Lp

≤c∥∥(ε2∇s − ε2∇2

t + ∆A

)α∥∥Lp

+ c‖πA(α)‖Lp ,

(9.12)

‖ψ‖Lp + ‖dAψ‖Lp + ‖d∗AdAψ‖Lp + ε‖∇tψ‖Lp + ε2‖∇t∇tψ‖Lp+ ε‖dA∇tψ‖Lp + ε‖∇tdAψ‖Lp + ε2‖∇sα‖Lp

≤c∥∥(ε2∇s − ε2∇2

t + ∆A

)ψ∥∥Lp.

(9.13)

1A i-form γ, i = 0, 1, is an element of W j,l,k;p, if j derivatives of γ in the Σ-direction, l derivativesin the S1-direction and k derivatives in the R direction are in Lp.

99

Theorem 56. We choose a regular value b of EH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , any 1-form α + ψdt ∈ W 1,1,1;p ∩ im dA+Ψdt, any 0-form φ ∈ W 1,1,1;p

and any 0 < ε < ε0

‖α‖Lp + ‖d∗Aα‖Lp + ε2‖∇sα‖Lp + ε‖∇tα‖Lp + ε‖ψ‖Lp + ε2‖∇tψ‖Lp+ ε3‖∇sψ‖Lp + ε2‖φ‖Lp + ε2‖dAφ‖Lp + ε3‖∇tφ‖Lp + ε4‖∇sφ‖Lp≤cε2 ‖∇sα− dAφ‖Lp + cε3 ‖∇sψ −∇tφ‖Lp

+ cε4

∥∥∥∥∇sφ−1

ε4d∗Aα +

1

ε2∇tψ

∥∥∥∥Lp.

(9.14)

Proof of theorem 54. By theorem 55 and the lemma 52

‖αk + ψkdt‖1,2;p,ε ≤c‖ε2∇sαk − ε2∇2tαk + ∆Aαk‖Lp + c‖πA(α)‖Lp

+ cε‖ε2∇sψk − ε2∇2tψk + d∗AdAψk‖Lp

≤cε2∣∣∣∣(1− Πim dA+Ψdt

)Dε(Ξ)(α + ψdt+ φds)

∣∣∣∣0,p,ε

+ c‖πA(α)‖Lp + cε‖α‖Lp + cε2‖ψk‖Lp + cε2‖∇tαk‖Lpand by theorem 56 and the lemma 52

‖αi + ψidt‖1,2;p,ε ≤cε2∣∣∣∣Πim dA+Ψdt

Dε(Ξ)(α + ψdt+ φds)∣∣∣∣

0,p,ε

+ cε2‖Dε3(Ξ)(α + ψdt+ φds)‖0,p,ε

+ cε‖α‖Lp + cε2‖ψk‖Lp + cε2‖∇tαk‖Lp ;(9.15)

for ε small enough we can conclude therefore that

‖ξ‖1,2;p,ε ≤ cε2 ‖Dε(Ξ)(ξ)‖0,p,ε + c‖πA(α)‖Lp .

The second estimate of the theorem follows from (9.15) and

‖(1− πA)αk + ψkdt‖1,2;p,ε

≤c‖ε2∇s(1− πA)αk − ε2∇2t (1− πA)αk + ∆A(1− πA)αk‖Lp

+ cε‖ε2∇sψk − ε2∇2tψk + d∗AdAψk‖Lp

≤cε2∣∣∣∣(1− Πim dA+Ψdt

)Dε(Ξ)(α + ψdt+ φds)

∣∣∣∣0,2,ε

+ cε‖α‖Lp

+ cε2‖ψk‖Lp + cε2‖∇tαk‖Lp + cε2‖(∇s −∇t∇t)πA(ξ)‖Lp .In order to prove the third estimate we need the following one. We choose q such that1p

+ 1q

= 1, then∣∣∣∣∣∣Πim dA+Ψdtd∗A∇tπA(α)dt

∣∣∣∣∣∣0,p,ε≤ sup

γ∈Ω0

ε2〈d∗A∇tπA(α),∇tγ〉‖dAγ +∇tγdt‖Lq

= supγ∈Ω0

ε2〈∇tπA(α), dA∇tγ〉‖dAγ +∇tγdt‖Lq

= supγ∈Ω0

ε2〈∇tπA(α),∇tdAγ〉 − ε2〈∇tπA(α), [(∂tA− dAΨ), γ]〉‖dAγ +∇tγdt‖Lq

≤ supγ∈Ω0

ε2‖∇t∇tπA(α)‖Lp‖dAγ‖Lq + cε2‖∇tπA(α)‖Lp‖γ‖Lq‖dAγ +∇tγdt‖Lq

≤ε2‖∇t∇tπA(α)‖Lp + cε2‖∇tπA(α)‖Lp .


where the last inequality follows from the estimate ‖γ‖Lq ≤ c‖dAγ‖Lq . Thus, sinceby (9.5)

ε2∥∥Πim dA+Ψdt

Restε(Ξ)(ξ + φds)∥∥

0,p,ε≤cε2‖α‖Lp + cε2‖ψ‖Lp + cε‖(1− πA)α‖Lp

+ c∣∣∣∣Πim dA+Ψdt

d∗A∇tπA(α)dt∣∣∣∣

0,p,ε,

‖αi + ψidt‖1,2;p,ε ≤cε2∥∥Πim dA+Ψdt

Dε(ξ + φds)∥∥

0,2,ε

+ cε2 ‖Dε3(ξ + φds)‖0,2,ε + cε2‖ψ‖Lp + cε2‖α‖Lp+ cε2‖∇tπA(α)‖Lp + cε2‖∇t∇tπA(α)‖Lp ,

‖αk − πA(α)‖1,2;p,ε ≤cε2∥∥(1− Πim dA+Ψdt

)Dε1((1− πA)ξ + φds)

∥∥Lp

≤cε2 (‖Dε1(ξ + φds)‖Lp + ‖ψ‖Lp + ‖α‖Lp)+ cε2 (‖∇sπA(α)‖Lp + ‖∇tπA(α)‖Lp + ‖∇t∇tπA(α)‖Lp)

and finally we can conclude that

‖(1− πA)α‖1,2;p,ε ≤cε2 (‖Dε(ξ)‖0,p,ε + ‖∇sπA(α)‖Lp + ‖πA(α)‖Lp)+ cε2 (‖∇tπA(α)‖Lp + ‖∇t∇tπA(α)‖Lp) .

The next goal is to improve the theorem 54 in the sense that we want to estimate thenorms using only the operator Dε(Ξ) (theorem 59) and in order to do this we need touse the properties of the geodesic flow (lemma 58). We define

ω(A) := dA (d∗AdA)−1 (∇t(∂tA− dAΨ) + ∗Xt(A)).

Lemma 57. We choose a regular value b of EH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , any 1-form ξ = α + ψdz + φds ∈ W 1,2;p and for 0 < ε < ε0

‖πA(Dε(Ξ)(ξ) + ∗[α, ∗ω(A)])−D0(Ξ)πA(ξ)‖Lp≤c‖(1− πA)α + ψdt‖Lp + c‖∇t(1− πA)α‖Lp + ε2‖ψ‖Lp

+ cε2‖∇t∇tπA(α)‖Lp + cε2‖∇tπA(α)‖Lp + cε2‖πA(α)‖Lp+ cε2‖∇sπA(α)‖Lp + cε2‖Dε2(Ξ)(ξ)‖Lp ,

(9.16)

‖πA((Dε(Ξ)∗(ξ) + ∗[α, ∗ω(A)])− (D0(Ξ))∗πA(ξ)‖Lp≤c‖(1− πA)α + ψdt‖Lp + c‖∇t(1− πA)α‖Lp + ε2‖ψ‖Lp

+ cε2‖∇t∇tπA(α)‖Lp + cε2‖∇tπA(α)‖Lp + cε2‖πA(α)‖Lp+ cε2‖∇sπA(α)‖Lp + cε2‖(Dε2(Ξ))∗(ξ)‖Lp .

(9.17)

Proof. By definition we have that

πADε(Ξ)(ξ) :=πA (∇sα−∇t∇tα− 2[ψ, (∂tA− dAΨ)]− d ∗Xt(A)α) ,

D0(Ξ)(πA(ξ)) :=πA(∇sπA(α)− 2[ψ0, (∂tA− dAΨ)]

−∇t∇tπA(α)− d ∗Xt(A)πA(α) + ∗[πA(α) ∧ ∗ω(A)])

101

where d∗AdAψ0 = −2 ∗ [πA(α) ∧ ∗(∂tA− dAΨ)]; therefore

‖πA(Dε(Ξ)(ξ) + ∗[α ∧ ∗ω])−D0(Ξ)πA(ξ)‖Lp≤‖πA (∇s(1− πA)α−∇t∇t(1− πA)α) ‖Lp + c‖(1− πA)α‖Lp

+ ‖πA (2[(ψ − ψ0), (∂tA− dAΨ)]) ‖Lp≤c‖(1− πA)α‖Lp + c‖∇t(1− πA)α‖Lp + c‖ψ − ψ0‖Lp

where we used the commutation formula and the uniform L∞ bound of the curvaturesin order to drop the derivative ∇s and a derivative ∇t. Next, we split the 1-formα+ ψdt = (αk + ψkdt) + (αi + ψidt) in the same way as (9.2). We can easily remarkthat

‖αi + ψidt‖Lp + ‖αk + ψkdt‖Lp ≤ 2‖α + ψdt‖Lp ,

‖αi + ψidt‖Lp ≤ ‖(1− πA)α + ψdt‖Lp .

Furthermore, since ‖ψi‖Lp ≤ c‖dAψi‖Lp , by the commutation formula

‖ψi‖Lp ≤‖(1− πA)α‖Lp + ‖∇tαi‖Lp≤‖(1− πA)α‖Lp + ‖∇t(1− πA)α‖Lp + ‖d∗A∇t(1− πA)αk‖Lp

αk + ψkdt lies in the kernel of dA+Ψds, thus

≤‖(1− πA)α‖Lp + ‖∇t(1− πA)α‖Lp + ε2‖∇t∇tψk‖Lp≤‖(1− πA)α‖Lp + ‖∇t(1− πA)α‖Lp

+ ε2‖∇t∇t(ψk − ψ0)‖Lp + ε2‖∇t∇tψ0‖Lp

by the definition of ψ0

≤‖(1− πA)α‖Lp + ‖∇t(1− πA)α‖Lp + ε2‖∇t∇t(ψk − ψ0)‖Lp+ ε2‖∇t∇tπAα‖Lp + ε2‖πA(α)‖Lp + ε2‖∇tπA(α)‖Lp .

Moreover, the theorem 55 yields to

‖ψk − ψ0‖Lp + ε2‖∇t∇t(ψk − ψ0)‖Lp≤c‖(ε2∇s − ε2∇t∇t + d∗AdA)(ψk − ψ0)‖Lp

by the definition of Dε2 and by lemma 52

≤cε2‖Dε2(Ξ)(α + ψdt)‖Lp + cε2‖∇t∇tψ0‖Lp+ c‖(1− πA)α‖Lp + cε2‖∇sψ0‖Lp + cε2‖α‖Lp+ cε2‖ψk‖Lp + cε2‖∇tα‖Lp + cε2‖∇t∇tπA(α)‖Lp≤cε2‖Dε2(Ξ)(α + ψdt)‖Lp + cε2‖∇t∇tπA(α)‖Lp

+ c‖(1− πA)α‖Lp + cε2‖∇tπA(α)‖Lp + cε2‖πA(α)‖Lp+ cε2‖∇sπA(α)‖Lp + cε2‖ψk‖Lp ,


where the last two steps follows from the definition of ψ0. Thus, by the last estimateswe can conclude that

‖πA(Dε(Ξ)(ξ) + ∗[α ∧ ∗ω])−D0(Ξ)πA(ξ)‖Lp≤c‖(1− πA)α‖Lp + c‖∇t(1− πA)α‖Lp + cε2‖∇t∇tπA(α)‖Lp

+ cε2‖∇tπA(α)‖Lp + cε2‖πA(α)‖Lp + cε2‖∇sπA(α)‖Lp+ cε2‖Dε2(Ξ)(α + ψdt)‖Lp + ε2‖ψ‖Lp

The second estimate of the lemma follows exactly in the same way.

The following lemma was proved by Salamon and Weber in [17] (lemma D.7).

Lemma 58. Assume EH is Morse-Smale and let Ξ = A+Ψdt+Φds ∈M0(Ξ−,Ξ+),Ξ± ∈ CritbEH . Then, for every p > 1, there is a constant c > 0 such that

‖α‖Lp + ‖∇sα‖Lp + ‖∇t∇tα‖ ≤ c‖D0(Ξ)∗(α)‖Lp , (9.18)

‖α‖Lp +‖∇sα‖Lp +‖∇t∇tα‖ ≤ c(‖α− (D0(Ξ))∗(η)‖Lp + ‖D0(Ξ)(α)‖Lp

)(9.19)

for all compactly supported vector fields α, η ∈ H1A.

Theorem 59. We choose a regular value b ofEH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , and any 0 < ε < ε0 the following estimates hold.

‖πA(α)‖1,2;p,1 ≤cε‖Dε(Ξ)(ξ)‖0,p,ε + c‖πA(Dε(Ξ)(ξ) + ∗[α, ∗ω(A)])‖Lp+ c‖πA(ξ − (D0(Ξ))∗(πA(η)))‖Lp ,

(9.20)

‖(1− πA)ξ‖1,2;p,ε ≤cε2‖Dε(Ξ)(ξ)‖0,p,ε + cε‖πA(Dε(Ξ)(ξ) + ∗[α, ∗ω(A)])‖Lp+ cε‖πA(ξ − (D0(Ξ))∗(πA(η)))‖Lp ,

(9.21)

‖(1− πA)α‖1,2;p,ε ≤cε2‖Dε(Ξ)(ξ)‖0,p,ε + cε2‖πA(ξ − (D0(Ξ))∗(πA(η)))‖Lp (9.22)

for all compactly supported 1-forms ξ, η ∈ W 1,2;p and 0 < ε ≤ ε0.

Proof. By theorem 54 and by lemma 58 we have that

‖ξ‖1,2;p,ε + ‖∇sπA(ξ)‖Lp + ‖∇tπA(ξ)‖Lp + ‖∇t∇tπA(α)‖Lp ≤ cε‖Dε(Ξ)(ξ)‖0,p,ε

+ c‖πA(α−D0(Ξ)∗(η))‖Lp + c‖D0(Ξ)(πA(α))‖Lp

and thus always by lemma 58 and by the last estimate

‖(1− πA)(ξ)‖1,2;p,ε ≤cε2‖Dε(Ξ)(ξ)‖0,p,ε + cε‖πA(α−D0(Ξ)∗(η))‖Lp+ cε‖D0(Ξ)(πA(α))‖Lp ,

‖(1− πA)α‖1,2;p,ε ≤cε2‖Dε(Ξ)(ξ)‖0,p,ε + cε2‖πA(α−D0(Ξ)∗(η))‖Lp+ cε2‖D0(Ξ)(πA(α))‖Lp .

9.1 Proof of the theorem 55 103

Therefore, with the lemma 57 we obtain

‖ξ‖1,2;p,ε + ‖∇sπA(ξ)‖Lp + ‖∇t∇tπA(α)‖Lp≤cε‖Dε(Ξ)(ξ)‖0,p,ε + c‖πA(Dε(Ξ)(ξ) + ∗[α, ∗ω])‖Lp

+ c‖πA(ξ − (Dε(Ξ))∗(η)− ∗[η, ∗ω])‖Lp

+c

ε‖(1− πA)ξ‖1,2;p,ε,

‖(1− πA)ξ‖1,2;p,ε ≤cε2‖Dε(Ξ)(ξ)‖0,p,ε + cε‖πA(Dε(Ξ)(ξ) + ∗[α, ∗ω])‖Lp+ cε‖πA(ξ − (Dε(Ξ))∗(η)− ∗[η, ∗ω])‖Lp+ c‖(1− πA)α‖1,2;p,ε,

‖(1− πA)α‖1,2;p,ε ≤cε2‖Dε(Ξ)(ξ)‖0,p,ε + cε2‖πA(Dε(Ξ)(ξ) + ∗[α, ∗ω])‖Lp+ cε2‖πA(ξ − (Dε(Ξ))∗(η)− ∗[η, ∗ω])‖Lp

and thus the theorem follows combining these last three estimates.

In the same way as theorem 59 one can prove the following theorem.

Theorem 60. We choose a regular value b ofEH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , and any 0 < ε < ε0 the following estimates hold.

‖(1− πA)ξ‖1,2;p,ε + ε‖πA(α)‖1,2;p,1

≤cε2‖(Dε(Ξ))∗(ξ)‖0,p,ε + cε‖πA((Dε(Ξ))∗(ξ) + ∗[α, ∗ω(A)])‖Lp ,(9.23)

‖(1− πA)α‖1,2;p,ε ≤cε2‖(Dε(Ξ))∗(ξ)‖0,p,ε (9.24)

for all compactly supported 1-forms ξ ∈ W 1,2;p and 0 < ε ≤ ε0.

9.1 Proof of the theorem 55We will use the following criterion to prove our estimates (cf. [25], theorem C.2).

Theorem 61 (Marcinkiewicz, Mihlin). Let m : Rn → C be a measurable functionthat for some constant c0 satisfies∣∣∣∣∣xi1xi2 ...xis ∂sm

∂xi1∂xi2 ...∂xis

∣∣∣∣∣ ≤ c0 (9.25)

for all integers 0 ≤ s ≤ n and 1 ≤ i1 < i2 < ... < is ≤ n. We define Tm : L2(R2)→L2(R2) by

Tmf := F−1(mF(f))

where f ∈ L2(R2) and F : L2(Rn,C) → L2(Rn,C) is the Fourier transformationgiven by

(Ff)(y1, ..., yn) :=1

2π

∫ ∞−∞

. . .

∫ ∞−∞

e−i(y0x0+···+ynxn)f(x1, . . . , xn)dx1 . . . dxn


for f ∈ L2(Rn,C) ∩ L1(Rn,C). Then m is an Lp-multiplier for all 1 < p < ∞, i.e.there exists a constant c such that whenever f ∈ Lp(Rn)∩L2(R2) then Tmf ∈ Lp(Rn)and

‖Tmf‖Lp ≤ c‖f‖Lp . (9.26)

Corollary 62. For every p > 1 there is a positive constant c such that

‖∂su‖Lp +n−1∑i,j=0

∥∥∂xi∂xju∥∥Lp ≤ c

∥∥∥∥∥(∂s −

n−1∑i=0

∂xi∂xi

)u

∥∥∥∥∥Lp

for every u ∈ W 1,2,p0 (R× Rn) ∩W 1,2,2

0 (R× Rn).

Proof. We define f ∈ Lp(R × Rn) ∩ L2(R × Rn) by f =(∂s −

∑n−1i=0 ∂xi∂xi

)u and

thus

F(f) =

(iσ +

n−1∑i=0

y2i

)F(u)

and therefore

F(∂su) =iσ

iσ +∑n−1

i=0 y2i

F(f) =: ms(σ, y0, ..., yn−1)F(f),

F(∂xi∂xju) =yiyj

iσ +∑n−1

i=0 y2i

F(f) =: myi(σ, y0, ..., yn−1)F(f).

The multipliers ms(σ, y0, ..., yn−1) and myi(σ, y0, ..., yn−1) satisfy the condition (9.25)and therefore we can apply the theorem 61 and conclude the proof.

Notation. We denote dAd∗A + d∗AdA by ∆A.

Lemma 63. We choose a connection A0 ∈ A0(P ), then there is a positive constant csuch that for any 0- or 1-form α with compact support:

‖∂sα‖Lp(Σ×R2) + ‖∂2t α‖Lp(Σ×R2) + ‖dA0d

∗A0α‖Lp(Σ×R2)

+ ‖d∗A0dA0α‖Lp(Σ×R2) + ‖∂td∗A0

α‖Lp(Σ×R2) + ‖∂tdA0α‖Lp(Σ×R2)

≤c∥∥(∂s − ∂2

t + ∆A0

)α∥∥Lp(Σ×R2)

+ c‖α‖Lp(Σ×R2)

+ c‖∂tα‖Lp(Σ×R2) + c‖dA0α‖Lp(Σ×R2) + c‖d∗A0α‖Lp(Σ×R2).

(9.27)

Proof. The previous corollary continues to holds if we consider a metric closed toa constant metric. Therefore we can pick a finite atlas R2 × Vi, ϕi : R2 × Vi →R2×Σi∈I and a partition of the unity ρii∈I ⊂ C∞(R2×Σ, [0, 1]),

∑i∈I ρi(x) = 1

for every x ∈ R2 × Σ and supp(ρi) ⊂ ϕi(R2 × Vi) for any i ∈ I , and apply thecorollary 62 for (ρi ϕi)αi where αi is the local representations of α on R2 × Vi.Summing all the estimates and considering the smooth constant connection A0(t, s)we obtain (9.27).


Lemma 64. We choose a flat connectionA0 ∈ A0(P ), then there is a positive constantc such that for any 0- or 1-form α with compact support:

ε2‖∂sα‖Lp(Σ×R2) + ε2‖∂2t α‖Lp(Σ×R2) + ‖dA0d

∗A0α‖Lp(Σ×R2)

+ ‖d∗A0dA0α‖Lp(Σ×R2) + ε‖∂td∗A0

α‖Lp(Σ×R2) + ε‖∂tdA0α‖Lp(Σ×R2)

≤cε2

∥∥∥∥(∂s − ∂2t +

1

ε2∆A0

)α

∥∥∥∥Lp(Σ×R2)

+ c‖α‖Lp(Σ×R2)

+ cε‖∂tα‖Lp(Σ×R2) + c‖dA0α‖Lp(Σ×R2) + c‖d∗A0α‖Lp(Σ×R2)

(9.28)

Proof. The lemma follows from the previous lemma 63 using the rescaling α(t, s) :=α(εt, ε2s) .

Lemma 65. We choose a flat connection A0 ∈ A0(P ) and a constant c0, then there isa positive constant c such that following holds. For any connection A ∈ A2,p(P ×R2)which satisfies

sup(s,t)∈R2

(‖A(s, t)− A0‖C1 + ε‖∂tA‖L∞) ≤ c0 (9.29)

and for any 0- or 1-form α with compact support:

ε2‖∂sα‖Lp(Σ×R2) + ε2‖∂2t α‖Lp(Σ×R2) + ‖dAd∗Aα‖Lp(Σ×R2)

+ ‖d∗AdAα‖Lp(Σ×R2) + ε‖∂td∗Aα‖Lp(Σ×R2) + ε‖∂tdAα‖Lp(Σ×R2)

≤cε2

∥∥∥∥(∂s − ∂2t +

1

ε2dAd

∗A +

1

ε2d∗AdA

)α

∥∥∥∥Lp(Σ×R2)

+ c‖α‖Lp(Σ×R2).

(9.30)

Proof. This lemma follow directly from the lemma 63 using the assumption (9.29) andthe lemma 10.

Lemma 66. We choose a regular value b ofEH , then there is a positive constant c suchthat the following holds. For any Ξ = A+ Ψdt+ Φds ∈M0(Ξ−,Ξ+), Ξ± ∈ CritbEH ,and any 0- or 1-form α ∈ W 2,2,1;p

‖α‖Lp + ‖dAα‖Lp + ‖d∗Aα‖Lp + ‖d∗AdAα‖Lp + ‖dAd∗Aα‖Lp + ε‖∇tα‖Lp+ ε2‖∇t∇tα‖Lp + ε2‖∇sα‖Lp + ε‖∇td

∗Aα‖Lp + ε‖∇tdAα‖Lp

≤c∥∥(ε2∇s − ε2∇2

t + ∆A

)α∥∥Lp

+ c‖α‖Lp .(9.31)

Proof. We choose a finite atlas Bi, ϕi : Bi → S1 × Ri∈I of S1 × R such that thecondition (9.29) is satisfied for every chart; we can cover the two ends of the cylinderwith two chart each because A(t, s) converges exponentially to A± as s → ±∞ andthus for s0 big enough

sup(s,t)∈S1×[s0,∞)

(‖A(s, t)− A+‖C1 + ε‖∂tA‖L∞) ≤c0,

sup(s,t)∈S1×(−∞,s0]

(‖A(s, t)− A−‖C1 + ε‖∂tA‖L∞) ≤c0.

Furthermore, we take a partition of the unity∑

i∈N ρi(t, s) = 1, ρi(t, s) ∈ [0, 1] andsupp(ρi) ⊂ ϕ(Bi); next, collecting the estimate given by the lemma 65 on every chart


Bi × Σ for (ρi ϕi)αi, where αi is the representation of α on Bi × Σ, we obtain

ε2‖∂sα‖Lp + ε2‖∂t∂tα‖Lp + ‖d∗AdAα‖Lp + ‖dAd∗Aα‖Lp+ ε‖∂td∗Aα‖Lp + ε‖∂tdAα‖Lp≤c∥∥(ε2∂s − ε2∂2

t + ∆A

)α∥∥Lp

+ c‖α‖Lp + cε2‖∂tα‖Lp .(9.32)

Since ‖Ψ‖L∞ + ‖∂tΨ‖L∞ + ‖Φ‖L∞ ≤ c1, we have

ε2‖∇sα‖Lp + ε2‖∇t∇tα‖Lp + ‖d∗AdAα‖Lp + ‖dAd∗Aα‖Lp+ ε‖∇td

∗Aα‖Lp + ε‖∇tdAα‖Lp

≤c∥∥(ε2∇s − ε2∇2

t + ∆A

)α∥∥Lp

+ c‖α‖Lp+ cε2‖∇tα‖Lp + cε‖d∗Aα‖Lp + cε‖dAα‖Lp .

(9.33)

The estimate (9.31) follows then from the lemmas 9 and 10.

Lemma 67. We choose a regular value b of EH , then there are two positive constantsc and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈ M0(Ξ−,Ξ+),Ξ± ∈ CritbEH , any i-form ξ ∈ W 2,2,1;p, i = 0, 1 and 0 < ε < ε0∫

S1×R‖ξ‖pL2(Σ) dt ds ≤c

∫S1×R

‖ε2∂sξ − ε2∂2t ξ + ∆Aξ‖pL2(Σ)dt ds

+ c

∫S1×R

‖πA(ξ)‖pL2(Σ)dt ds.

(9.34)

Proof. In this proof we denote the norm ‖ · ‖L2(Σ) by ‖ · ‖. If we consider only theLaplace part of the operator, we obtain that∫

S1×R‖ξ‖p−2〈ξ,−ε2∂2

t ξ + ∆Aξ〉ds dt

=

∫S1×R

‖ξ‖p−2(ε2‖∂tξ‖2 + ‖dAξ‖2 + ‖d∗Aξ‖2

)ds dt

+

∫S1×R

(p− 2)‖ξ‖p−4〈ξ, ∂tξ〉2ds dt

and thus∫S1×R‖ξ‖p−2

(ε2‖∂tξ‖2 + ‖dAξ‖2 + ‖d∗Aξ‖2

)ds dt

≤∫S1×R

‖ξ‖p−2〈ξ,−ε2∂2t ξ + ∆Aξ〉ds dt

=

∫S1×R

‖ξ‖p−2〈ξ, ε2∂sξ − ε2∂2t ξ + ∆Aξ〉ds dt

≤∫S1×R

‖ξ‖p−1‖ε2∂sξ − ε2∂2t ξ + ∆Aξ‖ ds dt

≤(∫

S1×R‖ξ‖pds dt

) p−1p(∫

S1×R‖ε2∂sξ − ε2∂2

t ξ + ∆Aξ‖pds dt) 1

p

(9.35)


where the second step follows from∫S1×R

‖ξ‖p−2〈ξ, ∂sξ〉ds dt =1

p

∫S1×R

∂s‖ξ‖pds dt = 0,

the third from the Cauchy-Schwarz inequality and the fourth from the Holder’s in-equality. Therefore, by lemma 8∫

S1×R‖ξ‖pds dt ≤

∫S1×R

‖ξ‖p−2(‖dAξ‖2 + ‖d∗Aξ‖2 + ‖πA(ξ)‖2

)ds dt

by (9.35) we have that

≤(∫

S1×R‖ξ‖pds dt

) p−1p(∫

S1×R‖ε2∂sξ − ε2∂2


p

+

∫S1×R

‖ξ‖p−1‖πA(ξ)‖ ds dt

and by the Holder’s inequality

≤(∫

S1×R‖ξ‖pds dt

) p−1p(∫

S1×R‖ε2∂sξ − ε2∂2


p

+

(∫S1×R

‖ξ‖pds dt) p−1

p(∫

S1×R‖πA(ξ)‖pds dt

) 1p

;

thus, we can conclude that∫S1×R

‖ξ‖pds dt ≤ c

∫S1×R

(‖ε2∂sξ − ε2∂2

t ξ + ∆Aξ‖p + ‖πA(ξ)‖p)ds dt.

and hence we finished the proof of the lemma.

Proof of theorem 55. By lemma 8, for any δ > 0 there is a c0 such that

‖α‖pLp ≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0

∫S1×R

‖α‖pL2dt ds

≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0c1

∫S1×R

‖πA(α)‖pL2dt ds

+ c0c1

∫S1×R

‖ε2∂sα− ε2∂2t α + ∆Aα‖pL2dt ds

≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0c1c2‖πA(α)‖pLp+ c0c1c2‖ε2∂sα− ε2∂2

t α + ∆Aα‖pLp≤δ (‖dAα‖pLp + ‖dA ∗ α‖pLp) + c0c1c2‖πA(α)‖pLp

+ c0c1c2‖ε2∇sα− ε2∇2tα + ∆Aα‖pLp + c3ε

2‖Ψ‖L∞‖∇tα‖pLp+ c3ε

2(‖Ψ‖2

L∞ + ‖∂tΨ‖L∞ + ‖Φ‖L∞)‖α‖pLp


where the second step follows form the lemma 67 and the third by the Holder’s in-

equality with c2 :=(∫

ΣdvolΣ

) p−2p . Therefore if we choose δ and ε small enough we

can improve the estimate (9.31) of the corollary 66 using the last estimate, i.e.

‖α‖Lp + ‖dAα‖Lp + ‖d∗Aα‖Lp + ‖d∗AdAα‖Lp + ‖dAd∗Aα‖Lp + ε‖∇tα‖Lp+ ε2‖∇t∇tα‖Lp + ε‖∇tdAα‖Lp + ε‖∇td

∗Aα‖Lp + ε2‖∇sα‖Lp

≤∥∥(ε2∇s − ε2∇2

t + ∆A

)α∥∥Lp

+ c‖πA(α)‖Lp ,

because ‖Ψ‖L∞+‖∂tΨ‖L∞+‖Φ‖L∞ is bounded by a constant. Furthermore, the termsε‖dA∇tα‖Lp and ε‖d∗A∇tα‖Lp can be estimated by

dA∇tα‖Lp + ε‖d∗A∇tα‖Lp ≤ ε‖∇tdAα‖Lp + ε‖∇td∗Aα‖Lp + cε‖α‖Lp

using the commutation formulas and because the curvature ∂tA−dAΨ is bounded; weproved therefore (9.12) and the second inequality of the theorem can be proved in thesame way.

9.2 Proof of the theorem 56Before proving the theorem we will show some preliminary results, in fact the theorem56 will then follow from the corollary 69 and the lemmas 8 and 70.

Corollary 68. For every p > 1 there is a positive constant c such that the followingholds. For every two maps γ ∈ W 2,p

0 (R4,R), φ ∈ W 1,p0 (R4,R) we have that

‖∂sα1‖Lp + λ‖∂x1α1‖Lp + ‖∂sα2‖Lp + λ‖∂x2α2‖Lp + λ‖∂tα1‖Lp+ λ‖∂tα2‖Lp + ‖∂sψ‖Lp + λ‖∂tψ‖Lp + ‖∂sφ‖Lp+ ‖∂x1φ‖Lp + ‖∂x2φ‖Lp + ‖∂tφ‖Lp≤c‖∂sα1 − ∂x1φ‖Lp + c‖∂sα2 − ∂x2φ‖Lp + c‖∂sψ − ∂tφ‖Lp

+ c‖∂sφ+ ∂x2α1 + ∂x1α2 + ∂tψ‖Lp + cλ‖∂sγ‖Lp ,

(9.36)

where α1 = ∂x1γ, α2 = ∂x2γ, ψ = ∂tγ and λ ∈ [0, 1].

Proof. In order to prove this corollary we need to apply the theorem 61 of Marcinkiewiczand Mihlin stated in the previous section and in order to do this we have to define themultipliers and prove the assumption (9.25); therefore, we look at the following systemof equations

f =

f1

f2

f3

f4

:=

∂s 0 0 −∂x1

0 ∂s 0 −∂x2

0 0 ∂s −∂t∂x1 ∂x2 ∂t ∂s

α1

α2

ψφ

. (9.37)

One can remark that the four lines of (9.37) correspond to the first four terms in theLp-norm in the right side of the estimate (9.36). Applying the Fourier transformationto (9.36) we obtain

F(f) =

σi 0 0 −y1i0 σi 0 −y2i0 0 σi −τiy1i y2i τ i σi

F(α1)F(α2)F(ψ)F(φ)


and thus computing its invers:

F(α1) =−i (σ2 + y2

2 + τ 2)

(σ2 + y12 + y2

2 + τ 2)σF(f1) +

iy2y1

(σ2 + y12 + y2

2 + τ 2)σF(f2)

+iτ y1

(σ2 + y12 + y2

2 + τ 2)σF(f3) +

−iy1

σ2 + y12 + y2

2 + τ 2F(f4),

F(α2) =iy2y1

(σ2 + y12 + y2

2 + τ 2)σF(f1) +

−i (σ2 + y12 + τ 2)

(σ2 + y12 + y2

2 + τ 2)σF(f2)

+iτ y2

(σ2 + y12 + y2

2 + τ 2)σF(f3) +

−iy2

σ2 + y12 + y2

2 + τ 2F(f4),

F(ψ) =iτ y1

(σ2 + y12 + y2

2 + τ 2)σF(f1) +

iτ y2

(σ2 + y12 + y2

2 + τ 2)σF(f2)

+−i (σ2 + y1

2 + y22)

(σ2 + y12 + y2

2 + τ 2)σF(f3) +

−iτσ2 + y1

2 + y22 + τ 2

F(f4),

F(φ) =iy1

σ2 + y12 + y2

2 + τ 2F(f1) +

iy2

σ2 + y12 + y2

2 + τ 2F(f2)

+iτ

σ2 + y12 + y2

2 + τ 2F(f3) +

−iσσ2 + y1

2 + y22 + τ 2

F(f4),

and then

F(∂sα1) =σ2 + y2

2 + τ 2

σ2 + y12 + y2

2 + τ 2F(f1) +

−y2y1

σ2 + y12 + y2

2 + τ 2F(f2)

+−τ y1

σ2 + y12 + y2

2 + τ 2F(f3) +

y1σ

σ2 + y12 + y2

2 + τ 2F(f4),

F(∂sα2) =−y2y1

σ2 + y12 + y2

2 + τ 2F(f1) +

σ2 + y12 + τ 2

σ2 + y12 + y2

2 + τ 2F(f2)

+−τ y2

σ2 + y12 + y2

2 + τ 2F(f3) +

y2σ

σ2 + y12 + y2

2 + τ 2F(f4),

F(∂sψ) =−τ y1

σ2 + y12 + y2

2 + τ 2F(f1) +

−τ y2

σ2 + y12 + y2

2 + τ 2F(f2)

+σ2 + y1

2 + y22

σ2 + y12 + y2

2 + τ 2F(f3) +

τσ

σ2 + y12 + y2

2 + τ 2F(f4),

F(∂sφ) =−y1σ

σ2 + y12 + y2

2 + τ 2F(f1) +

−y2σ

σ2 + y12 + y2

2 + τ 2F(f2)

+−τσ

σ2 + y12 + y2

2 + τ 2F(f3) +

σ2

σ2 + y12 + y2

2 + τ 2F(f4).


Since the multipliers for F(∂sα1), F(∂sα2), F(∂sψ) and F(∂sφ) satisfy the assump-tion (9.25) of the theorem 61, we can conclude that

‖∂sα1‖Lp + ‖∂sα2‖Lp + ‖∂sψ‖Lp + ‖∂sφ‖Lp + ‖∂x1φ‖Lp+ ‖∂x2φ‖Lp + ‖∂tφ‖Lp + ‖∂x1α1 + ∂x2α2 + ∂tψ‖Lp≤c‖∂sα1 − ∂x1φ‖Lp + c‖∂sα2 − ∂x2φ‖Lp + c‖∂sψ − ∂tφ‖Lp

+ c‖∂sφ+ ∂x2α1 + ∂x1α2 + ∂tψ‖Lp .

(9.38)

Next, we use that α1 = ∂x1γ, α2 = ∂x2γ, ψ = ∂tγ and thus

‖∂sγ − (∂2x1

+ ∂2x2

+ ∂2t )γ‖Lp ≤‖(∂2

x1+ ∂2

x2+ ∂2

t )γ‖Lp + ‖∂sγ‖Lp≤‖∂x1α1 + ∂x2α2 + ∂tψ‖Lp + ‖∂sγ‖Lp .

Therefore by corollary 62, it follow that

λ‖∂x1α1‖Lp+λ‖∂x2α2‖Lp + λ‖∂tα1‖Lp + λ‖∂tα2‖Lp + λ‖∂tψ‖Lp≤cλ‖∂sα1 − ∂x1φ‖Lp + cλ‖∂sα2 − ∂x2φ‖Lp + cλ‖∂sψ − ∂tφ‖Lp

+ cλ‖∂sφ+ ∂x2α1 + ∂x1α2 + ∂tψ‖Lp + cλ‖∂sγ‖Lp .(9.39)

Therefore the theorem follows combining (9.38) and (9.38).

Lemma 69. We choose a regular value b ofEH , then there is a positive constant c suchthat the following holds. For any Ξ = A+ Ψdt+ Φds ∈M0(Ξ−,Ξ+), Ξ± ∈ CritbEH ,and any 1-form α + ψdt = dA+Ψdtγ ∈ W 1,2;p ∩ im dA+Ψdt

‖α‖Lp + ‖d∗Aα‖Lp + ε2‖∇sα‖Lp + ε‖∇tα‖Lp + ε‖ψ‖Lp + ε2‖∇tψ‖Lp+ ε3‖∇sψ‖Lp + ε2‖φ‖Lp + ε2‖dAφ‖Lp + ε3‖∇tφ‖Lp + ε4‖∇sφ‖Lp≤cε2 ‖∇sα− dAφ‖Lp + cε3 ‖∇sψ −∇tφ‖Lp

+ cε4

∥∥∥∥∇sφ−1

ε4d∗Aα +

1

ε2∇tψ

∥∥∥∥Lp

+ c‖α‖Lp + cε2‖φ‖Lp .

(9.40)

Proof. In the same way that the lemma 66 follows from the corollary 62, the corollary68 implies

‖α‖Lp + ε2‖∇sα‖Lp + λ‖d∗Aα‖Lp + λε‖∇tα‖Lp + ε3‖∇sψ‖Lp + λε2‖∇tψ‖Lp+ ε2‖φ‖Lp + ε4‖∇sφ‖Lp + ε2‖dAφ‖Lp + ε3‖∇tφ‖Lp≤cε2‖∇sα− dAφ‖Lp + cε3‖∇sψ −∇tφ‖Lp + cλε2‖∇sγ‖Lp

+ cε4

∥∥∥∥∇sφ+1

ε4d∗Aα1 +

1

ε2∇tψ

∥∥∥∥Lp

+ c‖α‖Lp + cε2‖φ‖Lp

The term ε2‖∇sγ‖Lp can be estimate by cε2‖∇sα‖Lp + cε2‖α‖Lp by the lemma 6 andthe commutation formula and thus using the last estimate for λ = 0

ε2‖∇sγ‖Lp ≤cε2‖∇sα− dAφ‖Lp + cε3‖∇sψ −∇tφ‖Lp

+ cε4

∥∥∥∥∇sφ+1

ε4d∗Aα1 +

1

ε2∇tψ

∥∥∥∥Lp

+ c‖α‖Lp + cε2‖φ‖Lp .


Furthermore, by the lemma 6 and the commutation formula

ε‖ψ‖Lp ≤ cε‖dAψ‖Lp ≤ c‖α‖Lp + cε‖∇tα‖Lp

and finally collecting the last thee estimates we obtain (9.40).

Lemma 70. We choose a regular value b ofEH and a δ > 0, then there are two positiveconstants c and ε0 such that the following holds. For any Ξ = A + Ψdt + Φds ∈M0(Ξ−,Ξ+), Ξ± ∈ CritbEH , any 1-form ξ := α + ψdt + φdt ∈ W 1,1,1;p, whereα + ψdt ∈ im dA+Ψdt, and any 0 < ε < ε0∫

S1×R‖ξ‖p−2

L2(Σ)

(‖α‖2

L2(Σ) + ε4‖φ‖2L2(Σ)

)dt ds

≤∫S1×R

(c‖ε2∂sα− ε2dAφ‖pL2(Σ) + cεp‖ε2∂sψ − ε2∂tφ‖pL2(Σ)

)dt ds

+

∫S1×R

(cε2p

∥∥∥∥ε2∂sφ−1

ε2d∗Aα + ∂tψ

∥∥∥∥pL2(Σ)

+ δ‖ξ‖pL2(Σ)

)dt ds.

(9.41)

Proof. In this proof we denote the norm ‖ · ‖L2(Σ) by ‖ · ‖. We consider ξ = α+ψdt+φds where α + ψdt = dA+Ψdtγ and

ξ = α + ψdt+ φds = Dξ =

ε2∂s 0 −ε2dA0 ε2∂s −ε2∂t

− 1ε2d∗A ∂t ε2∂s

αψφ

;

thus D∗Dξ can be written in the following way

D∗ξ =

−ε2∂s 0 −ε2dA0 −ε2∂s −ε2∂t

− 1ε2d∗A ∂t −ε2∂s

ε2∂s 0 −ε2dA0 ε2∂s −ε2∂t

− 1ε2d∗A ∂t ε2∂s

αψφ

=

−ε4∂2s + dAd

∗A −ε2dA∂t ε4(∂sdA − dA∂s)

∂td∗A −ε4∂2

s − ε2∂2t 0

−d∗A∂s + ∂sd∗A 0 d∗AdA − ε2∂2

t − ε4∂2s

αψφ

.

We define

B :=1

2‖d∗Aα− ε2∂tψ‖2 + ε4‖∂sα‖2 + ε6‖∂sψ‖2 + ε4‖dAφ‖2

+1

2

(‖d∗Aα‖2 + ‖ε2∂tψ‖2

)+ ε6‖∂tφ‖2 + ε8‖∂sφ‖2

Gp := ‖ε2∂sα− ε2dAφ‖p + εp‖ε2∂sψ − ε2∂tφ‖p + ε2p

∥∥∥∥ε2∂sφ−1

ε2d∗Aα + ∂tψ

∥∥∥∥p


Using the partial integration we obtain∫S1×R‖ξ‖p−2〈α, (−ε4∂2

s + dAd∗A)α− ε2dA∂tψ〉dt ds

+

∫S1×R

‖ξ‖p−2ε2〈ψ, (−ε4∂2s − ε2∂2

t )ψ + ∂td∗Aα〉dt ds

+

∫S1×R

‖ξ‖p−2ε4〈φ, (d∗AdA − ε2∂2t − ε4∂2

s )φ〉dt ds

=

∫S1×R

‖ξ‖p−2B dt ds

+ (p− 2)

∫S1×R

‖ξ‖p−4(〈α, ε2∂sα〉+ ε2〈ψ, ε2∂sψ〉+ ε4〈φ, ε2∂sφ〉

)2dt ds

+

∫S1×R

‖ξ‖p−2ε2∂t〈ψ,−ε2∂tψ〉dt ds+

∫S1×R

‖ξ‖p−2ε2〈dAψ, ∂tα〉dt ds

(9.42)

whose last term, using that α + ψdt = dAγ + ∂tγdt, can be estimate as follows∫S1×R‖ξ‖p−2ε2〈dAψ, ∂tα〉dt ds =

∫S1×R

‖ξ‖p−2ε2〈dA∇tγ, ∂tdAγ〉dt ds

≥∫S1×R

‖ξ‖p−2ε2‖dAψ‖2 − cε2

∫S1×R

‖ξ‖p−1(‖α‖+ ‖d∗Aα‖)dt ds.(9.43)

Since the penultimate line of (9.42) is positive, (9.42) and (9.43) yield∫S1×R‖ξ‖p−2Bdt ds

≤∫S1×R

‖ξ‖p−2(〈α, (D∗ξ)1〉+ ε2〈ψ, (D∗ξ)2〉+ ε4〈φ, (D∗ξ)3〉

)dt ds

−∫S1×R

‖ξ‖p−2ε4 (〈α, [∂sA, φ]〉 − 〈φ, ∗[∂sA ∧ ∗α]〉) dt ds

+ cε2

∫S1×R

‖ξ‖p−1(‖α‖+ ‖d∗Aα‖)dt ds

integrating by parts the first line after the inequality and using the Cauchy-Schwarzinequality, we obtain

≤c∫S1×R

‖ξ‖p−2B12

(‖ε2α‖+ ε‖ε2ψ‖+ ε2

∥∥ε2φ∥∥) dt ds

+ cε2

∫S1×R

‖ξ‖pdt ds+ cε2

∫S1×R

‖ξ‖p−1‖d∗Aα‖ dt ds

Since 2ab ≤ a2 + b2 for any a, b ∈ R, choosing ε small enough

≤c∫S1×R

‖ξ‖p−2(‖ε2α‖+ ε‖ε2ψ‖+ ε2

∥∥ε2φ∥∥)2

dt ds

+1

2

∫S1×R

‖ξ‖p−2Bdt ds+ cε2

∫S1×R

‖ξ‖pdt ds.


The last estimate implies that1

2

∫S1×R‖ξ‖p−2B dt ds ≤ c

∫S1×R

‖ξ‖p−2G2ds+ ε2

∫S1×R

‖ξ‖pdt ds

≤c(∫

S1×R‖ξ‖pds

)1− 2p(∫

S1×RGpds

) 2p

+ cε2

∫S1×R

‖ξ‖pdt ds

≤c∫S1×R

Gpds+ δ

∫S1×R

‖ξ‖pdt ds,

where in the second step we use the Holder’s estimate and in the third the estimateab ≤ ar

r+ bq

q, 1r

+ 1q

= 1, with r = p2. Finally,∫

S1×R‖ξ‖p−2

(‖α‖2 + ε4‖φ‖2

)dt ds ≤

∫S1×R

‖ξ‖p−2(‖d∗Aα‖2 + ε4‖dAφ‖2

)ds

≤c∫S1×R

Gpds+ δ

∫S1×R

‖ξ‖pdt ds

and thus the lemma is proved.

Proof of theorem 56. By lemma 8, for any δ > 0 there is a positive constant c0 suchthat

‖α‖pLp + ε2p‖φ‖pLp ≤ δ(‖dAα‖pLp + ‖d∗Aα‖

pLp + ε2p‖dAφ‖pLp

)+ c0

∫S1×R

(‖α‖pL2 + ε2p‖φ‖pL2

)dt ds

since α = dAγ and the connection is flat on Σ, ‖dAα‖Lp vanishes. By the lemma 70we have then

≤δ(‖d∗Aα‖

pLp + ε2p‖dAφ‖pLp

)+ c0c1

∫S1×R

(‖ε2∂sα− ε2dAφ‖pL2(Σ) + εp‖ε2∂sψ − ε2∂tφ‖pL2(Σ)

)dt ds

+

∫S1×R

(c0c1ε

2p

∥∥∥∥ε2∂sφ−1

ε2d∗Aα + ∂tψ

∥∥∥∥pL2(Σ)

+ δ‖ξ‖pL2(Σ)

)dt ds

and by the Holder’s inequality with c2 =(∫

ΣdvolΣ

) p−2p

≤δ(‖d∗Aα‖

pLp + ε2p‖dAφ‖pLp + c2‖ξ‖pLp

)+ c0c1c2ε

2p‖∂sα− dAφ‖pL2

+ c0c1c2ε3p‖∂sψ − ∂tφ‖pLp + c0c1c2ε

4p

∥∥∥∥∂sφ− 1

ε4d∗Aα +

1

ε2∂tψ

∥∥∥∥pLp

since ‖Ψ‖L∞ + ‖Φ‖L∞ ≤ c3, for c0c1c3εp ≤ δ

≤δ(‖d∗Aα‖

pLp + ε2p‖dAφ‖pLp + 2c2‖ξ‖pLp

)+ c0c1c2ε

2p‖∇sα− dAφ‖pL2

+ c0c1c2ε3p‖∇sψ −∇tφ‖pLp + c0c1c2ε

4p

∥∥∥∥∇sφ−1

ε4d∗Aα +

1

ε2∇tψ

∥∥∥∥pLp.

Therefore, the theorem follows from the lemma 69 and the last estimate choosing δsmall enough.

Quadratic estimates II 10In this chapter we prove the following quadratic estimates.

Lemma 71. For any c0 > 0 there are two positive constants c and ε0 such that, forany 0 < ε < ε0, the following holds. If two connections Ξ = A + Ψdt + Φds,Ξ = A+ Ψdt+ Φds, with α+ ψdt+ φds := Ξ− Ξ, satisfies ‖α+ ψdt‖∞,ε ≤ c0, then

ε2∣∣∣∣(Dε(Ξ)−Dε(Ξ)

)(α + ψdt+ φds)

∣∣∣∣0,p,ε

≤c‖α + ψdt+ φds‖∞,ε‖α + ψdt+ φds‖1,p,ε

+ c‖α + ψdt+ φds‖∞,ε‖α + ψdt‖1,p,ε,

ε2∣∣∣∣(Dε(Ξ)−Dε(Ξ)

)(α + ψdt+ φds)

∣∣∣∣0,p,ε

≤c‖(1− πA)α + ψdt+ φds‖∞,ε‖α + ψdt+ φds‖1,p,ε

+ c‖α + ψdt+ φds‖∞,ε‖(1− πA)α + ψdt+ φds‖1,p,ε

+ c‖α + ψdt+ φds‖0,p,ε (‖dAα‖L∞ + ‖d∗Aα‖L∞ + ε‖∇tα‖L∞)

+ c‖α + ψdt+ φds‖0,p,ε

(ε‖dAψ‖L∞ + ε2‖∇tψ‖L∞

),

ε2∣∣∣∣πA(Dε(Ξ)− λ ∗ [α ∧ ω(A)]−Dε(Ξ)

)(α + ψdt+ φds)

∣∣∣∣Lp

≤c‖α + ψdt+ φds‖∞,ε‖(1− πA)α + ψdt+ φds‖1,p,ε + cε2‖α‖L∞‖α‖Lp+ cε2‖ψdt‖Lp‖∇tα‖L∞ + cε2‖ψ‖L∞‖∇tα‖Lp + cε2‖∇tψ‖L∞‖α‖Lp+ cε2

(‖φ‖L∞ + ‖α‖2

L∞

)‖πA(α)‖Lp + c‖α‖L∞‖dAα− λε2ω(A)‖Lp ,

for any α + ψdt+ φds ∈ W 1,2;p and where λ ∈ 0, 1.

Proof. The lemma can be proved directly estimating term by term the following iden-tities.

(Dε1(Ξ)−Dε1(Ξ)

)(α + ψdt+ φds) = [φ, α]− [α, φ]− 1

ε2[α, ∗dAα + [α ∧ α]]

+1

ε2∗[α, ∗

(dAα +

1

2[α ∧ α]

)]+

1

ε2d∗A[α ∧ α]−

[ψ, (∇tα + [ψ, α])

]+ [α, (∇tψ + [ψ, ψ])] + dA[ψ, ψ]− 2[ψ, (∇tα− dAψ − [α, ψ])]

−∇t[ψ, α]− (d ∗Xt(A)− d ∗Xt(A))α,

115

116 10. Quadratic estimates II

πA(Dε1(Ξ)− 1

ε2∗ [α, ∗ω(A)]−Dε1(Ξ)

)(α + ψdt+ φds) = πA

([φ, α]− [α, φ]

− 1

ε2[α, ∗dAα + [α ∧ α]]−

[ψ, (∇tα + [ψ, α])

]+

1

ε2∗[α, ∗

(dAα− λε2ω(A) +

1

2[α ∧ α]

)]+ [α, (∇tψ + [ψ, ψ])]

− 2[ψ, (∇tα− dAψ − [α, ψ])]−∇t[ψ, α]− (d ∗Xt(A)− d ∗Xt(A))α),(

Dε2(Ξ)−Dε2(Ξ))(α + ψdt+ φds) = [φ, ψ]− [ψ, φ]− 1

ε2∗ ∇t[α ∧ ∗α]

− 2

ε2∗ [α, ∗(∇tα− dAψ − [α, ψ])] +

1

ε2[ψ, (d∗Aα− ∗[α ∧ ∗α])]

− ∗ 1

ε2[α ∧ ∗(dAψ + [α, ψ])] +

1

ε2d∗A[α, ψ],(

Dε3(Ξ)−Dε3(Ξ))(α + ψdt+ φds) = [φ, φ] + ∗ 1

ε4[α ∧ ∗α] +

1

ε2[ψ, ψ].

We choose a connection Ξ = A+Ψdt+Φds and a 1-form ξ = α+ψdt+φds ∈ W 1,2;p,then F ε(Ξ + ξ) = F ε(Ξ) +Dε(Ξ)(ξ) + Cε(Ξ)(ξ) and we denote by Cε1(Ξ), Cε2(Ξ) andCε3(Ξ) the three components of Cε(Ξ) = Cε1(Ξ) + Cε2(Ξ) dt+ Cε3(Ξ) ds; in this case wehave the following estimates.

Lemma 72. For any c0 > 0 there are constants c > 0 and ε0 > 0 such that, for any0 < ε < ε0,

ε2∣∣∣∣Cε(Ξ)(ξ)

∣∣∣∣0,p,ε≤ c‖ξ‖∞,ε‖ξ‖1,p,ε (10.1)

ε2∣∣∣∣πACε1(Ξ)(ξ)

∣∣∣∣0,p,ε≤c‖(1− πA)ξ‖∞,ε (‖(1− πA)α + ψdt‖1,p,ε + ε‖∇tα‖Lp)

+ c‖πAα‖L∞‖(1− πA)α + ψdt‖1,p,ε

+ c‖πAα‖L∞(‖πAα‖L∞ + ε2)‖πAα‖Lp(10.2)

for any ξ := α + ψdt+ φds ∈ W 1,2;p and where we assume that ‖α + ψdt‖∞,ε ≤ c0.

Proof. Also this lemma can be showed estimating term by term the identities:

Cε1(Ξ)(ξ) = [φ, α]− [α, φ] +1

ε2d∗A[α ∧ α]

− 1

ε2∗ [α ∧ ∗(dAα + [α ∧ α])] +

1

ε2∗[α ∧ ∗

(dAα +

1

2[α ∧ α]

)]− [ψ, (∇tα + [ψ, α])]− 2[ψ, (∇tα− dAψ − [α, ψ])]

−∇t[ψ, α] + [α,∇tψ]− (∗Xt(A+ α)− ∗Xt(A)− d ∗Xt(A)α),

πACε1(Ξ)(ξ) = πA

([φ, α]− [α, φ]− [ψ, (∇tα + [ψ, α])]

− 1

ε2∗ [α ∧ ∗(dAα + [α ∧ α])] +

1

ε2∗[α ∧ ∗

(dAα +

1

2[α ∧ α]

)]− 2[ψ, (∇tα− dAψ − [α, ψ])]−∇t[ψ, α] + [α,∇tψ]

− (∗Xt(A+ α)− ∗Xt(A)− d ∗Xt(A)α)),

(10.3)

117

Cε2(Ξ)(ξ) = [φ, ψ]− [ψ, φ] +2

ε2∗ [α ∧ ∗(∇tα− dAψ − [α, ψ])]

− 1

ε2[ψ, d∗Aα]− 1

ε2∗ [α ∧ ∗(dAψ + [α, ψ])] +

1

ε2d∗A[α, ψ],

(10.4)

Cε3(Ξ)(α + ψdt+ φds) = 0.

The map Kε2: A firstApproximation 11

In the next chapter, we will construct, using a Newton’s iteration, a perturbed Yang-Mills flow Ξε ∈ Mε

(T ε,b(Ξ−), T ε,b(Ξ+)

)for any perturbed geodesic flow Ξ0 ∈

M0 (Ξ−,Ξ+) and every pair of connections Ξ± ∈ CritbEHwith index difference 1,where b is a regular value of EH . For this purpose, we need to define a connectionKε2(Ξ0) which is an approximate solution of the perturbed Yang-Mills flow equation.Kε2(Ξ0) is constructed in two steps: First, we add to Ξ0 a 1-form αε0(s)+ψε0(s)dtwhichsatisfies the limit conditions lims→±∞ Ξ0 + αε0 + ψε0dt = T ε,b(Ξ±) and then we add asecond 1-form in order to have an approximated solution of the Yang-Mills flow equa-tions.

First, we recall that a connection Ξ0 := A0 + Ψ0dt + Φ0ds descends to a flow linebetween two perturbed geodesics Ξ± = A± + Ψ±dt when it satisfies the equations(8.2) and (8.1), i.e.

∂sA0 − dA0Φ0 − πA0

(∇t(∂tA

0 − dA0Ψ0) + ∗Xs(A0))

= 0,

d∗A0(∂tA0 − dA0Ψ0) = d∗A0(∂sA

0 − dA0Φ0) = 0.

Next, we choose a smooth positive function θ such that θ(s) = 0 for s ≤ 1, θ(s) = 1when s ≥ 2, 0 ≤ θ ≤ 1 and 0 ≤ ∂sθ ≤ c0 with c0 > 0. Thus, we define αε0 + ψε0dt as

αε0(s) + ψε0(s)dt :=θ(−s)g(s)−1(T ε,b(A− + Ψ−dt)− (A− + Ψ−dt))g(s)

+ θ(s)g(s)−1(T ε,b(A+ + Ψ+dt)− (A+ + Ψ+dt))g(s),(11.1)

where g ∈ G0(P × S1 × R) satisfies

g−1∂sg = Φ0, lims→−∞

g = 1; (11.2)

we introduce g in order to make the definition of Ξ0 + αε0 + ψε0dt gauge-invariant.

Lemma 73. We choose two constants b > 0, p > 2. There are positive constantsε0, c such that the following holds. For every ε ∈ (0, ε0), every pair Ξ± := A± +Ψ±dt ∈ CritbEH that are perturbed closed geodesics of index difference one, thereexists a unique equivariant map

Kε2 :M0(Ξ−,Ξ+)→ W 1,2;p(T ε,b(Ξ−), T ε,b(Ξ+)

)such that for any Ξ0 := A0 + Ψ0dt + Φ0ds ∈ M0(Ξ−,Ξ+), with αε0 + ψε0dt and gdefined as in (11.1) and in (11.2),

Kε2(Ξ0)− (Ξ0 + αε0 + ψε0dt) ∈ im d∗A0 (11.3)

119

120 11. The map Kε2: A first Approximation

and

1

ε2d∗A0dA0

(Kε2(Ξ0)− (Ξ0 + αε0 + ψε0dt)

)=d∗A0 (dA0d∗A0)−1 dA0

(∇t(∂tA

0 − dA0Ψ0) + ∗Xt(A0))

− θ(−s)d∗A0dA0g−1(dA−d

∗A−

)−1(∇Ψ−t

(∂tA− − dA−Ψ−

)+ ∗Xt(A−)

)g

− θ(s)d∗A0dA0g−1(dA+d

∗A+

)−1(∇Ψ+

t

(∂tA+ − dA+Ψ+

)+ ∗Xt(A+)

)g

− θ(−s)d∗A0dA0πg−1(A−)g(αε0)− θ(s)d∗A0dA0πg−1(A+)g(α

ε0)

(11.4)

In addition, it satisfies

‖Kε2(Ξ0)− (Ξ0 + αε0 + ψε0dt)‖1,2;p,1 ≤ cε2, (11.5)

‖F ε1(Kε2(Ξ0))‖Lp ≤ cε2, ‖F ε2(Kε2(Ξ0))‖Lp ≤ c. (11.6)

Ξ−Ξ+

Ξε−

Ξε+

Ξ0

αε− + ψε−dtαε+ + ψε+dt

Ξ0 + αε0 + φε0dt

Kε2(Ξ0)

Figure 11.1: Situation of the lemma 73.

The critical points of our two functionals EH und YMε,H do not coincide and there-fore the difference Ξ − Ξ0 between a geodesic flow Ξ0 ∈ M0 (Ξ−,Ξ+) and any con-nection Ξ which converges to T ε,b(Ξ±) as s goes to ±∞ can not be estimates usingthe norm defined in the section 8.4. We need therefore another reference connectionin order to compute the norms and for this purpose we use Kε2(Ξ0).

Proof of lemma 73. Kε2(Ξ0) is uniquely defined by (11.3) and (11.4) because FA0 = 0and

d∗A0dA0 : im d∗A0Ω2(Σ, gP )→ im d∗A0Ω2(Σ, gP )

is bijective. Furthermore, the lemma 6, the commutation formulas (1.20), (1.21) andthe estimates of the geodesic flow (8.7)-(8.10) yield to (11.5). Therefore we need onlyto prove (11.6).

We define Ξε1 := Aε1 + Ψε

1dt+ Φε1ds := Ξ0 + αε0 + ψε0dt and we consider

A±(s) + Ψ±(s)dt =g(s)∗(A± + Ψ±dt),

α(s) + ψ(s)dt =θ(−s)((A0(s) + Ψ0(s)dt)− (A−(s) + Ψ−(s)dt))

+ θ(s)((A0(s) + Ψ0(s)dt)− (A+(s) + Ψ+(s)dt))

(11.7)

121

and

αε1 :=θ(−s)(dA−d

∗A−

)−1(∇Ψ−t

(∂tA− − dA−Ψ−

)+ ∗Xt(A−)

)θ(s)

(dA+d

∗A+

)−1(∇Ψ+

t

(∂tA+ − dA+Ψ+

)+ ∗Xt(A+)

).

(11.8)

Furthermore, we consider Aε2 + Ψε2dt+ Φε

2ds := Ξε1 +αε1 := Kε2(Ξ0). If we look at the

expansion

F ε1(Kε2(Ξ0)

)=∂sA

ε2 − dAε2Φε

2 +1

ε2d∗Aε2FA

ε2−∇Ψε2

t (∂tAε2 − dAε2Ψε

2)− ∗Xt(Aε2)

=∂sA0 − dA0Φ0 − πA0

(∇t(∂tA

0 − dA0Ψ0) + ∗Xs(A0))

+(∇s −∇2

t

)αε1 − ∗Xt(A

2) + ∗Xt(A0 + αε0)

− 1

ε2∗[αε1 ∧ ∗

(dA0αε1 +

1

2[αε1 ∧ αε1]

)]+

1

2ε2d∗A0 [αε1 ∧ αε1]

+1

ε2d∗A0dA0αε1 − (1− πA0)

(∇t(∂tA

0 − dA0Ψ0) + ∗Xt(A0))

+ F ε1 + F ε

2 + F ε3

(11.9)

we remark that the second line vanishes because Ξ0 is a geodesic flow, the third andthe fourth can be estimates by cε2 because the ‖ · ‖1,2;p,1-norm of αε1 can be estimatedby the same factor by (11.5). The fifth line of (11.9) can be written by the definitionsas

− 1

ε2d∗A0dA0αε1 − θ(−s)dA0dA0πA−(αε0)− θ(s)dA0dA0πA+(αε0).

Therefore we have that∥∥F ε1 (Kε2(Ξ0))∥∥

Lp≤cε2 +

∥∥∥∥ 1

ε2d∗A0dA0(αε1 + πA−(αε0) + πA+(αε0))− F ε

1

∥∥∥∥Lp

+ ‖F ε2 ‖Lp + ‖F ε

3 ‖Lp .

For this purpose, we need to investigate F ε1 , F ε

2 and F ε3 . In order to simplify the

exposition we evaluate F εi (s) for s ≤ 0; for s > 0 the computation is the same, we

only need to substitute A−+Ψ−dt with A+ +Ψ+dt and θ(−s) with θ(s). If we denote∂tA− − dA−Ψ− by B−t , we have

F ε1 :=

1

ε2d∗A−

(dA−α

ε0 +

1

2[αε0 ∧ αε0]

)− [ψε0, B

−t ]

− 1

ε2∗[αε0 ∧

(dA−α

ε0 +

1

2[αε0 ∧ αε0]

)]− [ψε0, (∇

Ψ−t αε0 − dA−ψε0 − [αε0, ψ

ε0])]

−∇Ψ−t

(∇Ψ−t αε0 − dA−ψε0 − [αε0, ψ

ε0])

− ∗Xt(A− + αε0) + ∗Xt(A−)

− 1

ε2∗[α ∧ ∗dA−αε0

]+

1

ε2d∗A0 [α ∧ αε0]


and since T ε,b(Ξ−) is a perturbed Yang-Mills connection and by the definition of αε0 +ψε0dt we get that

0 =θ(−s)(−∇Ψ−

t B−t − ∗Xt(A−))

+1

ε2d∗A−

(dA−α

ε0 +

1

2[αε0 ∧ αε0]

)− 1

ε2∗[αε0 ∧

(dA−α

ε0 +

1

2[αε0 ∧ αε0]

)]− [ψε0, B

−t ]

− [ψε0, (∇Ψ−t αε0 − dA−ψε0 − [αε0, ψ

ε0])]−∇Ψ−

t

(∇Ψ−t αε0 − dA−ψε0 − [αε0, ψ

ε0])

− ∗Xt(A− + αε0) + ∗Xt(A−) + F ε[−2,0]

where F ε[−2,0] contain only quadratic terms with support in [−2, 0]. Therefore we can

write F ε1 in the following way by the definition of αε1

F ε1 =θ(−s)

(∇Ψ−t B−t + ∗Xt(A

−))

+ F ε[−2,0]

− 1

ε2∗[α ∧ ∗dA−αε0

]+

1

ε2d∗A0 [α ∧ αε0]

=1

ε2d∗A0dA0(αε1 + πA−(αε0)) + F ε

[−2,0]

− 1

ε2∗[α ∧ ∗dA−(αε0 − αε1)

]+

1

ε2d∗A0 [α ∧ ((1− πA−)αε0 − αε1)]

The two terms are

F ε2 :=− 1

ε2∗[α ∧ ∗1

2[αε0 ∧ αε0]

]−[ψε0,(∇Ψ−t α− dA−ψ − [α, ψ]

)]− [ψε0, [ψ, α

ε0]− [α, ψε0]]

−[ψ,(∇Ψ−t αε0 − dA0ψε0 − [αε0, ψ

ε0])]−∇t ([ψ, αε0]− [α, ψε0])

− ∗Xt(A0 + αε0) + ∗Xt(A− + αε0) + ∗Xt(A

0)− ∗Xt(A−),

F ε3 :=−∇t[ψ

ε0, α

ε1]− [ψε0,∇tα

ε1]

− 1

ε2∗[αε0 ∧ ∗

(dA0αε1 +

1

2[αε1 ∧ αε1]

)]+

1

ε2d∗A0 [αε0 ∧ αε1]

− 1

ε2∗ [αε1 ∧ ∗ (dA0αε0 + [αε0 ∧ αε1])] .

This allows us to conclude that, by the apriori estimates (8.7)-(8.10),

∥∥F ε1 (Kε2(Ξ0))∥∥

Lp≤cε2 +

∥∥∥∥ 1

ε2d∗A0dA0αε1 − F ε

1

∥∥∥∥Lp

+ ‖F ε2 ‖Lp + ‖F ε

3 ‖Lp ≤ cε2

in order to see this we need that

‖dA−(αε0 − αε1)‖Lp(Σ×S1) + ‖(1− πA−)αε0 − αε1‖Lp(Σ×S1) ≤ cε4

123

which holds by the theorem 24. The second Yang-Mills flow equation can be writtenas

F ε2(Kε2(Ξ0)

)=∂sΨ

2 −∇tΦ2 − 1

ε2d∗A2(∂tA

2 − dA2Ψ2)

=∂sΨ0 −∇tΦ

0 − 1

ε2d∗A0(∂tA

0 − dA0Ψ0)

+1

ε2∗[(αε0 + α1) ∧ ∗(∂tA0 − dA0Ψ0)

]+

1

ε2∗[(αε0 + α1) ∧ ∗

(∇Ψ0

t αε0 − dA0ψε0 − [αε0, ψε0])]

+1

ε2∗[(αε0 + α1) ∧ ∗

(∇Ψ0

t α1 − [α1, ψε0])]

+ d∗A0

((∂tA

2 − dA2Ψ2 − (∂tA0 − dA0Ψ0

)and therefore by the lemma 53 and the identity d∗A0(∂tA

0 − dA0Ψ0) = 0∥∥F ε2 (Kε2(Ξ0))∥∥

Lp≤∥∥2[(∂sA

0 − dA0Φ0) ∧ ∗(∂tA0 − dA0Ψ0]∥∥

Lp

+c

ε2‖αε0 + α1‖2,p,ε + ‖d∗A0dA0ψε0‖Lp ≤ c.

Theorem 74. We choose a regular value b of EH , p > 2, then there are two positiveconstants c and ε0 such that the following holds. For any Ξ0 ∈ M0(Ξ−,Ξ+) withΞ± ∈ CritbEH the estimates

‖πA(ξ)‖1,2;p,1 ≤cε‖Dε(Kε2(Ξ0))ξ‖0,p,ε + c‖πADε(Kε2(Ξ0))ξ‖Lp ,‖(1− πA)ξ‖1,2;p,ε ≤cε2‖Dε(Kε2(Ξ0))ξ‖0,p,ε + cε‖πADεKε2(Ξ0))ξ‖Lp ,‖(1− πA)α‖1,2;p,ε ≤cε2‖Dε(Kε2(Ξ0))ξ‖0,p,ε + cε2‖πADεKε2(Ξ0))ξ‖Lp

(11.10)

hold for all compactly supported 1-form ξ = α + ψdt + φds ∈ W 1,2;p, η ∈ W 1,2;p,ξ ∈ im (Dε(Kε2(Ξ0)))∗ and 0 < ε ≤ ε0.

Proof. By the existence theorem 24 and by the Sobolev theorem 14 we have that

ε ‖(1− πA)αε0 + ψε0dt‖∞,ε + ‖dAαε0‖L∞ + ‖d∗Aαε0‖L∞ ≤ cε4− 1p ,

ε‖ψε0‖L∞ + ε2‖∇tψε0‖L∞ ≤ cε3− 1

p , ‖∇tαε0‖L∞ + ‖πA(αε0)‖L∞ ≤ cε2;

in addition, by the previous lemma 73 we know that∥∥Kε2(Ξ0)− (Ξ0 + αε0 + ψε0dt)∥∥

1,2;p,1≤ cε2.

Thus by the quadratic estimates of the lemma 71, we obtain

ε2∥∥(Dε (Kε2 (Ξ0

))−Dε

(Ξ0))ξ∥∥

0,p,ε

≤ε2∥∥(Dε (Kε2 (Ξ0

))−Dε

(Ξ0 + αε0 + ψε0dt

))ξ∥∥

0,p,ε

+ ε2∥∥(Dε (Ξ0 + αε0 + ψε0dt

)−Dε

(Ξ0))ξ∥∥

0,p,ε

≤cε2− 1p‖ξ‖1,2;p,ε,


∥∥πA0

(Dε(Kε2(Ξ0))ξ −Dε

(Ξ0)ξ − ∗[α, ∗ω(A0)]

)∥∥0,p,ε≤ cε1− 1

p‖ξ‖1,2;p,ε (11.11)

where we used that

ω(A0) = dA0 (d∗A0dA0)−1 (∇t

(∂tA

0 − dA0Ψ0)

+ ∗Xt(A0))

and, with the notation of the proof of theorem 73,∥∥∥∥ 1

ε2dA0

(Kε2(Ξ0)− Ξ0

)− ω(A0)

∥∥∥∥L∞

≤θ(−s)∥∥dA0

(T ε,b(Ξ−)− Ξ− − αε1

)∥∥L∞

+ cθ(−s)∥∥πA0

(T ε,b(Ξ−)− Ξ−

)∥∥L∞

+ θ(s)∥∥dA0

(T ε,b(Ξ+)− Ξ+ − αε1

)∥∥L∞

+ cθ(s)∥∥πA0

(T ε,b(Ξ+)− Ξ+

)∥∥L∞

which is smaller than cε2. Furthermore, for ξ = (Dε(Kε2))∗η, η = η1 + η2dt + η3ds,we have by the lemmas 71 (for the adjoint operators) and 57 as well as by the theorem60 that

‖πA(ξ − (D0(Ξ0))∗(πA(η)))‖Lp

≤‖πA((Dε(Ξ0))∗(η)− ∗[η1 ∧ ∗ω(A0)]− (D0(Ξ0))∗(πA(η)))‖Lp + cε1− 1p‖ξ‖0,p,ε

≤cε1− 1p‖ξ‖0,p,ε.

(11.12)

The theorem follows then from the theorem 59 and the last computations.

The map Rε,b betweenflows 12

In this chapter we will show that, for any pair Ξ0± ∈ CritEH , any geodesic flow Ξ0 ∈

M0(Ξ0−,Ξ

0+

)can be approximated by a Yang-Mills flow

Ξε ∈Mε(T ε,b

(Ξ0−), T ε,b

(Ξ0

+

))provided that ε is small enough. In addition, Ξε will turn out to be the unique Yang-Mills flow in a ball around Ξ0 of radius δε. Therefore we can define an injective mapRε,b between the flows of the two functionals provided that we choose an energy boundb for the critical connections and for ε small enough.

δε

cε2Ξ0

existenceuniqueness

Figure 12.1: Existence and uniqueness.

Theorem 75 (Existence). We assume that the energy functional EH is Morse-Smaleand we choose two constants b > 0, p > 2. There are constants ε0, c > 0 such that thefollowing holds. For every ε ∈ (0, ε0), every pair Ξ0

± := A0± + Ψ0

±dt ∈ CritbEH thatare perturbed closed geodesics of index difference one and every Ξ0 := A0 + Ψ0dt +Φ0ds ∈ M0(Ξ0

−,Ξ0+), there exists a connection Ξε ∈ Mε

(T ε,b

(Ξ0−), T ε,b

(Ξ0

+

))which satisfies

d∗εΞ0

(Ξε −Kε2

(Ξ0))

= 0, Ξε −Kε2(Ξ0)∈ im

(Dε(K2

(Ξ0)))∗

, (12.1)∥∥(1− πA0)(Ξε −Kε2

(Ξ0))∥∥

1,2;p,ε+ ε

∥∥πA0

(Ξε −Kε2

(Ξ0))∥∥

1,2;p,1≤ cε3. (12.2)

Theorem 76 (Local uniqueness). We choose p > 3. For every pair Ξ0± := A0

± +Ψ0±dt ∈ CritbEH that are perturbed closed geodesics of index difference one, every

Ξ0 := A0 + Ψ0dt + Φ0ds ∈ M0(Ξ0−,Ξ

0+

)and any c > 0 there are an ε0 > 0 and a

δ > 0 such that the following holds. If Ξε, Ξε ∈Mε(T ε,b

(Ξ0−), T ε,b

(Ξ0

+

))satisfy

d∗εΞ0

(Ξε −Kε2

(Ξ0))

= d∗εΞ0

(Ξε −Kε2

(Ξ0))

= 0,

125

126 12. The mapRε,b between flows

Ξε −Kε2(Ξ0), Ξε −Kε2

(Ξ0)∈ im

(Dε(Kε2(Ξ0)))∗

and the estimates ∥∥Ξε −Kε2(Ξ0)∥∥

1,2;p,ε≤ cε2, (12.3)∥∥Ξε −Kε2

(Ξ0)∥∥

1,2;p,ε+∥∥Ξε −Kε2

(Ξ0)∥∥

L∞≤ δε, (12.4)

then Ξε = Ξε.

Definition 77. We choose p > 3. For every regular value b > 0 of the energy EH

there are positive constants ε0, δ and c such that the assertion of the theorems 75

and 76 hold with these constants. Shrink ε0 such that cε0 + c0cε1− 3

p

0 < δ, wherec0 is the constant of the Sobolev’s theorem 51. Theorems 75 and 76 assert that, forevery pair Ξ0

± := A0± + Ψ0

±dt ∈ CritbEH that are perturbed closed geodesics of indexdifference one, every Ξ0 ∈ M0

(Ξ0−,Ξ

0+

)and every ε with 0 < ε < ε0, there is a

unique Ξε ∈Mε(T ε,b

(Ξ0−), T ε,b

(Ξ0

+

))satisfying

d∗εΞ0

(Ξε −Kε2

(Ξ0))

= 0, Ξε −Kε2(Ξ0)∈ im

(Dε(K2

(Ξ0)))∗ (12.5)

and ∥∥Ξε −Kε2(Ξ0)∥∥

1,2;p,ε≤ cε2. (12.6)

We define the map

Rε,b :M0(Ξ0−,Ξ

0+

)→Mε

(T ε,b

(Ξ0−), T ε,b

(Ξ0

+

))by Rε,b (Ξ0) := Ξε where Ξε ∈ Mε

(T ε,b

(Ξ0−), T ε,b

(Ξ0

+

))is the unique Yang-Mills

flow satisfying (12.5) and (12.5).

Proof of theorem 75. We choose Ξε2 := Kε2(Ξ0). By induction we define, for k ≥ 3,

Ξεk := Ξε

k−1 + ηεk−1, ηεk = αεk + ψεkdt+ φεkds, where ηk is defined by

Dε(Kε2(Ξ0))(ηεk−1) = −F ε(Ξεk−1), ηεk−1 ∈ im (Dε(Kε2(Ξ0)))∗. (12.7)

In addition, one can remark that

F ε(Ξεk−1) = F ε(Ξε

k−2) +Dε(Ξεk−2)(ηεk−2) + Cε(Ξε

k−2)(ηεk−2). (12.8)

By theorem 74 we have the estimate

‖(1− πA0)ηεk−1‖1,2;p,ε + ε‖πA(αεk−1)‖1,2;p,1

≤cε2‖Dε(Kε2(Ξ0))(ηεk−1)‖0,p,ε + cε∥∥πA0(Dε(Kε2(Ξ0))(ηεk−1))

∥∥Lp

=cε2‖F ε(Ξεk−1)‖0,p,ε + cε

∥∥πA0(F ε(Ξεk−1))

∥∥Lp

where the last step follows from (12.7) and by (12.8) we obtain

ε2‖F ε(Ξεk−1)‖0,p,ε + cε

∥∥πA0(F ε(Ξεk−1))

∥∥Lp

≤cε2‖Cε(Ξεk−2)(ηεk−2)‖0,p,ε + cε

∥∥πA0(Cε(Ξεk−2)(ηεk−2))

∥∥Lp

+ cε2∥∥(Dε(Ξε

k−1)−Dε(Kε2(Ξ0)))ηεk−1

∥∥0,p,ε

+ cε∥∥πA0

(Dε(Ξε

k−1)−Dε(Kε2(Ξ0)))ηεk−1

∥∥Lp

127

and finally using the lemmas 71 and 72, we can conclude

≤c‖ηεk−2‖∞,ε‖αεk−2 + ψεk−2‖1,1;p,ε

+c

ε||(1− πA)αεk−2 + ψεk−2dt+ φεk−2ds‖∞,ε‖αεk−2 + ψεk−2dt‖1,p,ε

+c

ε‖πA(αεk−2)‖L∞‖(1− πA)αεk−2 + ψεk−2dt+ φεk−2ds‖1,p,ε

+c

ε‖πA(αεk−2)‖L∞(‖πA(αεk−2)‖L∞ + ε2)‖πA(αεk−2)‖Lp .

Next, by the estimates of lemma 73:

‖F ε1(Kε2(Ξ0))‖Lp ≤ cε2, ‖F ε2(Kε2(Ξ0))‖Lp ≤ c,

there is a positive constant c0 such that

‖(1− πA0)ηε2‖1,2;p,ε + ε‖πA(αε2)‖1,2;p,1 ≤ c0ε3.

Using the Sobolev’s theorem 51, one can easily see that, for ε small enough and k > 3,by induction there are two positive constants c1, c such that

‖(1− πA0)ηε3‖1,2;p,ε + ε‖πA(αε3)‖1,2;p,1 ≤cε4− 3p ,

‖(1− πA0)ηεk‖1,2;p,ε + ε‖πA(αεk)‖1,2;p,1 ≤2−(k−2)c0ε3,

ε2‖F ε(Ξεk)‖0,p,ε + ε ‖πA0(F ε(Ξε

k))‖Lp ≤2−(k−2)c1ε3.

Therefore F ε(Ξk) converges to 0 and we can choose Ξε := K2(Ξ0) +∑∞

k=2 ηεk. Then

Ξε −Kε2(Ξ0) ∈ im (Dε(Kε2(Ξ0)))∗, F ε(Ξε) = 0

and ∥∥(1− πA0)(Ξε −Kε2(Ξ0)

)∥∥1,2;p,ε

+ ε∥∥πA0

(Ξε −Kε2(Ξ0)

)∥∥1,2;p,1

≤ cε3.

Since F ε(Ξε) = 0, by the definition of F ε3 , d∗εΞ0 (Ξε −Kε2(Ξ0)) = 0 holds and thus weconcluded the proof of the theorem 75.

Proof of theorem 76. First, we improve the estimate (12.4) and we show that

‖Ξε −Kε2(Ξ0)‖1,2;p,ε + ε‖πA0(Ξε −Kε2(Ξ0))‖1,2;p,1 ≤ cε3. (12.9)

In order to fulfill this task, we consider the identity

Dεi (Kε2(Ξ0))(Ξε −Kε2(Ξ0)) = −Cεi (Kε2(Ξ0))(Ξε −Kε2(Ξ0))−F εi (Kε2(Ξ0)) (12.10)

then, by theorem 74

‖Ξε −Kε2(Ξ0)‖1,2;p,ε + ε‖πA0(Ξε −Kε2(Ξ0))‖1,2;p,1

≤cε2‖Dε(Kε2(Ξ0))(Ξε −Kε2(Ξ0))‖0,p,ε

+ cε‖πA0Dε(Kε2(Ξ0))(Ξε −Kε2(Ξ0))‖Lp

128 12. The mapRε,b between flows

next, we can apply (12.10), i.e.

≤cε2‖Cε(Kε2(Ξ0))(Ξε −Kε2(Ξ0))‖0,p,ε

+ cε‖πA0Cε(Kε2(Ξ0))(Ξε −Kε2(Ξ0))‖Lp+ cε2‖F ε(Kε2(Ξ0))‖0,p,ε + cε‖πA0F ε(Kε2(Ξ0))‖Lp

and the quadratic estimates (71) and (72):

≤cε3 + c‖Ξε −Kε2(Ξ0)‖∞,ε‖Ξε −Kε2(Ξ0)‖1,1;p,ε

+c

ε‖(1− πA0)(Ξε −Kε2(Ξ0))‖∞,ε‖(1− πA0)(Ξε −Kε2(Ξ0))‖1,1;p,ε

+ c‖(1− πA0)(Ξε −Kε2(Ξ0))‖∞,ε‖∇tπA0(Ξε −Kε2(Ξ0))‖Lp

+c

ε‖πA0(Ξε −Kε2(Ξ0))‖L∞‖(1− πA0)(Ξε −Kε2(Ξ0))‖1,1;p,ε

+c

ε‖πA0(Ξε −Kε2(Ξ0))‖2

L∞‖πA0(Ξε −Kε2(Ξ0))‖Lp

+ cε‖πA0(Ξε −Kε2(Ξ0))‖L∞‖πA0(Ξε −Kε2(Ξ0))‖Lp≤cε3 + cδ‖(1− πA0)(Ξε −Kε2(Ξ0))‖1,1;p,ε

+ cδε‖πA0(Ξε −Kε2(Ξ0))‖1,1;p,1

where the last inequality follows from the assumptions. Therefore for δ small enough(12.9) holds. Furthermore, by (12.3) and (12.9), ‖Ξε − Ξε‖1,2;p,ε ≤ cε2. Thus, alwaysby theorem 74,

‖Ξε − Ξε‖1,2;p,ε + ε‖πA0(Ξε − Ξε)‖1,2;p,1

≤cε2‖Dε(Kε2(Ξ0))(Ξε − Ξε)‖0,p,ε + cε‖πA0Dε(Kε2(Ξ0))(Ξε − Ξε)‖Lp

and since F ε(Ξε)F ε(Ξε) = 0, Dεi (Ξε)(Ξε − Ξε) = −Cεi (Ξ

ε)(Ξε − Ξε) and thus weobtain

≤c(ε2‖Cε

i (Ξε)(Ξε − Ξε)‖0,p,ε + ε‖πA0C

εi (Ξ

ε)(Ξε − Ξε)‖Lp)

+ cε2‖(Dε(Ξε)−Dε(Kε2(Ξ0)))(Ξε − Ξε)‖0,p,ε

+ cε‖πA0(Dε(Ξε)−Dε(Kε2(Ξ0)))(Ξε − Ξε)‖Lp

≤cε1− 3p(‖(1− πA0)(Ξε − Ξε)‖1,1;p,ε + ε‖πA0(Ξε − Ξε)‖1,1;p,1

)where the last inequality follows from the quadratic estimates of the lemmas 71 and72. Hence for p > 3 and ε small enough Ξε = Ξε.

A priori estimates forthe Yang-Mills flow 13

In this chapter we will prove some a priori estimates, that will be stated in theorem 78,on the curvature for a perturbed Yang-Mills flow. These will then be used to prove thesurjectivity of the mapRε,b in the chapter 17.

In order to simplify the exposition we denote by ‖ · ‖ the L2-norm over Σ and weintroduce the following notation. We choose two perturbed Yang-Mills connectionsΞε± ∈ CritbYMε,H where b > 0. For any Yang-Mills flow Ξε := A + Ψdt + Φds ∈Mε(Ξε

−,Ξε+) we define

Bt := ∂tA− dAΨ, Bs := ∂sA− dAΦ, C := ∂tΨ− ∂sΦ− [Ψ,Φ]; (13.1)

thus, the Yang-Mills flow equations (8.12) can be written as

Bs +1

ε2d∗AFA −∇tBt − ∗Xt(A) = 0, C − 1

ε2d∗ABt = 0; (13.2)

with this notation, the Bianchi identities are

∇tFA = dABt, ∇sFA = dABs, ∇tBs −∇sBt = dAC (13.3)

and the commutation formulas

[∇t, dA] = Bt, [∇s, dA] = Bs, [∇s,∇t] = C. (13.4)

Furthermore, we have the identity

‖Bs + C dt‖20,2,ε = YMε,H(Ξε

−)− YMε,H(Ξε+) (13.5)

which can be showed by the direct computation:

‖Bs + C dt‖20,2,ε =

∫R‖Bs + C dt‖2

0,2,ε ds

and by the Yang-Mills flow equations (8.12)

=

∫R

∫Σ×S1

〈Bs,−1

ε2d∗AFA +∇tBt + ∗Xt(A)〉 dvolΣ dt ds

+

∫R

∫Σ×S1

〈C, d∗ABt〉 dvolΣ dt ds

=

∫R

∫Σ×S1

(− 1

ε2〈dABs, FA〉 − 〈∇tBs, Bt〉

)dvolΣ dt ds

−∫R∂sH(A) ds+

∫R

∫Σ×S1

〈dAC,Bt〉 dvolΣ dt ds

129

130 13. A priori estimates for the Yang-Mills flow

and by the Bianchi identity (13.3)

=−∫R

∫Σ×S1

1

ε2〈∇sFA, FA〉dvolΣ dt ds−

∫R∂sH(A) ds

−∫R

∫Σ×S1

〈∇sBt, Bt〉dvolΣ dt ds

=YMε,H(A− + Ψ−dt)− YMε,H(A+ + Ψ+dt).

Furthermore, we denote by a1i , i = 1, 2, 3, the three operators dA, d∗A and ε∇t, by a2

i ,i = 1, . . . , 9, the nine operators defined combining two operators between dA, d∗A andε∇t, by a3

i , i = 1, . . . , 27, the 27 operators defined combining three operators betweendA, d∗A, ε∇t and finally we denote by a4

i , i = 1, . . . , 81, the 81 operators definedcombining four operators between dA, d∗A, ε∇t. In addition we denote by Nj(t, s) thenorms

Nj(t, s) :=∑

i=1,...,3j

(‖ajiBs‖2 + ε4‖ajiC‖2

).

Theorem 78. We choose an open interval Ω ⊂ R, a compact set Q ⊂ Ω, p > 4 andtwo constants b, c0 > 0. There are two positive constants ε0, c such that the followingholds. If a perturbed Yang-Mills flow Ξ = A + Ψdt + Φds ∈ Mε(Ξ−,Ξ+), withΞ−,Ξ+ ∈ CritbYMε,H and 0 < ε < ε0, satisfies

sup(t,s)∈S1×R

(‖∂tA− dAΨ‖L4(Σ) + ‖∂sA− dAΦ‖L∞(Σ)

)≤ c0, (13.6)

then ∫S1×Q

(‖FA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)dt ds ≤ cε4, (13.7)

supS1×Q

(ε2‖Bt‖2 + ‖d∗AdABt‖2 + ‖d∗AdAd∗ABt‖2

)≤c∫S1×Ω

(ε2‖Bs‖2 + ε2‖Bt‖2 + ‖FA‖2 + ε2cXt(A) + ε4‖C‖2

)dt ds,

(13.8)

supS1×Q

(ε2‖Bs‖2 + ‖d∗AdABs‖2 + ‖dAd∗ABs‖2

)≤c∫S1×Ω

(ε2‖Bs‖2 + ε2‖Bt‖2 + ‖FA‖2 + ε2cXt(A) + ε4‖C‖2

)dt ds,

(13.9)

supS1×Q

(‖FA‖+ ε‖∇tFA‖+ ε2‖∇t∇tFA‖

)≤ cε2, (13.10)

sup(t,s)∈S1×R

‖FA‖L∞(Σ) ≤ cε2 (13.11)

where cXt(A) is a constant which estimates the norm ‖Xt(A)‖2L∞(Σ). The constant c

depends on Ω and on Q, but only on their length and on the distance between theirboundaries. Furthermore,

sup(t,s)∈S1×Q

(ε2‖Bs‖2 + ε4‖C‖2 +N1 +N2 +N3 +N4

)≤ε2c

∫S1×Ω

(‖Bs‖2 + ε2‖C‖2

)dt ds.

(13.12)

131

Remark. In the theorem we assume that the L4(Σ)-norm of the curvature term ∂tA−dAΨ and the L∞(Σ)-norm of ∂sA − dAΦ are uniformly bounded; this condition is,for a Yang-Mills flow, always satisfied if we choose ε small enough as we will see inchapter 15.

Before starting to prove the last theorem we consider the following three lemmas. Thefirst one show a regularity result for the curvature terms Bs, C. The last two wereproved by Salamon and Weber (cf. [17], lemmas B.1. and B.4). For an interval Q ⊂ Rand a 0- or 1-form α we define the norm ‖ · ‖1,2,2;2,ε,Q by

‖α‖21,2,2;2,ε,Q :=

∫S1×Q

(‖α‖2 + ε4‖∇sα‖2 + ‖dAα‖2 + ‖d∗Aα‖2

)dt ds

+

∫S1×Q

(ε2‖∇tα‖2 + ‖d∗AdAα‖2 + ‖dAd∗Aα‖2 + ε4‖∇t∇tα‖2

)dt ds.

Lemma 79. We choose a positive constant b, then there are two positive constants ε0, csuch that the following holds. For any perturbed Yang-Mills flow Ξ = A+Ψdt+Φds ∈Mε(Ξ−,Ξ+), with Ξ−,Ξ+ ∈ CritbYMε,H and 0 < ε < ε0, satisfies

‖Bs‖21,2,2;2,ε,R + ε2‖C‖2

1,2,2;2,ε,R ≤ c.

Proof. By the Yang-Mills flow equation (13.2), the Bianchi identity (13.3) and thecommutation formaula (13.2) we have

0 =∇sBs +1

ε2∇sd

∗AFA −∇s∇tBt −∇s ∗Xt(A)

=∇sBs +1

ε2d∗AdABs −∇t∇tBs −∇tdAC

− d ∗Xt(A)Bs −1

ε2∗ [Bs, ∗FA]− [C,Bt]

=∇sBs +1

ε2d∗AdABs −∇t∇tBs − dAd∗ABs

− d ∗Xt(A)Bs −1

ε2∗ [Bs, ∗FA]− 2[C,Bt],

0 =∇sC −1

ε2∇sd

∗ABt

=∇sC +1

ε2d∗AdAC +

1

ε2d∗A∇tBs +

1

ε2∗ [Bs, ∗Bt]

=∇sC +1

ε2d∗AdAC +∇t∇tC +

2

ε2∗ [Bs, ∗Bt].

Furthermore, choosing s0 ∈ R and a smooth cut-off function with support in [s0 −1, s0 + 2] and with value 1 on [s0, s0 + 1], one can prove that, for Ω1(s0) := Σ× S1 ×


[s0 − 1, s0 + 2],

‖Bs‖21,2,2;2,ε,[s0,s0+1] + ε2‖C‖2

1,2,2;2,ε,[s0,s0+1]

≤ε4

∥∥∥∥∇sBs +1

ε2d∗AdABs −∇t∇tBs +

1

ε2dAd

∗ABs

∥∥∥∥2

L2(Ω1(s0))

+ ε6

∥∥∥∥∇sC +1

ε2d∗AdAC +∇t∇tC

∥∥∥∥2

L2(Ω1(s0))

+ ‖Bs‖2L2(Ω1(s0)) + ε2‖C‖2

L2(Ω1(s0))

=∥∥−ε2d ∗Xt(A)Bs − ∗[Bs, ∗FA]− 2ε2[C,Bt]

∥∥2

L2(Ω1(s0))

+ ε2 ‖∗[Bs, ∗Bt]‖2L2(Ω1(s0)) + ‖Bs‖2

L2(Ω1(s0)) + ε2‖C‖2L2(Ω1(s0))

≤‖Bs‖2L2(Ω1(s0)) + ε2‖C‖2

L2(Ω1(s0))

+ cε‖Bs‖21,2,2;2,ε,[s0−1,s0+2] + ε3‖C‖2

1,2,2;2,ε,[s0−1,s0+2]

(13.13)

In order to prove the last estimate we use the energy bound∫Σ×S1

(1

ε2‖FA‖2 + ‖Bt‖2

)dt ≤ b

combined with the Sobolev inequality for 1-forms on Σ × S1: For example for theterm ‖∗[Bs, ∗Bt]‖2

L2(Ω1(s0)) we proceed in the following way.

ε2 ‖∗[Bs, ∗Bt]‖2L2(Ω1) = ε2

∫[s0−1,s0+2]

‖Bt‖2L2(Σ×S1) ‖Bs‖2

L∞(Σ×S1) ds

≤cε2

∫[s0−1,s0+2]

‖Bs‖2L∞(Σ×S1) ds

≤cε2

∫[s0−1,s0+2]

1

ε

(‖Bs‖2

L2(Σ×S1) + ‖d∗AdABs‖2L2(Σ×S1)

)ds

+ cε2

∫[s0−1,s0+2]

1

ε

(‖dAd∗ABs‖2

L2(Σ×S1) + ε4 ‖∇t∇tBs‖2L2(Σ×S1)

)ds

≤cε‖Bs‖1,2,2;2,ε,[s0−1,s0+2].

Finally since∞∑i=0

ε|i|2

(‖Bs‖2

1,2,2;2,ε,[s0+i,s0+i+1] + ε2‖C‖21,2,2;2,ε,[s0+i,s0+i+1]

)≤∞∑i=0

ε|i|2

(‖Bs‖2

L2(Σ×S1×[s0+i−1,s0+i+2]) + ε2‖C‖2L2(Σ×S1×[s0+i−1,s0+i+2])

)+ cε

∞∑i=0

ε|i|2

(‖Bs‖2

1,2,2;2,ε,[s0+i−1,s0+i+2] + ε2‖C‖21,2,2;2,ε,[s0+i−1,s0+i+2]

)≤2‖Bs‖2

L2(Ω1(s0)) + 2ε2‖C‖2L2(Ω1(s0))

+ cε12

∞∑i=0

ε|i|2

(‖Bs‖2

1,2,2;2,ε,[s0+i,s0+i+1] + ε2‖C‖21,2,2;2,ε,[s0+i,s0+i+1]

),

133

we can conclude that

‖Bs‖21,2,2;2,ε,R + ε2‖C‖2

1,2,2;2,ε,R

≤∑s0∈Z

∞∑i=0

ε|i|2

(‖Bs‖2

1,2,2;2,ε,[s0+i,s0+i+1] + ε2‖C‖21,2,2;2,ε,[s0+i,s0+i+1]

)≤5 ‖Bs‖2

L2 + 5ε2‖C‖2L2 ≤ 5b

for ε small enough.

Remark. We can prove that the curvature of a connection which rappresent a Yang-Mills flow is smooth with an analogous argument as for the previous lemma. Ourconnection satisfies the perturbed Yang-Mills equation and thus, by the Bianchi iden-tity (13.3) and the commutation formula (13.3),

0 =dABs +1

ε2dAd

∗AFA − dA∇tBt − dA ∗Xt(A)

=∇sFA +1

ε2dAd

∗AFA −∇tdABt + [Bt ∧Bt]− dA ∗Xt(A)

=∇sFA +1

ε2dAd

∗AFA −∇t∇tFA + [Bt ∧Bt]− dA ∗Xt(A),

0 =∇tBs +1

ε2∇td

∗AFA −∇t∇tBt −∇t ∗Xt(A)

=∇sBt + dAC +1

ε2d∗A∇tFA −∇t∇tBt

− d ∗Xt(A)Bt − ∗Xt(A)− 1

ε2∗ [Bt, ∗FA]

=∇sBt +1

ε2dAd

∗ABt +

1


− d ∗Xt(A)Bt − ∗Xt(A)− 1

ε2∗ [Bt, ∗FA]

Thus in the same way as for the last lemma we can estimate all the first derivarives ofFA and Bt in a set Ωs0 := Σ × S1 × [s0, s0 + 1], s0 ∈ R. Then, by a bootstrappingargument one can prove that the curvature terms are in W k,2 for any k.

We denote by ∆ := ∂21 + ∂2

2 + · · · + ∂2n the standard Laplacian on Rn, n > 0, and we

define Pr := (−r2, 0)×Br(0).

Lemma 80. For every n ∈ N there is a constant cn > 0 such that the following holdsfor every r ∈ (0, 1]. If a ≥ 0 and w : R × Rn ⊃ Pr → R is C1 in the s-variable andC2 in the x-variable such that

(∆− ∂s)w ≥ −aw, w ≥ 0, (13.14)

then

w(0) ≤ cnear2

rn+2

∫Pr

w. (13.15)


Lemma 81. Let R, r > 0 and u : R × Rn ⊃ PR+r → R be C1 in the s-variable andC2 in the x-variable and f, g : PR+r → R be continuous functions such that

(∆− ∂s)u ≥ g − f, u ≥ 0, f ≥ 0, g ≥ 0. (13.16)

Then ∫PR

g ≤∫PR+r

f +

(4

r2+

1

Rr

)∫PR+r\PR

u. (13.17)

Proof of the theorem 78. The proof will be divided in some steps. In the first one wewill prove that the L2-norm over Σ of FA can be bounded by any positive constantprovided we choose ε small enough. This allows us to apply the lemmas 6 and 7 forp = 2. The next three steps provide bounds of the L2-norm over Σ for some curvatureterms (step 2), their derivatives (step 3) and their second derivatives (step 4). In thelast two steps we will prove the estimates (13.11) and (13.10).

Step 1. We choose a positive constant δ0 < 1. There is a constant ε0 > 0 such that thefollowing holds. If 0 < ε < ε0, then sup(t,s)∈S1×R ‖FA‖L2(Σ) ≤ δ0.

Proof of step 1. The idea is to use lemma 80 and therefore we need an estimate frombelow of (∂2

t − ∂s) ‖FA‖2. First, using the Bianchi identity (13.3) in the second andin the fourth equality, the commutation formula (13.4) in the third and the Yang-Millsflow equation (13.2) in the fourth, we obtain that

1

2

(∂2t − ∂s

)‖FA‖2 = ‖∇tFA‖2 + 〈FA,∇t∇tFA〉 − 〈FA,∇sFA〉

=‖∇tFA‖2 + 〈FA,∇tdABt〉 − 〈FA,∇sFA〉=‖∇tFA‖2 + 〈FA, dA∇tBt〉+ 〈FA, [Bt ∧Bt]〉 − 〈FA,∇sFA〉

=‖∇tFA‖2 +1

ε2〈FA, dAd∗AFA〉+ 〈FA, dABs〉

− 〈FA, dA ∗Xt(A)〉+ 〈FA, [Bt ∧Bt]〉 − 〈FA, dABs〉

=‖∇tFA‖2 +1

ε2‖d∗AFA‖2 + 〈FA, [Bt ∧Bt]〉 − 〈d∗AFA, ∗Xt(A)〉;

(13.18)

therefore, applying the Cauchy-Schwarz inequality

|〈d∗AFA, ∗Xt(A)〉| ≤ c‖d∗AFA‖ ≤1

2ε2‖d∗AFA‖2 + 2c2ε2,

for any positive γ1 we have1

2

(∂2t − ∂s

)‖FA‖2 ≥− ‖Bt‖2

L4(Σ)‖FA‖ − cε2 ≥ − c

γ1

‖FA‖2 − γ1 − cε2

≥− c

γ1

(‖FA‖2 +

γ21

c− ε2γ1

) (13.19)

where c ≥ 1 depends on ‖Bt‖L4(Σ) and on ‖Xt‖L2(Σ). Thus, by lemma 80, for everyr ∈ (0, 1], Pr := (−r2, 0)×Br(0),

‖FA‖2 ≤c1ecγ1r2

r3

∫Pr

(‖FA‖2 +

γ21

c+ ε2γ1

)dt ds

≤4c1ecγ1r2

bε2

r+ 2c1e

cγ1r2

(γ2

1

c+ ε2γ1

) (13.20)

135

and where we use that∫ 1

0‖FA‖2dt ≤ 2bε2; next, we choose γ1 := δ0

2(c1e)12

and r =(γ1

c

) 12 , then ‖FA‖2 < 4

√2c

541 c

12 e

54 bδ− 1

20 ε2 + 1

2δ2

0 + (c1e)12 δ0ε

2 and finally with ε2 <

12

δ520

4√

2c541 c

12 e

54 b+(c1e)

12 δ

320

, it follows that ‖FA‖L2(Σ) < δ0 and we end the proof of the first

step.

For the rest of the proof we choose δ0 satisfying the condition of the theorems 6 and 7for p = 2.

Step 2. There are two constants ε0, c > 0 such that the following holds. If 0 < ε < ε0,then

sup(s,t)∈S1×Q

(‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2

)≤c∫S1×Ω

(ε2‖Bt‖2 + ε2cXt + ‖FA‖2 + ε2‖Bs‖2 + ε4‖C‖2

)dt ds,∫

S1×Q

(‖FA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)≤ cε2

∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2

)dt,∫

S1×Q

(ε2‖∇tBt‖2 + ‖dABt‖2 + ‖d∗ABt‖2

)dt ds

≤c∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds,∫

S1×Q

(ε2‖∇tBs‖2 + ‖dABs‖2 + ‖d∗ABs‖2

)dt ds

≤c∫S1×Ω

(‖FA‖2 + ε2‖Bs‖2 + ε2‖Bt‖2 + ε4‖C‖2

)dt ds.

Proof of step 2. Analougously to the first step we need to compute 12

(∂2t − ∂s) ‖Bt‖2,

12

(∂2t − ∂s) ‖Bs‖2 and 1

2(∂2t − ∂s) ‖C‖2 in order to apply the lemmas 80 and 81. On

the one hand, we consider the following computation1

2(∂2t − ∂s)‖Bt‖2 = ‖∇tBt‖2 − 〈∇sBt, Bt〉+ 〈∇t∇tBt, Bt〉

=‖∇tBt‖2 − 〈∇sBt, Bt〉+1

ε2〈∇td

∗AFA, Bt〉

+ 〈∇tBs, Bt〉 − 〈∇t ∗Xt(A), Bt〉

=‖∇tBt‖2 +1

ε2〈d∗A∇tFA, Bt〉+

1

ε2〈∗[Bt, ∗FA], Bt〉

+ 〈dAC,Bt〉 − 〈d ∗Xt(A)Bt + Xt(A), Bt〉

where for the second equality we use the Yang-Mills flow equation (13.2) and for thethird the commutation formula (13.4) and the Bianchi identity (13.3). By the Yang-Mills flow equation (13.2) and the Bianchi identity (13.3) we finally obtain that

=‖∇tBt‖2 +1

ε2‖dABt‖2 +

1

ε2‖d∗ABt‖2 +

1

ε2〈∗[Bt, ∗FA], Bt〉

− 〈d ∗Xt(A)Bt + Xt(A), Bt〉.


Thus, since by the Cauchy-Schwarz inequality and the Sobolev estimate

|〈∗[Bt, ∗FA], Bt〉| ≤c (‖Bt‖+ ‖dABt‖+ ‖d∗ABt‖) ‖Bt‖L4(Σ)‖FA‖

≤cε2‖Bt‖2 +1

ε2‖FA‖2 +

1

2‖dABt‖2 +

1

2‖d∗ABt‖2

|〈d ∗Xt(A)Bt + Xt(A), Bt〉| ≤ c‖Bt‖2 + ‖Xt(A)‖2,

we have (∂2t − ∂s

)‖Bt‖2 ≥‖∇tBt‖2 +

1

ε2‖dABt‖2 +

1

ε2‖d∗ABt‖2

− c

ε4‖FA‖2 − c‖Bt‖2 − c‖Xt(A)‖2.

(13.21)

On the other hand, using the Bianchi identity (13.3) in the second equality and thecommutation formula (13.4) in the third, it follows that

1

2(∂2t − ∂s)‖Bs‖2 = ‖∇tBs‖2 + 〈Bs,∇t∇tBs〉 − 〈Bs,∇sBs〉

=‖∇tBs‖2 + 〈Bs,∇t∇sBt〉+ 〈Bs,∇tdAC〉 − 〈Bs,∇sBs〉=‖∇tBs‖2 + 〈Bs,∇s∇tBt〉 − 〈Bs, [C,Bt]〉

+ 〈d∗ABs,∇tC〉+ 〈Bs, [Bt, C]〉 − 〈Bs,∇sBs〉

the next equality follows from the Yang-Mills flow equation (13.2) and the after onefrom the commutation formula (13.4)

=‖∇tBs‖2 +1

ε2〈Bs,∇sd

∗AFA〉 − 〈Bs,∇s ∗Xt(A)〉+ 〈Bs,∇sBs〉

− 2〈Bs, [C,Bt]〉+1

ε2〈d∗ABs,∇td

∗ABt〉 − 〈Bs,∇sBs〉

=‖∇tBs‖2 +1

ε2〈dABs,∇sFA〉 −

1

ε2〈Bs, ∗[Bs, ∗FA]〉

− 〈Bs,∇s ∗Xt(A)〉 − 2〈Bs, [C,Bt]〉+1

ε2〈d∗ABs, d

∗A∇tBt〉

finally, applying the Bianchi identity (13.3) and the Yang-Mills flow equation (13.2)one more time, we can conclude that

=‖∇tBs‖2 +1

ε2‖∇sFA‖ −

1

ε2〈Bs, ∗[Bs, ∗FA]〉 − 〈Bs,∇s ∗Xt(A)〉

− 2〈Bs, [C,Bt]〉+1

ε2〈d∗ABs, d

∗ABs〉

+1

ε4〈d∗ABs, d

∗Ad∗AFA〉+

1

ε2〈d∗ABs, dAXt(A)〉

=‖∇tBs‖2 +1

ε2‖∇sFA‖+

1

ε2‖d∗ABs‖2 − 2〈Bs, [C,Bt]〉

− 1

ε2〈Bs, ∗[Bs, ∗FA]〉 − 〈Bs, d ∗Xt(A)Bs〉

137

where we use that [FA, ∗FA] = 0 and dAXt(A) = 0. Thus, since ‖Bs‖L∞(Σ) is boundedby a constant by the assumptions,(

∂2t − ∂s

)‖Bs‖2 ≥‖∇tBs‖2 +

1

ε2‖dABs‖2 +

1

ε2‖d∗ABs‖2

− c

ε4‖FA‖2 − c‖Bs‖2 − c‖Bt‖2 − ‖C‖2.

(13.22)

If we consider

1

2

(∂2t − ∂s

)ε4‖C‖2 =

1

2

(∂2t − ∂s

)‖d∗ABt‖2

=‖∇td∗ABt‖2 − 〈∇sd

∗ABt, d

∗ABt〉+ 〈∇t∇td

∗ABt, d

∗ABt〉

by the Yang-Mills flow equation (13.2) and the commutation formula (13.4) we have

=‖∇td∗ABt‖2 − 〈∇sd

∗ABt, d

∗ABt〉+

1

ε2〈∇td

∗Ad∗AFA, d

∗ABt〉

+ 〈∇td∗ABs, d

∗ABt〉 − 〈∇td

∗A ∗Xt(A), d∗ABt〉 − 〈∗∇t[Bt ∧ ∗Bt], d

∗ABt〉

and using the commutation formula (13.2) one more time

=‖∇td∗ABt‖2 − 〈∇sd

∗ABt, d

∗ABt〉+ 〈d∗A∇tBs, d

∗ABt〉

− 〈∗[Bt ∧ ∗Bs], d∗ABt〉 − 〈∗∇tdAXt(A), d∗ABt〉

=‖∇td∗ABt‖2 +

2

ε2‖dAd∗ABt‖2

− 2〈∗[Bt ∧ ∗Bs], d∗ABt〉 − 〈∗∇tdAXt(A), d∗ABt〉

where the last step follows from the Bianchi identity (13.3) and the commutation for-mula (13.4). Using the Cauchy-Schwarz inequality and that ‖Bs‖L∞(Σ) is uniformlybounded, one can easily see that(

∂2t − ∂s

)ε4‖C‖2 ≥ ε2‖dAC‖2 + cε2‖Bt‖2 + cε2‖Xt(A)‖2 (13.23)

and therefore, by (13.21), (13.22), (13.23) and ‖C‖ ≤ c‖dAC‖ by the lemma 6 wehave for two positive constants c, c0

(∂2t − ∂s)

(‖Bt‖2 + ‖Bs‖2 + c0ε

2‖C‖2)

≥‖∇tBt‖2 +1

ε2‖dABt‖2 +

1

ε2‖d∗ABt‖2 + ‖∇tBs‖2 +

1

ε2‖dABs‖

+1

ε2‖d∗ABs‖2 − c

ε4‖d∗AFA‖2 − c‖Bt‖2 − c‖Bs‖2 − ‖Xt(A)‖2.

(13.24)

Since ‖FA‖ ≤ δ, ‖FA‖L4(Σ) ≤ c‖d∗AFA‖ for a constant c by the theorems 6 and 7 thereis a A1 ∈ A0(P ) such that ‖A− A1‖ ≤ ‖FA‖ and thus we can write

dA∗Xt(A) = dA (∗Xt(A)− ∗Xt(A1))

+ dA1 ∗Xt(A1) + [(A− A1) ∧ ∗Xt(A1)](13.25)


where dA1 ∗Xt(A1) = 0. Therefore, by (13.18)

1

2

(∂2t − ∂s

)‖FA‖2 ≥ 1

4ε2‖d∗AFA‖2 +

1

4‖∇tFA‖2 − cε2‖Bt‖2 (13.26)

and with (13.24) it follows that for a constant c0 big enough

1

2

(∂2t − ∂s

) (c0‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2 + ε2cXt + c0ε

4‖C‖2)

≥− c(c0‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2 + ε2cXt + c0ε

4‖C‖2).

Finally by lemma 80, for a fix r we can conclude that

sup(t,s)∈S1×Q

(‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2 + c0ε

4‖C‖2)

≤c∫S1×Ω

(ε2‖Bt‖2 + ε2cXt + ‖FA‖2 + ε2‖Bs‖2 + c0ε

4‖C‖2)dt ds.

Next, we can apply lemma 81 to the inequality (13.26) and we obtain∫S1×Q

(‖FA‖2 + ε2‖∇tFA‖2 + ‖d∗AFA‖2

)dt ds

≤cε2

∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2

)dt ds ≤ cε4,

(13.27)

where the estimate of ‖FA‖2L2(S1×Q) follows from ‖FA‖ ≤ c‖d∗AFA‖. Using the lemma

81 for the inequalities (13.21) and (13.24) we can conclude the last two estimates ofthe third step: ∫

S1×Q

(ε2‖∇tBt‖2 + ‖dABt‖2 + ‖d∗ABt‖2

)dt ds

≤c∫S1×Ω

(1

ε2‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds,

(13.28)

∫S1×Q

(ε2‖∇tBs‖2 + ‖dABs‖2 + ‖d∗ABs‖2

)dt ds

≤c∫S1×Ω

(1

ε2‖FA‖2 + ε2‖Bs‖2 + ε2‖Bt‖2 + c0ε

4‖C‖2

)dt ds

(13.29)

and the second step follows combining the last two estimates with (13.27).

Step 3.There are two constants ε0, c > 0 such that the following holds. If 0 < ε < ε0,then ∫

S1×Q

(ε2‖∇tdABt‖2 + ‖d∗AdABt‖2 + ε2‖∇td

∗ABt‖2 + ‖dAd∗ABt‖2

)≤cε2

∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds,

(13.30)

∫S1×Q

(ε2‖∇tdABs‖2 + ‖d∗AdABs‖2 + ε2‖∇td

∗ABs‖2 + ‖dAd∗ABs‖2

)≤c∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2 + ε4cXt + ε6‖C‖2

)dt ds.

(13.31)

139

Proof of step 3. Like in the previous steps we will prove this one using the lemmas 80and 81 and therefore we need to compute 1

2

(∂2t − ∂s

)||dABt||2, 1

2

(∂2t − ∂s

)||d∗ABt||2,

12

(∂2t − ∂s

)||dABs||2, 1

2

(∂2t − ∂s

)||d∗ABt||2 and 1

2

(∂2t − ∂s

)||d∗AFA||2. First, by the first

two claims of the section 13.1, we consider

1

2

(∂2t − ∂s

) (‖dABt‖2 + ‖d∗ABt‖2

)≥ 1

2‖∇tdABt‖2 +

1

2ε2‖d∗AdABt‖2

+1

2‖∇td

∗ABt‖2 +

1

2ε2‖dAd∗ABt‖2 − c

ε2‖d∗AFA‖2

− cε2‖Bt‖2 − cε2‖Xt(A)‖2L∞ − cε2‖∇tBt‖2.

(13.32)

and combined with (13.21) yield to(∂2t − ∂s

) (‖dABt‖2 + ‖d∗ABt‖2 + c1‖FA‖2 + c0ε

2‖Bt‖2)

≥‖∇tdABt‖2 +1

ε2‖d∗AdABt‖2 + ‖∇td

∗ABt‖2 +

1

ε2‖dAd∗ABt‖2

− cε2‖Bt‖2 − cε2‖Xt(A)‖2L∞(Σ)

(13.33)

for two positive constants c0 and c1. Therefore by lemma 81 we have, for an open setΩ1 with Q ⊂ Ω1 ⊂⊂ Ω∫

S1×Q

(ε2‖∇tdABt‖2 + ‖d∗AdABt‖2 + ε2‖∇td

∗ABt‖2 + ‖dAd∗ABt‖2

)≤cε2

∫S1×Ω1

(‖dABt‖2 + ‖d∗ABt‖2 + ‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds

≤cε2

∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds

(13.34)

where the second estimate follows from step 2. Moreover, by the third and the fourthclaim of the section 13.1, the Bianchi identity dABs = ∇sFA and the identity ∇tC =1ε2d∗ABs which follows from the Yang-Mills flow equation, for a positive c1

1

2(∂2t − ∂s)(‖dABs‖2 + c1‖d∗ABs‖2) ≥ ‖∇tdABs‖2 +

1

ε2‖d∗AdABs‖2

+ ‖∇td∗ABs‖2 +

1

ε2‖dAd∗ABs‖2

− c

ε2‖d∗ABs‖2 − ε2‖∇tBs‖2 − ε2‖∇tBt‖2 − c 1

ε2‖dAd∗ABt‖2

− cε2‖Bt‖2 − cε2‖Bs‖2 − c

ε2‖d∗AFA‖2 − c‖FA‖

2

ε2‖dAd∗ABs‖2.

(13.35)

Collecting all the estimates, for a constant c0 > 0, we obtain

1

2

(∂2t − ∂s

) (c0‖FA‖2 + c0ε

2‖Bt‖2 + c0ε2‖Bs‖2

)+

1

2

(∂2t − ∂s

) (c1‖d∗ABs‖2 + ‖dABs‖2 + ‖d∗ABt‖2

)≥ 1

ε2‖dAd∗ABs‖2 +

1

ε2‖d∗AdABs‖2 − cε2‖Bt‖2

− cε2‖Xt(A)‖L∞ − cε2‖Bs‖2


By the lemma 81 we have then,∫S1×Q

(‖dAd∗ABs‖2 + ‖d∗AdABs‖2

)≤cε2

∫S1×Ω1

(ε2‖Bt‖2 + ε2cXt + ε2‖Bs‖2

)+ cε2

∫S1×Ω1

(‖FA‖2 + ‖d∗ABs‖2 + ‖dABs‖2 + ‖d∗ABt‖2

)≤c∫S1×Ω

(ε2‖Bt‖2 + ε4cXt + ε2‖Bs‖2 + ε2‖FA‖2 + ε6‖C‖2

).

(13.36)

Step 4. There are two constants ε0, c > 0 such that the following holds.If 0 < ε < ε0,then

sup(t,s)∈S1×Q

(‖dAd∗ABt‖2 + ‖d∗AdABt‖2

)≤c∫S×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds,

sup(t,s)∈S1×Q

(‖d∗AdABs‖2 + ‖dAd∗ABs‖2

)≤∫S1×Ω

(c‖FA‖2 + cε2‖Bt‖2 + ε2‖Bs‖2 + ε2cXt(A) + ε4‖C‖2)dt ds,

sup(t,s)∈S1×Q

‖d∗AdAd∗ABt‖2

≤c∫S1×Ω

(ε2‖Bs‖2 + ε2‖Bt‖2 + ‖FA‖2 + ε4cXt + ε4‖C‖2

)dt ds.

Proof of step 4. Analogously to the previous steps we need to compute

1

2

(∂2t − ∂s

)‖d∗AdABt‖2,

1

2

(∂2t − ∂s

)‖dAd∗ABt‖2,

as we will show in the section 13.1, and we obtain(∂2t − ∂s

) (‖dAd∗ABt‖2 + ‖d∗AdABt‖2

)≥ 1

ε2‖d∗AdAd∗ABt‖2 − cε2‖Bt‖2 − cε2‖d∗ABs‖2

− c 1

ε2‖d∗AFA‖2 − cε2‖dABs‖2 − cε2‖∇tBt‖2

and thus there is a constant c0 such that(∂2t − ∂s

) (‖dAd∗ABt‖2 + ‖dAd∗ABt‖2 + c0‖FA‖2 + c0ε

2‖Bt‖+ c0ε2‖Bs‖2

)≥− cε2‖Bt‖2 − cε2‖Bs‖2 + εcXt .

141

We can therefore conclude by the lemma 80 and the previous step that

sup(t,s)∈S1×Q

(‖dAd∗ABt‖2 + ‖d∗AdABt‖2

)≤c∫S×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2cXt(A)

)dt ds.

Furthermore, by the computation of the section 13.1

1

2

(∂2t − ∂s

)‖dAd∗ABs‖2 ≥ ‖∇tdAd

∗ABs‖2 +

1

ε2‖d∗AdAd∗ABs‖2

− ε2‖∇tBt‖2 − ε2‖∇tBs‖2 − ε2‖∇td∗ABt‖2

− 1

ε2‖d∗AdAd∗ABt‖2 − ε2‖∇sd

∗ABt‖2 − ε2‖FA‖2 − cε2‖dABs‖2,

1

2

(∂2t − ∂s

)‖d∗AdABs‖2 ≥ ‖∇td

∗AdABs‖2 +

1

ε2‖dAd∗AdABs‖2

− ε2‖Bt‖2 − ε2‖∇tBt‖2 − ε2‖∇tBs‖2 − ε2‖∇tdABt‖2

− ε2‖d∗AdABt‖2 − ε2‖∇sBt‖2 − ε2‖∇sdABt‖2 − 1

ε2‖d∗AFA‖2

− 1

ε2‖d∗AdAd∗ABt‖2 − 〈∇sd

∗AdA ∗Xt(A), d∗AdABs〉+

1

ε2‖∇tFA‖2

and hence for two positive constants c and c0

1

2

(∂2t − ∂s

)(‖d∗AdABs‖2 + ‖dAd∗ABs‖2 + c0‖FA‖2 + c0‖dABt‖2 + c0‖dAd∗ABt‖2

+ c0‖d∗AdABt‖2 + c0‖d∗ABt‖2 + ε2‖Bt‖2 + ε2‖Bs‖2)

≥c(‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2 + cXt(A)

)Thus, by the lemma 80 and the previous steps we can comnclude that

sup(t,s)∈S1×Q

(‖d∗AdABs‖2 + ‖dAd∗ABs‖2)

≤c∫S1×Ω1

(‖d∗AdABs‖2 + ‖dAd∗ABs‖2 + ‖FA‖2 + ‖dABt‖2

)dt ds

≤c∫S1×Ω1

(‖dAd∗ABt‖2 + ‖d∗ABt‖2

)dt ds

≤c∫S1×Ω1

(ε2‖Bt‖2 + ε2‖Bs‖2 + ε2cXt(A)

)dt ds

≤c∫S1×Ω

(‖FA‖2 + ε2‖Bt‖2 + ε2‖Bs‖2 + ε2cXt(A)

)dt ds.

Moreover,

1

2

(∂2t − ∂s

)‖d∗AdAd∗ABt‖2 ≥ ‖∇td

∗AdAd

∗ABt‖2 +

1

ε2‖dAd∗AdAd∗ABt‖2

− cε2(‖Bs‖2 + ‖dABs‖2 + ‖dABt‖2

)− cε2‖dAd∗ABs‖ −

1

ε2‖d∗AFA‖2


and hence by the lemma 80 we can conclude the proof of the fourth step:

sup(t,s)∈S1×Q

‖d∗AdAd∗ABt‖2

≤c∫S1×Ω1

(ε2‖Bs‖2 + ε2‖dABs‖2 + ε2‖dABt‖2

)dt ds

+ c

∫S1×Ω1

(ε2‖Bt‖2 + ‖FA‖2 + ‖d∗AdAd∗ABt‖2

)dt ds

≤c∫S1×Ω

(ε2‖Bs‖2 + ε2‖Bt‖2 + ‖FA‖2 + ε4cXt + ε4‖C‖2

)dt ds.


supS1×Q

(‖FA‖+ ε‖∇tFA‖+ ‖dAd∗AFA‖) ≤ cε2− 1p . (13.37)

Proof of step 5. To prove the fifth step we need the folloving observation:

f(t, s) = ‖FA‖2 + ε2‖∇tFA‖2 + ε4‖dAd∗AFA‖2 + ε2‖d∗ABt‖2

thenε2(∂2

t − ∂s)f ≥ c0f − cε4

and since for p ∈ N, p ≤ 2

∂2t f

p = pfp−1∂2t f + p(p− 1) (∂tf)2 ≥ pfp−1∂2

t f,

ε2(∂2t − ∂s)fp ≥pfp−1(∂2

t − ∂s)f ≥ pc0fp − pcε4fp−1 ≥ c0f

p − cε4p

where we used that ab ≤ (p−1)app−1

p+ bp

pfor any positive nampers a and b. Therefore

by lemma 81, for a sequence of open sets Ωi ⊂ R, i = 1, . . . , 2p, Q ⊂ Ω0 ⊂⊂ Ω1 ⊂⊂Ω2 ⊂⊂ · · · ⊂⊂ Ω2p ⊂⊂ Ω∫

S1×Ω0

fpdt ds ≤ cε4p + cε2

∫S1×Ω1

fpdt ds ≤ cε4p + cε4p

∫S1×Ω2p

fpdt ds ≤ cε4p

where the last step follows by inerating the first one and from supΩ2pf ≤ cε. By

lemma 80supS1×Q

ε2fp ≤ cε4p + ε2

∫S1×Ω0

fpdt ds ≤ cε4p

and thereforesupS1×Q

(‖FA‖+ ε‖∇tFA‖+ ‖dAd∗AFA‖) ≤ cε2− 1p .

143


supS1×Q

(‖FA‖L∞(Σ) + ‖dAd∗AFA‖+ ε2‖∇t∇tFA‖

)≤ cε2. (13.38)

Proof. By the previous steps and the claim 90 there are three constants c, c0 and c1

such that

1

2

(∂2t − ∂s

) (‖∇t∇tFA‖2 +

c0

ε2‖FA‖2 + c1‖Bt‖2

)≥− c

(‖∇t∇tFA‖2 +

c0

ε2‖FA‖2 + c1‖Bt‖2

)and thus by the lemma 80

sups∈Q‖∇t∇tFA‖2 ≤ c

∫S1×Ω

(‖∇t∇tFA‖2 +

c0

ε2‖FA‖2 + c1‖Bt‖2

)dt ds ≤ c.

Furthermore, by the Bianchi identity (13.3) and the Yang-Mills flow equation (13.2)

sups∈Q‖dAd∗AFA‖2 ≤ sup

s∈Qε4‖dA∇tBt‖2 + cε4

≤ sups∈Q

ε4‖∇tdABt‖2 + ε4‖[Bt ∧Bt]‖2 + cε4

≤ sups∈Q

ε4‖∇t∇tFA‖2 + cε4 ≤ cε4

and by the lemma 6

‖FA‖L∞(Σ) ≤ c‖dAd∗AFA‖L(Σ) ≤ cε2.


sup(t,s)∈S1×Q

(ε2‖Bs‖2 + ε4‖C‖2 +N1 +N2 +N3 +N4

)≤ε2c

∫S1×Ω

(‖Bs‖2 + ε2‖C‖2

)dt ds.

Proof. Since we know now that, for a positive constant c0,

ε2‖Bt‖L∞(Σ) + ‖FA‖L∞(Σ) ≤ c0ε2,

using the computations of the second step we can obtain the following estimate apositive constant c

1

2

(∂2t − ∂s

) (‖Bs‖2 + ε2‖C‖2

)≥‖∇tBs‖2 +

1

ε2‖dABs‖2 +

1

ε2‖d∗ABs‖+ ε2‖∇tC‖2 + ‖dAC‖2

− c(‖Bs‖2 + ε2‖C‖2

).


Analogously, for seven positive constants ci, i = 1, . . . , 7, we have(∂2t − ∂s

) (N1 + c1ε

2‖Bs‖2 + c1ε4‖C‖2

)≥c5

ε2N2 − c

(N1 + c1ε

2‖Bs‖2 + c1ε4‖C‖2

),

(∂2t − ∂s

) (N2 + c2N1 + c2c1ε

2‖Bs‖2 + c2c1ε4‖C‖2

)≥c6

ε2N3 − c

(N2 + c2N1 + c2c1ε

2‖Bs‖2 + c2c1ε4‖C‖2

),

(∂2t − ∂s

) (N3 + c3N2 + c3c2N1 + c3c2c1ε

2‖Bs‖2 + c3c2c1ε4‖C‖2

)≥c7

ε2N4 − c (N4 + c4N3 + c4c3N2 + c4c3c2N1)

− c(c4c3c2c1ε

2‖Bs‖2 + c4c3c2c1ε4‖C‖2

)(∂2t − ∂s

)(N4 + c4N3 + c4c3N2 + c4c3c2N1)

+(∂2t − ∂s

) (c4c3c2c1ε

2‖Bs‖2 + c4c3c2c1ε4‖C‖2

)≥− c (N4 + c4N3 + c4c3N2 + c4c3c2N1)

− c(c4c3c2c1ε

2‖Bs‖2 + c4c3c2c1ε4‖C‖2

).

The seventh step follows then from the lemmas 80 and 81 and the last estimates.

With the seventh step we concluded the proof of the theorem 78.

13.1 ComputationsUnder the assumptions of the theorem 78 we prove the following estimates that areused to prove the first six steps of the theorem.


1

2

(∂2t − ∂s

)||dABt||2

=||∇tdABt||2 +1

ε2||d∗AdABt||2 +

1


− 1

ε2〈∗[Bt, ∗FA], d∗AdABt〉+ 〈[FA, C], dABt〉+ 〈[Bt ∧Bs], dABt〉

− 〈∇tdA ∗Xt(A), dABt〉+ 〈2[∇tBt ∧Bt], dABt〉.


Proof. In order to prove the claim we consider

1

2

(∂2t − ∂s

)||dABt||2 = ||∇tdABt||2 − 〈∇sdABt, dABt〉+ 〈∇t∇tdABt, dABt〉

13.1 Computations 145

using the commutation formula (13.4) and the Yang-Mills flow equation (13.2), wehave

=||∇tdABt||2 − 〈∇sdABt, dABt〉+1

ε2〈∇tdAd

∗AFA, dABt〉

+ 〈∇tdABs, dABt〉 − 〈∇tdA ∗Xt(A), dABt〉+ 〈∇t[Bt ∧Bt], dABt〉

by the commutation formula (13.4)

=||∇tdABt||2 − 〈∇sdABt, dABt〉+1

ε2〈dAd∗A∇tFA, dABt〉

+1

ε2〈[Bt ∧ d∗AFA], dABt〉 −

1

ε2〈∗[Bt, ∗FA], d∗AdABt〉+ 〈dA∇tBs, dABt〉

+ 〈[Bt ∧Bs], dABt〉 − 〈∇tdA ∗Xt(A), dABt〉+ 〈∇t[Bt ∧Bt], dABt〉

the Bianchi identity (13.3) yields to

=||∇tdABt||2 +1

ε2||d∗AdABt||2 +

1


− 1

ε2〈∗[Bt, ∗FA], d∗AdABt〉+ 〈[FA, C], dABt〉+ 〈[Bt ∧Bs], dABt〉

− 〈∇tdA ∗Xt(A), dABt〉+ 〈2[∇tBt ∧Bt], dABt〉.

Claim 83. There are two constants c, ε0 > 0 such that1

2

(∂2t − ∂s

)||d∗ABt||2 = ||∇td

∗ABt||2 +

2

ε2||dAd∗ABt||2 − 2〈∗[Bt, ∗Bs], d

∗ABt〉


Proof. The claim follows from1

2

(∂2t − ∂s

)||d∗ABt||2 = ||∇td

∗ABt||2 − 〈∇sd

∗ABt, d

∗ABt〉+ 〈∇t∇td

∗ABt, d

∗ABt〉

and if we combine the commutation formula (13.4) and the Yang-Mills flow equations(13.2), then

=||∇td∗ABt||2 − 〈∇sd

∗ABt, d

∗ABt〉

+1

ε2〈∇td

∗Ad∗AFA, d

∗ABt〉+ 〈∇td

∗ABs, d

∗ABt〉

− 〈∇td∗A ∗Xt(A), d∗ABt〉 − 〈∗∇t[Bt ∧ ∗Bt], d

∗ABt〉

applying the (13.4) one more time and recalling that [FA, ∗FA] = 0

=||∇td∗ABt||2 − 〈∇sd

∗ABt, d

∗ABt〉+ 〈d∗A∇tBs, d

∗ABt〉

− 〈∗[Bt ∧ ∗Bs], d∗ABt〉 − 〈∗∇tdAXt(A), d∗ABt〉

then, the Bianchi identity allows us to conclude that

=||∇td∗ABt||2 +

2

ε2||dAd∗ABt||2 − 2〈∗[Bt, ∗Bs], d

∗ABt〉.



2(∂2t − ∂s)||∇sFA||2 =||∇t∇sFA||2 +

1

ε2||d∗A∇sFA||2 + 〈∇sFA,∇t[C,FA]〉

+ 〈∇sFA,∇t[Bs ∧Bt]〉+ 〈∇sFA, [Bt ∧∇sBt]〉+ 〈d∗A∇sFA, [C,Bt]〉 − 〈d∗A∇sFA,∇s ∗Xt(A)〉

− 〈∇sFA, [Bs ∧Bs]〉 −1

ε2〈d∗A∇sFA, ∗[Bs ∧ ∗FA]〉;


Proof. In order to prove the claim we consider1

2(∂2t − ∂s)||∇sFA||2

=||∇t∇sFA||2 − 〈∇s∇sFA,∇sFA〉+ 〈∇sFA,∇t∇t∇sFA〉

the commutation formula (13.4) yields to

=||∇t∇sFA||2 − 〈∇s∇sFA,∇sFA〉+ 〈∇sFA,∇t[C,FA]〉+ 〈∇sFA,∇t∇s∇tFA〉


=||∇t∇sFA||2 − 〈∇s∇sFA,∇sFA〉+ 〈∇sFA,∇t[C,FA]〉+ 〈∇sFA,∇t∇sdABt〉

using the commutation formulas (13.4) we obtain

=||∇t∇sFA||2 − 〈∇s∇sFA,∇sFA〉+ 〈∇sFA,∇t[C,FA]〉+ 〈∇sFA,∇t[Bs ∧Bt]〉+ 〈∇sFA, [Bt ∧∇sBt]〉+ 〈∇sFA, dA∇t∇sBt〉

=||∇t∇sFA||2 − 〈∇s∇sFA,∇sFA〉+ 〈∇sFA,∇t[C,FA]〉+ 〈∇sFA,∇t[Bs ∧Bt]〉+ 〈∇sFA, [Bt ∧∇sBt]〉+ 〈d∗A∇sFA, [C,Bt]〉+ 〈∇sFA, dA∇s∇tBt〉

applying the Yang-Mills flow equations (13.2)

=||∇t∇sFA||2 − 〈∇s∇sFA,∇sFA〉+ 〈∇sFA,∇t[C,FA]〉+ 〈∇sFA,∇t[Bs ∧Bt]〉+ 〈∇sFA, [Bt ∧∇sBt]〉+ 〈d∗A∇sFA, [C,Bt]〉

− 〈d∗A∇sFA,∇s ∗Xt(A)〉+ 〈∇sFA, dA∇sBs〉+1

ε2〈∇sFA, dA∇sd

∗AFA〉

finally by the commutation formula (13.4) and the Bianchi identity

=||∇t∇sFA||2 +1

ε2||d∗A∇sFA||2 + 〈∇sFA,∇t[C,FA]〉

+ 〈∇sFA,∇t[Bs ∧Bt]〉+ 〈∇sFA, [Bt ∧∇sBt]〉+ 〈d∗A∇sFA, [C,Bt]〉 − 〈d∗A∇sFA,∇s ∗Xt(A)〉

− 〈∇sFA, [Bs ∧Bs]〉 −1

ε2〈d∗A∇sFA, ∗[Bs ∧ ∗FA]〉.



2(∂2t − ∂s)||d∗ABs||2 = ||∇td

∗ABs||2 − 〈∗∇t[Bt ∧ ∗Bs], d

∗ABs〉

− 〈∗[Bt ∧ ∗∇tBs], d∗ABs〉+ 〈[C,Bt], dAd

∗ABs〉

+ 〈∗[C,Bs], d∗ABs〉 −

1

ε2〈∗[Bs, ∗FA], dAd

∗ABs〉

+1

ε2〈d∗AdABs, dAd

∗ABs〉 − 〈∇s ∗Xt(A), dAd

∗ABs〉+

1

ε2||dAd∗ABs||2

− 1

ε2〈dAd∗A ∗Xt(A), dAd

∗ABs〉+

1

ε2〈[Bt, d

∗ABt], dAd

∗ABs〉;


Proof. In order to prove the claim we computer1

2(∂2t − ∂s)||d∗ABs||2 = ||∇td

∗ABs||2 − 〈∇sd

∗ABs, d

∗ABs〉+ 〈∇t∇td

∗ABs, Bs〉

the commutation formulas (13.4) yields to

=||∇td∗ABs||2 − 〈∇sd

∗ABs, d

∗ABs〉 − 〈∗∇t[Bt ∧ ∗Bs], d

∗ABs〉

− 〈∗[Bt ∧ ∗∇tBs], d∗ABs〉+ 〈∇t∇tBs, dAd

∗ABs〉

and the Bianchi identity (13.3) to

=||∇td∗ABs||2 − 〈∇sd

∗ABs, d

∗ABs〉 − 〈∗∇t[Bt ∧ ∗Bs], d

∗ABs〉

− 〈∗[Bt ∧ ∗∇tBs], d∗ABs〉+ 〈∇t∇sBt, dAd

∗ABs〉+ 〈∇tdAC, dAd

∗ABs〉

next, if we use the commutation formulas (13.4) and the Yang-Mills equations (13.2)

=||∇td∗ABs||2 − 〈∇sd

∗ABs, d

∗ABs〉 − 〈∗∇t[Bt ∧ ∗Bs], d

∗ABs〉


∗ABs〉

+ 〈∇s∇tBt, dAd∗ABs〉+

1

ε2〈∇tdAd

∗ABt, dAd

∗ABs〉

=||∇td∗ABs||2 − 〈∇sd

∗ABs, d

∗ABs〉 − 〈∗∇t[Bt ∧ ∗Bs], d

∗ABs〉


∗ABs〉+ 〈∇sBs, dAd

∗ABs〉

+1

ε2〈∇sd

∗AFA, dAd


∗ABs〉

+1

ε2〈dAd∗A∇tBt, dAd

∗ABs〉+

1

ε2〈[Bt, d

∗ABt], dAd

∗ABs〉

finally, using the Yang-Mills flow equations (13.2) for the last time

=||∇td∗ABs||2 − 〈∗∇t[Bt ∧ ∗Bs], d

∗ABs〉 − 〈∗[Bt ∧ ∗∇tBs], d

∗ABs〉

+ 〈[C,Bt], dAd∗ABs〉+ 〈∗[C,Bs], d

∗ABs〉 −

1

ε2〈∗[Bs, ∗FA], dAd

∗ABs〉

+1

ε2〈d∗AdABs, dAd


∗ABs〉+

1

ε2||dAd∗ABs||2

− 1

ε2〈dAd∗A ∗Xt(A), dAd

∗ABs〉+

1

ε2〈[Bt, d

∗ABt], dAd

∗ABs〉.


Claim 86. There are two constants c, ε0 > 0 such that(∂2t − ∂s


)≥ 1


− c 1



Proof. On the one hand,

1

2

(∂2t − ∂s

)||d∗AdABt||2 = ||∇td

∗AdABt||2 − 〈∇sd

∗AdABt, d

∗AdABt〉

+ 〈∇t∇td∗AdABt, d

∗AdABt〉

by the commutation formula (13.4) and the Yang-Mills flow equations (13.2) we obtain

=||∇td∗AdABt||2 − 〈∇sd

∗AdABt, d

∗AdABt〉

+1

ε2〈∇td

∗AdAd

∗AFA, d

∗AdABt〉+ 〈∇td

∗AdABs, d

∗AdABt〉

− 〈∇td∗AdA ∗Xt(A), d∗AdABt〉

+ 〈∇t (− ∗ [Bt∧, ∗dABt] + d∗A[Bt ∧Bt]) , d∗AdABt〉

and applying one more time (13.4)

=||∇td∗AdABt||2 − 〈∇sd

∗AdABt, d

∗AdABt〉

+1

ε2〈d∗AdAd∗A∇tFA, d

∗AdABt〉

+1


− 1


∗AdABt〉+ 〈d∗AdA∇tBs, d

∗AdABt〉

+ 〈d∗A[Bt ∧Bs]− ∗[Bt, ∗dABs], d∗AdABt〉 − 〈dA ∗ ∇tXt(A), dAd

∗AdABt〉


∗AdABt〉

finally, by the Bianchi identity (13.3) we can conclude that


1

ε2||dAd∗AdABt||2

+1


− 1


∗AdABt〉+ 〈d∗AdAdAC, d∗AdABt〉

− 〈d∗A[Bs ∧Bt], d∗AdABt〉+ 〈[Bs ∧ dABt], d

∗AdABt〉

+ 〈d∗A[Bt ∧Bs]− ∗[Bt, ∗dABs], d∗AdABt〉 − 〈dA ∗ ∇tXt(A), dAd

∗AdABt〉


∗AdABt〉;


on the other hand,

1

2

(∂2t − ∂s

)||dAd∗ABt||2 = ||∇tdAd

∗ABt||2 − 〈∇sdAd

∗ABt, dAd

∗ABt〉

+ 〈∇t∇tdAd∗ABt, dAd

∗ABt〉

the commutation formula (13.4) and the Yang-Mills flow equations (13.2) yield to

=||∇tdAd∗ABt||2 − 〈∇sdAd

∗ABt, dAd

∗ABt〉

+1

ε2〈∇tdAd

∗Ad∗AFA, dAd

∗ABt〉+ 〈∇tdAd

∗ABs, dAd

∗ABt〉

− 〈∇tdAd∗A ∗Xt(A), dAd

∗ABt〉

+ 〈∇t[Bt, d∗ABt], dAd

∗ABt〉

that can be reformulated by (13.4) and [FA, ∗FA] = 0, i.e.

=||∇tdAd∗ABt||2 − 〈∇sdAd

∗ABt, dAd

∗ABt〉

+ 〈dAd∗A∇tBs, dAd∗ABt〉

+ 〈[Bt, d∗ABs], dAd

∗ABt〉 − 〈dA[Bt ∧ ∗Bs], dAd

∗ABt〉


∗ABt〉


∗ABt〉

next, using the commutation formula (13.4) and the Bianchi identity we have that

=||∇tdAd∗ABt||2 +

1

ε2||d∗AdAd∗ABt||2

+ 〈dA ∗ [Bs ∧ ∗Bt]− [Bs, d∗ABt], dAd

∗ABt〉

+ 〈[Bt, d∗ABs], dAd

∗ABt〉 − 〈dA[Bt ∧ ∗Bs], dAd

∗ABt〉


∗ABt〉


∗ABt〉.

The last two computations imply(∂2t − ∂s


)≥‖∇tdAd

∗ABt‖2 +

1

ε2‖d∗AdAd∗ABt‖2 + ‖∇td

∗AdABt‖2

+1

ε2‖dAd∗AdABt‖2 − cε2‖Bt‖2 − cε2‖d∗ABs‖2

− c 1


≥ 1


− c 1

ε2‖d∗AFA‖2 − cε2‖dABs‖2 − cε2‖∇tBt‖2.



1

2

(∂2t − ∂s


∗ABs‖2 +

1


− ε2‖∇tBt‖2 − ε2‖∇tBs‖2 − ε2‖∇td∗ABt‖2

− 1



1

2

(∂2t − ∂s


∗AdABs‖2 +

1




ε2‖d∗AFA‖2

− 1



1

ε2‖∇tFA‖2


Proof. On the one hand we consider

1

2

(∂2t − ∂s

)‖dAd∗ABs‖2 = ‖∇tdAd

∗ABs‖2 − 〈∇sdAd

∗ABs, dAd

∗ABs〉

+ 〈∇t∇tdAd∗ABs, dAd

∗ABs〉

which can be reformulated using the commutation formula (13.4) in the following way

=‖∇tdAd∗ABs‖2 − 〈∇sdAd

∗ABs, dAd

∗ABs〉+ 〈∇t[Bt, d

∗ABs], dAd

∗ABs〉

− 〈∇tdA ∗ [Bt ∧ ∗Bs], dAd∗ABs〉+ 〈∇tdAd

∗A∇tBs, dAd

∗ABs〉



∗ABs, dAd


∗ABs], dAd

∗ABs〉

− 〈∇tdA ∗ [Bt ∧ ∗Bs], dAd∗ABs〉+ 〈∇tdAd

∗A∇sBt, dAd

∗ABs〉

+1

ε2〈∇tdAd

∗AdAd

∗ABt, dAd

∗ABs〉

next, using the commutation formula and the Bianchi identity one more time


∗ABs, dAd


∗ABs], dAd

∗ABs〉

− 〈∇tdA ∗ [Bt ∧ ∗Bs], dAd∗ABs〉+ 〈∇tdA ∗ [Bs ∧ ∗Bt], dAd

∗ABs〉

− 〈∇t[Bs, d∗ABt], dAd

∗ABs〉+ 〈[C, dAd∗ABt], dAd

∗ABs〉

+ 〈∇s[Bt, d∗ABt], dAd

∗ABs〉+ 〈∇sdAd

∗A∇tBt, dAd

∗ABs〉

+1

ε2〈[Bt, d

∗AdAd

∗ABt], dAd

∗ABs〉 −

1

ε2〈∗[Bt ∧ ∗dAd∗ABt], d

∗AdAd

∗ABs〉

+1

ε2〈dAd∗A[Bt, d

∗ABt], dAd

∗ABs〉+

1

ε2〈dAd∗AdAd∗A∇tBt, dAd

∗ABs〉


finally, the Yang-Mills equations (13.2) allows us to conclude that

=‖∇tdAd∗ABs‖2 + 〈∇t[Bt, d

∗ABs], dAd

∗ABs〉 − 〈∇tdA ∗ [Bt ∧ ∗Bs], dAd

∗ABs〉

+ 〈∇tdA ∗ [Bs ∧ ∗Bt], dAd∗ABs〉 − 〈∇t[Bs, d

∗ABt], dAd

∗ABs〉

+ 〈[C, dAd∗ABt], dAd∗ABs〉+ 〈∇s[Bt, d

∗ABt], dAd

∗ABs〉

− 〈∇sdAd∗A ∗Xt(A), dAd

∗ABs〉+

1

ε2〈[Bt, d

∗AdAd

∗ABt], dAd

∗ABs〉

− 1

ε2〈∗[Bt ∧ ∗dAd∗ABt], d

∗AdAd

∗ABs〉+

1

ε2〈dAd∗A[Bt, d

∗ABt], dAd

∗ABs〉

+1

ε2‖d∗AdAd∗ABs‖2 − 1

ε2〈d∗AdAd∗A ∗Xt(A), d∗AdAd

∗ABs〉.

On the other hand,

1

2

(∂2t − ∂s

)‖d∗AdABs‖2 = ‖∇td

∗AdABs‖2 − 〈∇sd

∗AdABs, d

∗AdABs〉

+ 〈∇t∇td∗AdABs, d

∗AdABs〉

by the commutation formula (13.4) we obtain

=‖∇td∗AdABs‖2 − 〈∇sd

∗AdABs, d

∗AdABs〉 − 〈∇t ∗ [Bt, ∗dABs], d

∗AdABs〉

+ 〈∇td∗A[Bt ∧Bs], d

∗AdABs〉+ 〈∇td

∗AdA∇tBs, d

∗AdABs〉



∗AdABs, d


∗AdABs〉


∗AdABs〉+ 〈∇td

∗AdA∇sBt, d

∗AdABs〉

+1

ε2〈∇td

∗A[FA, d

∗ABt], dAd

∗ABs〉

next, if we use the commutation formula (13.4) one more time, then


∗AdABs, d


∗AdABs〉


∗AdABs〉 − 〈∇td

∗A[Bs ∧Bt], d

∗AdABs〉

+ 〈∇t ∗ [Bs, ∗dABt], d∗AdABs〉+ 〈[C, d∗AdABt], d

∗AdABs〉

− 〈∇s ∗ [Bt, ∗dABt], d∗AdABs〉+ 〈∇sd

∗A[Bt, Bt], d

∗AdABs〉

+ 〈∇sd∗AdA∇tBt, d

∗AdABs〉 −

1

ε2〈∗[Bt, ∗[FA, d∗ABt]], d

∗AdABs〉

+1

ε2〈∇t[FA, d

∗ABt], dAd

∗AdABs〉


applying the Yang-Mills equations (13.2)

=‖∇td∗AdABs‖2 − 〈∇t ∗ [Bt, ∗dABs], d

∗AdABs〉



∗A[Bs ∧Bt], d

∗AdABs〉


∗AdABs〉


∗A[Bt, Bt], d

∗AdABs〉

+1

ε2〈∇sd

∗AdAd

∗AFA, d

∗AdABs〉 − 〈∇sd

∗AdA ∗Xt(A), d∗AdABs〉

− 1


∗AdABs〉+

1

ε2〈∇t[FA, d

∗ABt], dAd

∗AdABs〉

finally, by the commutation formula (13.4) and the Bianchi identity (13.3) we canconclude that

=‖∇td∗AdABs‖2 − 〈∇t ∗ [Bt, ∗dABs], d

∗AdABs〉



∗A[Bs ∧Bt], d

∗AdABs〉


∗AdABs〉


∗A[Bt, Bt], d

∗AdABs〉

− 1

ε2〈∗[Bs, ∗dAd∗AFA], d∗AdABs〉+

1

ε2〈[Bs ∧ d∗AFA], dAd

∗AdABs〉

− 1

ε2〈dA ∗ [Bs, ∗FA], dAd

∗AdABs〉+

1


− 〈∇sd∗AdA ∗Xt(A), d∗AdABs〉

− 1


∗AdABs〉+

1

ε2〈∇t[FA, d

∗ABt], dAd

∗AdABs〉.

The last two computation implies by the Cauchy-Schwarz inequality that1

2

(∂2t − ∂s


∗ABs‖2 +

1


− ε2‖∇tBt‖2 − ε2‖∇tBs‖2 − ε2‖∇td∗ABt‖2

− 1



1

2

(∂2t − ∂s


∗AdABs‖2 +

1




ε2‖d∗AFA‖2

− 1



1

ε2‖∇tFA‖2.


2

(∂2t − ∂s


∗AdAd

∗ABt‖2 +

1


− cε2(‖Bs‖2 + ‖dABs‖2 + ‖dABt‖2


1

ε2‖d∗AFA‖2



Proof. In order to prove the claim we consider

1

2

(∂2t − ∂s

)||d∗AdAd∗ABt||2 = ||∇td

∗AdAd

∗ABt||2 + 〈∇sd

∗AdAd

∗ABt, d

∗AdAd

∗ABt〉

+ 〈∇t∇td∗AdAd

∗ABt, d

∗AdAd

∗ABt〉

the commutation formula (13.4) and the Yang-Mills flow identity (13.2) imply

=||∇td∗AdAd

∗ABt||2 + 〈∇sd

∗AdAd

∗ABt, d

∗AdAd

∗ABt〉

+1

ε2〈∇td

∗AdAd

∗Ad∗AFA, d

∗AdAd

∗ABt〉+ 〈∇td

∗AdAd

∗ABs, d

∗AdAd

∗ABt〉

− 〈∇td∗AdAd

∗A ∗Xt(A), d∗AdAd

∗ABt〉

− 〈∇t ∗ [Bt, ∗dAd∗ABt], d∗AdAd

∗ABt〉+ 〈∇td

∗A[Bt, d

∗ABt], d

∗AdAd

∗ABt〉

and using the commutation formula one more time combined with [FA ∧∗FA] = 0 weobtain

=||∇td∗AdAd

∗ABt||2 + 〈∇sd

∗AdAd

∗ABt, d

∗AdAd

∗ABt〉

+ 〈d∗AdAd∗A∇tBs, d∗AdAd

∗ABt〉 − 〈∗[Bt, ∗dAd∗ABs], d

∗AdAd

∗ABt〉

+ 〈d∗A[Bt, d∗ABs], d

∗AdAd

∗ABt〉 − 〈d∗AdA[Bt ∧ ∗Bs], d

∗AdAd

∗ABt〉


∗ABt〉+ 〈∇td

∗A[Bt, d

∗ABt], d

∗AdAd

∗ABt〉

finally, by the Bianchi identity (13.3) and (13.4), we have

=||∇td∗AdAd

∗ABt||2 +

1

ε2||dAd∗AdAd∗ABt||2

+ 〈d∗AdA ∗ [Bs ∧ ∗Bt], d∗AdAd

∗ABt〉 − 〈d∗A[Bs, d

∗ABt], d

∗AdAd

∗ABt〉

+ 〈∗[Bs, ∗dAd∗ABt], d∗AdAd

∗ABt〉 − 〈∗[Bt, ∗dAd∗ABs], d

∗AdAd

∗ABt〉

+ 〈d∗A[Bt, d∗ABs], d

∗AdAd

∗ABt〉 − 〈d∗AdA[Bt ∧ ∗Bs], d

∗AdAd

∗ABt〉


∗ABt〉+ 〈∇td

∗A[Bt, d

∗ABt], d

∗AdAd

∗ABt〉.

We can therefore conclude that

1

2

(∂2t − ∂s


∗AdAd

∗ABt‖2 +

1


− cε2(‖Bs‖2 + ‖dABs‖2 + ‖dABt‖2


1

ε2‖d∗AFA‖2.


1

2(∂2t − ∂s)‖d∗AFA‖2 ≥ ‖∇td

∗AFA‖2 +

1

ε2‖dAd∗AFA‖2 − cε2‖Bt‖2. (13.39)



Proof. In order to prove the claim we consider1

2(∂2t − ∂s)||d∗AFA||2

=||∇td∗AFA||2 − 〈∇sd

∗AFA, d

∗AFA〉+ 〈d∗AFA,∇t∇td

∗AFA〉

by the commutation formula (13.4) we have

=||∇td∗AFA||2 − 〈∇sd

∗AFA, d

∗AFA〉+ 〈d∗AFA,−∇t ∗ [Bt ∧ ∗FA]〉

− 〈d∗AFA, ∗[Bt ∧ ∗∇tFA]〉+ 〈d∗AFA, d∗A∇t∇tFA〉

next, by the Bianchi identity (13.3)

=||∇td∗AFA||2 − 〈∇sd

∗AFA, d

∗AFA〉+ 〈d∗AFA,−∇t ∗ [Bt ∧ ∗FA]〉

− 〈d∗AFA, ∗[Bt ∧ ∗∇tFA]〉+ 〈d∗AFA, d∗A∇tdABt〉

and by the commutation formula (13.4)

=||∇td∗AFA||2 − 〈∇sd

∗AFA, d

∗AFA〉+ 〈d∗AFA,−∇t ∗ [Bt ∧ ∗FA]〉

− 〈d∗AFA, ∗[Bt ∧ ∗∇tFA]〉+ 〈d∗AFA, d∗A[Bt ∧Bt]〉+ 〈d∗AFA, d∗AdA∇tBt〉

if we use the Yang-Mills flow equations (13.2), then

=||∇td∗AFA||2 − 〈∇sd

∗AFA, d

∗AFA〉+ 〈d∗AFA,−∇t ∗ [Bt ∧ ∗FA]〉

− 〈d∗AFA, ∗[Bt ∧ ∗∇tFA]〉+ 〈d∗AFA, d∗A[Bt ∧Bt]〉+ 〈d∗AFA, d∗AdABs〉

+1

ε2〈d∗AFA, d∗AdAd∗AFA〉 − 〈d∗AFA, d∗AdA ∗Xt(A)〉

finally, combining the commutation formula (13.4) and the Bianchi identity

=||∇td∗AFA||2 +

1

ε2||dAd∗AFA||2 + 〈d∗AFA,−∇t ∗ [Bt ∧ ∗FA]〉

− 〈d∗AFA, ∗[Bt ∧ ∗∇tFA]〉+ 〈d∗AFA, d∗A[Bt ∧Bt]〉+ 〈d∗AFA, ∗[Bs ∧ ∗FA]〉 − 〈d∗AFA, d∗AdA ∗Xt(A)〉

and hence1

2(∂2t − ∂s)‖d∗AFA‖2 ≥ ‖∇td

∗AFA‖2 +

1

ε2‖dAd∗AFA‖2 − cε2‖Bt‖2. (13.40)

Claim 90. We assume that there is ε1 such that sup ‖FA‖ ≤ cε for any positive ε < ε1.Then there are two constants c, ε0 > 0 such that

1

2

(∂2t − ∂s

)‖∇t∇tFA‖2

≥||∇t∇t∇tFA||2 +1

ε2||∇t∇td

∗AFA||2 −

c

ε2‖dABt‖2

−c‖FA‖2

L∞(Σ)

ε2‖∇tBt‖2 − cε2‖∇tBs‖2 − cε2‖∇tFA‖2

− c‖dAd∗ABs‖2

ε2‖FA‖2 − c

ε2‖∇td

∗AFA‖2 − c




Proof. If we consider the computation

1

2(∂2t − ∂s)||∇t∇tFA||2 = ||∇t∇t∇tFA||2 + 〈∇t∇tFA,∇t∇t∇t∇tFA〉

− 〈∇t∇tFA,∇s∇t∇tFA〉

by the Bianchi identity (13.3)

=||∇t∇t∇tFA||2 + 〈∇t∇tFA,∇t∇t∇tdABt〉− 〈∇t∇tFA,∇s∇t∇tFA〉

and by the commutation formula (13.4)

=||∇t∇t∇tFA||2 + 〈∇t∇tFA, dA∇t∇t∇tBt〉− 〈∇tFA,∇s∇t∇tFA〉+ 〈∇tFA,∇t∇t[Bt ∧Bt]〉+ 〈∇tFA,∇t[Bt ∧∇tBt]〉+ 〈∇tFA, [Bt ∧∇t∇tBt]〉

next, we can apply the Yang-Mills flow equation (13.2)

=||∇t∇t∇tFA||2 +1

ε2〈∇t∇tFA, dA∇t∇td

∗AFA〉

+ 〈∇t∇tFA, dA∇t∇tBs〉 − 〈∇t∇tFA, dA∇t∇t ∗Xt(A)〉− 〈∇t∇tFA,∇s∇t∇tFA〉+ 〈∇tFA,∇t∇t[Bt ∧Bt]〉+ 〈∇tFA,∇t[Bt ∧∇tBt]〉+ 〈∇tFA, [Bt ∧∇t∇tBt]〉

and the commutation formula (13.4)

=||∇t∇t∇tFA||2 +1

ε2||∇t∇td

∗AFA||2

+1

ε2〈∗[Bt ∧ ∗∇tFA],∇t∇td

∗AFA〉+

1

ε2〈∗∇t[Bt ∧ ∗FA],∇t∇td

∗AFA〉

+ 〈∇t∇tFA,∇t∇tdABs〉 − 〈∇t∇tFA, [Bt ∧∇tBs]〉− 〈∇t∇tFA,∇t[Bt ∧Bs]〉 − 〈∇t∇tFA, dA∇t∇t ∗Xt(A)〉− 〈∇t∇tFA,∇t∇t∇sFA〉+ 〈∇t∇tFA, [C,∇tFA]〉+ 〈∇t∇tFA,∇t[C,FA]〉+ 〈∇tFA,∇t∇t[Bt ∧Bt]〉+ 〈∇tFA,∇t[Bt ∧∇tBt]〉+ 〈∇tFA, [Bt ∧∇t∇tBt]〉

finally, if we estimate term by term, we obtain

≥||∇t∇t∇tFA||2 +1

ε2||∇t∇td

∗AFA||2

− c

ε2||∇tFA|| · ||∇t∇td

∗AFA|| −

c

ε2||FA||L∞||∇tBt|| · ||∇t∇td

∗AFA||

− c||∇t∇tFA|| · ||∇tBs|| − c||∇t∇tFA|| · ||∇tBt||− c||∇t∇tFA|| · ||∇tFA|| − c||∇t∇tFA||L4(Σ) · ||∇tC||L4(Σ)||FA||− c||∇tFA|| · ||∇t∇tBt|| − c||∇tFA|| · ||d∗A∇tBt||− c||d∗A∇tFA|| · ||∇tBt||


and thus using the Cauchy-Schwarz inequality and the lemma 6

1

2

(∂2t − ∂s

)‖∇t∇tFA‖2

≥||∇t∇t∇tFA||2 +1

ε2||∇t∇td

∗AFA||2 −

c

ε2‖dABt‖2

−c‖FA‖2

L∞(Σ)

ε2‖∇tBt‖2 − cε2‖∇tBs‖2 − cε2‖∇tFA‖2

− c‖dAd∗ABs‖2

ε2‖FA‖2 − c

ε2‖∇td

∗AFA‖2 − c


Exponentialconvergence 14

In this chapter we will prove the exponential convergence of the curvature terms Bs

and C stated in the next theorem. In order to simplify the exposition, for this chapter,we denote by ‖ · ‖ the norm ‖ · ‖L2(Σ×S1).

Theorem 91. We assume that every perturbed closed geodesic on Mg(P ) is non-degenerate. Then for every c0, b > 0 there are four positive constants δ, ε0, c, ρ suchthat the following holds. If Ξ ∈ Mε(Ξε

−,Ξε+), Ξε

−,Ξε+ ∈ CritbYMε,H with 0 < ε ≤ ε0,

satisfies, for Bt := ∂tA− dAΨ, Bs := ∂sA− dAΦ, C := ∂sΨ− ∂tΦ− [Ψ,Φ],

‖Bs‖L∞(Σ×S1×R) + ‖Bt‖L∞(Σ×S1×R) + ε‖C‖L∞(Σ×S1×R) ≤ c0 (14.1)

and‖Bs‖2

L2(Σ×S1×[0,∞)) + ε2‖C‖2L2(Σ×S1×[0,∞)) ≤ δ, (14.2)

then, for S ≥ 1,

‖Bs‖2L2(Σ×S1×[S,∞)) + ε2‖C‖2

L2(Σ×S1×[S,∞)) ≤ ce−ρS. (14.3)

Lemma 92. We choose a positive constant c0 and we assume that every perturbedclosed geodesic onMg(P ) is non-degenerate. Then there are positive constants δ0, ε0

and c such that the following holds. If A + Ψdt is a connection on P × S1 whichsatisfies, for 0 < ε ≤ ε0, Bt := ∂tA− dAΨ,∥∥∥∥ 1


∥∥∥∥L∞

+1

ε‖|d∗ABt‖L∞ + sup

t∈S1

‖FA‖L2(Σ) ≤ δ0, (14.4)

‖Bt‖L∞ ≤ c0, (14.5)

where Bt := ∂tA− dAΨ, then

‖α + ψdt‖2,2,ε ≤c(ε ‖Dε(A,Ψ)(α, ψ)‖0,2,ε + ‖πA0Dε(A,Ψ)(α, ψ)‖L2

), (14.6)

‖(1− πA0)α + ψdt‖2,2,ε ≤cε2 ‖Dε(A,Ψ)(α, ψ)‖0,2,ε

+ cε ‖πA0Dε(A,Ψ)(α, ψ)‖L2

(14.7)

for every 1-form α + ψdt on P × S1.

Proof of lemma 92. We suppose that the lemma does not hold. Then there are se-quences Aν + Ψνdt, εν → 0, such that, for Bν

t := ∂tAν − dAνΨν , and

δν :=

∥∥∥∥ 1

ε2d∗AνFAν −∇Ψν

t Bνt − ∗Xt(A

ν)

∥∥∥∥L∞

+

∥∥∥∥ 1

ενd∗AνB

νt

∥∥∥∥L∞

+ supt∈S1

‖FAν‖L2(Σ),

157

158 14. Exponential convergence

with δν → 0 and (14.6) or (14.7) are not satisfied forA+Ψdt = Aν+Ψνdt and ε = εν .

Claim: The following two estimates hold:

‖FA‖1,2,ε ≤∥∥d∗AFA − ε2∇tBt − ε2 ∗Xt(A)

∥∥+ cε2,

‖Bt‖1,2,1 ≤‖d∗ABt‖+

∥∥∥∥ 1


∥∥∥∥+ ‖Bt‖+ c.

Proof. By the identity∥∥d∗AFA − ε2∇tBt − ε2 ∗Xt(A)∥∥2

+ ε2‖∇tFA − dABt‖2

= ‖d∗AFA‖2 + ε4 ‖∇tBt‖2 + ε4 ‖Xt(A)‖2 + ε2 ‖∇tFA‖2

+ ε2 ‖dABt‖2 − 2⟨ε2 ∗Xt(A), d∗AFA − ε2∇tBt

⟩− 2ε2 〈d∗AFA,∇tBt〉 − 2ε2 〈∇tFA, dABt〉 ,

‖d∗ABt‖2 +

∥∥∥∥ 1


∥∥∥∥2

+1

ε2‖∇tFA − dABt‖2

= ‖d∗ABt‖2 +1

ε4‖d∗AFA‖

2 + ‖∇tBt‖2 + ‖Xt(A)‖2

+1

ε2‖∇tFA‖2 +

1

ε2‖dABt‖2 − 2

⟨∗Xt(A),

1

ε2d∗AFA −∇tBt

⟩− 2

1

ε2〈d∗AFA,∇tBt〉 − 2

1

ε2〈∇tFA, dABt〉 ,

−2 〈d∗AFA,∇tBt〉 − 2 〈∇tFA, dABt〉 = 2 〈FA, [Bt ∧Bt]〉

and since ∇tFA − dABt = 0 by the Bianchi identity, the lemma holds.

Thus, the Lp-norm of the curvatures FAν +Bνt dt is uniformly bounded for any p which

satisfies the Sobolev’s condition −3p< 1 − 3

2and hence for p < 6. Therefore, by the

weak Uhlenbeck compactness theorem (see [20] or theorem 7.1. in [25]), we canassume that Aν + Ψνdt converges weakly to a A0 + Ψ0dt in W 1,p and hence alsostrongly in L∞. In addition we have that, for B0

t := ∂tA0 − dA0Ψ0,

FA0 = 0, d∗A0B0t = dA0B0

t = 0, ∇0tB

0t + ∗Xt(A

0) ∈ im d∗A0 ;

thus, A0 + Ψ0dt satisfies the equations of a perturbed closed geodesic and thereforefor any 1-form α on P × S1 with α(t) ∈ Ω1(Σ, gP )

‖πA0(α)‖1,2,1 ≤ c‖D0(A0)(πA0(α), ψ0)‖L2

where ψ satisfies −2 ∗ [πA0(α) ∧ ∗B0t ]− d∗A0dA0ψ0 = 0. Then

‖πA0(α)‖1,2,1 ≤ c∣∣∣∣∣∣πA0

(2[ψ0, B

νt ] + d ∗Xt(A

ν)πA0(α)

+∇Ψν

t ∇Ψν

t πA0(α) +1

ε2ν

∗ [πA0(α) ∧ ∗FAν ])∣∣∣∣∣∣

L2

159

where we used∥∥∥∥[πA0(α) ∧ ∗(dA0(d∗A0dA0)−1(∇0

tB0t − ∗Xt(A

0))− 1

ε2ν

FAν

)]∥∥∥∥≤ ‖πA0(α)‖L∞

∥∥∥∥ 1

ε2ν

d∗AνFAν −∇0tB

0t − ∗Xt(A

0)

∥∥∥∥+ ‖πA0(α)‖L∞

∥∥∇0tB

0t + ∗Xt(A

0)∥∥L∞

∥∥Aν − A0∥∥

≤ ‖πA0(α)‖L∞∥∥∥∥ 1

ε2ν

d∗AνFAν −∇νtB

νt − ∗Xt(A

ν)

∥∥∥∥+ ‖πA0(α)‖L∞

∥∥∇0tB

0t + ∗Xt(A

0)−∇νtB

νt − ∗Xt(A

ν)∥∥

≤ 1

2‖πA0(α)‖1,2,1

for ν big enough. Therefore, analogously to the theorem 15 and lemma 16, one canprove that

‖α‖2,2,εν ≤ c(εν ‖Dεν (Aν ,Ψν)(α, ψ)‖0,2,εν

+ ‖πA0Dεν1 (Aν ,Ψν)(α, ψ)‖L2

),

‖α + ψds− πA0(α)‖2,2,εν ≤ c(ε2ν ‖Dεν (Aν ,Ψν)(α, ψ)‖0,2,εν

+ εν ‖πA0Dεν1 (Aν ,Ψν)(α, ψ)‖L2

).

Thus, we have a contradiction and hence we finished the proof of the lemma 92.

Proof of theorem 91. To prove this theorem we proceed as Dostoglou and Salamon didfor the theorem 7.4 in [7]. The idea is to find a positive bounded function f(s) suchthat it satisfies

f ′′(s) ≥ ρ2f(s) (14.8)

for s ≥ 1. Then, this implies that f has an exponential decay, because, since

∂s(e−ρs (f ′(s) + ρf(s))

)= e−ρs

(−ρ2f(s) + f ′′(s)

)≥ 0,

f ′(s) + ρf(s) < 0 (otherwise e−ρs (f ′(s) + ρf(s)) would be positive and increase;thus, since f(s) is bounded, e−ρsf(s) would decrease and hence f ′(s) would increase.Therefore f(s) would be unbounded which is a contradiction.) and hence eρsf(s) isdecreasing. Therefore, if a function f satisfies (14.8), then

f(s) ≤ e−ρ(s−1)c1. (14.9)

with c1 = f(1). By the a priori estimate (13.12) and the lemma 6 for any δ

‖Bs‖L∞(Σ×S1×s) + ε‖C‖L∞(Σ×S1×s) ≤ δ (14.10)

holds for s sufficiently big. We define

f(s) :=1

2

∫ 1

0

(‖Bs(t, s)‖2

L2(Σ) + ε2‖C(t, s)‖2L2(Σ)

)dt; (14.11)


then, as we will show later,

f ′′(s) =1

2∂2s

(‖Bs‖2

L2(Σ×S1) + ε2‖C‖2L2(Σ×S1)

)≥∥∥∥ 1

ε2(d∗AdABs + dAd

∗ABs ∗+ ∗ [Bs, ∗FA])

−∇t∇tBs − d ∗Xt(A)Bs − 2[C,Bt]∥∥∥2

+1

ε2

∥∥d∗AdAC − ε2∇t∇tC + ∗[Bs ∧ ∗Bt]∥∥2

(14.12)

Next, for s ≥ 1 and δ sufficiently small we apply the lemma 91 for α+ψdt := Bs+Cdtand thus,

f(s) =1

2

(‖Bs‖2

2,2,ε + ε2‖C‖22,2,ε

)≤c∥∥∥ 1

ε2(d∗AdABs + dAd

∗ABs ∗+ ∗ [Bs, ∗FA])

−∇t∇tBs − d ∗Xt(A)Bs − 2[C,Bt]∥∥∥2

+ cε6

∥∥∥∥ 1

ε2d∗AdAC −∇t∇tC +

2

ε2∗ [Bs ∧ ∗Bt]

∥∥∥∥2

≤cf ′′(s).

Therefore, ρ2f(s) ≤ f ′′(s) for ρ > 0 small enough. Thus, by (14.9),∫ ∞S

(‖Bs‖2

L2(Σ×S1) + ε2‖C‖2L2(Σ×S1)

)ds ≤ ce−ρS

for S ≥ 1.

Proof of (14.12). First, we consider the following two short computations that are con-sequence of the commutation formula (13.4) and of the Yang-Mills flow equation(13.2):

∇tdAC =dA∇tC + [Bt, C] =1

ε2dA∇td

∗ABt + [Bt, C]

=1

ε2dAd

∗A∇tBt + [Bt, C] =

1

ε2dAd

∗ABs + [Bt, C],

(14.13)

d∗A∇tBs =∇td∗ABs + ∗[Bt ∧ ∗Bs] = ∇td

∗A∇tBt + ∗[Bt ∧ ∗Bs]

=∇t∇td∗ABt + ∗[Bt ∧ ∗Bs] = ε2∇t∇tC + ∗[Bt ∧ ∗Bs],

(14.14)

for in both cases we use [Bt ∧ ∗Bt] = 0 and [FA, ∗FA] = 0. In the following we usethe notation

D1 :=1

ε2d∗AdABs −∇t∇tBs +∇tdAC − d ∗Xt(A)Bs −

1

ε2∗ [Bs, ∗FA] + [Bt, C],

D2 :=ε2∇t∇tC − d∗AdAC − ∗2[Bs ∧ ∗Bt].

161

Next, we can compute the second derivative of ‖Bs‖2 + ‖C‖2, i.e.1

2∂2s

(‖Bs‖2 + ε2‖C‖2

)=‖∇sBs‖2 + ε2‖∇sC‖2 + 〈∇s∇sBs, Bs〉+ ε2〈∇s∇sC,C〉

=

∥∥∥∥ 1

ε2∇sd

∗AFA −∇s∇tBt − ∗∇sXt(A)

∥∥∥∥2

+1

ε2‖∇sd

∗ABt‖2

−⟨∇s∇s

(1


), Bs

⟩+ 〈∇s∇sd

∗ABt, C〉

where in the second step we use the Yang-Mills flow equation (13.2). Then by thecommutation formula (13.4)

=

∥∥∥∥ 1

ε2d∗A∇sFA −∇t∇sBt − d ∗Xt(A)Bs −

1

ε2∗ [Bs, ∗FA]− [C,Bt]

∥∥∥∥2

+1

ε2‖d∗A∇sBt − ∗[Bs ∧ ∗Bt]‖2

−⟨∇s

(1

ε2d∗A∇sFA −∇t∇sBt − d ∗Xt(A)Bs

), Bs

⟩−⟨∇s

(− 1

ε2∗ [Bs, ∗FA]− [C,Bt]

), Bs

⟩+ 〈∇s (d∗A∇sBt − ∗[Bs ∧Bt]) , C〉

and by the Bianchi identity

= ‖D1‖2 +1

ε2‖d∗A∇tBs − d∗AdAC − ∗[Bs ∧ ∗Bt]‖2

−⟨∇s

(1

ε2d∗AdABs −∇t∇tBs +∇tdAC − d ∗Xt(A)Bs

), Bs

⟩−⟨∇s

(− 1

ε2∗ [Bs, ∗FA]− [C,Bt]

), Bs

⟩+ 〈∇s (d∗A∇tBs − d∗AdAC − ∗[Bs ∧Bt]) , C〉

in addition, if we apply (14.13) and (14.14), then

= ‖D1‖2 +1

ε2‖D2‖2 − 〈∇sD1, Bs〉+ 〈∇sD2, C〉

if we permute the derivatives in D1 and D2 with ∇s and we apply the partial integra-tion, then

= ‖D1‖2 +1

ε2‖D2‖2 − 〈∇sBs, D1〉+ 〈∇sC,D2〉

−⟨[∇s,

(1

ε2d∗AdA −∇t∇t +

1

ε2dAd

∗A

)]Bs, Bs

⟩+

⟨1

ε2∗ [Bs, ∗∇sFA] + 2 [C,∇sBt] + d2 ∗Xt(A)[Bs, Bs], Bs

⟩+⟨[∇s,

(ε2∇t∇t − d∗AdA

)]C,C

⟩− 2 〈∗ [Bs ∧ ∗∇sBt] , C〉


The last three lines can be estimates by

∣∣∣∣⟨[∇s,1

ε2d∗AdA

]Bs, Bs

⟩∣∣∣∣ =1

ε2|2 〈[Bs ∧Bs] , dABs〉|

≤cδε2‖Bs‖ · ‖dABs‖ ≤

cδ

ε4‖dABs‖2 + δ‖Bs‖2,

|〈[∇s,∇t∇t]Bs, Bs〉| = |〈[C,∇tBs] , Bs〉+ 〈∇t [C,Bs] , Bs〉|= |2 〈[C,∇tBs] , Bs〉|≤cδ‖C‖ · ‖Bs‖ ≤ cδ‖C‖2 + δ‖Bs‖2,∣∣∣∣⟨[∇s,

1

ε2dAd

∗A

]Bs, Bs

⟩∣∣∣∣ =0,∣∣∣∣⟨ 1

ε2∗ [Bs, ∗∇sFA] , Bs

⟩∣∣∣∣ ≤cδε2‖dABs‖ · ‖Bs‖ ≤

cδ

ε4‖dABs‖2 + δ‖Bs‖2,

|〈2 [C,∇sBt] , Bs〉| ≤cδ‖dABs‖ · ‖C‖ ≤cδ

ε2‖dABs‖2 + ε2δ‖C‖2,∣∣⟨[∇s, ε

2∇t∇t

]C,C

⟩∣∣ =0,

|〈[∇s, d∗AdA]C,C〉| = |2 〈[C, dAC] , Bs〉| ≤ cδ‖dAC‖2,

|〈2 ∗ [Bs ∧ ∗∇sBt] , C〉| ≤cδ‖C‖ · ‖dABs‖ ≤ ε2δ‖C‖2 +cδ

ε2‖dABs‖2,

|〈d2 ∗Xt(A)[Bs, Bs], Bs〉| ≤cδ‖Bs‖2;

we can therefore conclude that

1

2∂2s

(‖Bs‖2 + ε2‖C‖2

)≥2 ‖D1‖2 +

2

ε2‖D2‖2 − cδ

ε4‖dABs‖2

− δ‖Bs‖2 − cδ‖C‖2 + cδ‖dAC‖

≥‖D1‖2 +1

ε2‖D2‖2

where the last step follows from the lemma 92 and choosing δ and ε0 small enoughand thus, we concluded the proof of the identity (14.12).

We concluded therefore the proof of the theorem 91.

Next, we use the notation of the chapter 13.

Theorem 93. We choose four constants b, c0 > 0, p, s1 ≥ 2. There are three positiveconstants ε0, c and ρ such that the following holds. If a perturbed Yang-Mills flowΞ = A + Ψdt + Φds ∈ Mε(Ξ−,Ξ+), with Ξ± = A± + Ψ±dt ∈ CritbYMε,H and0 < ε < ε0, satisfies

‖∂tA− dAΨ‖L4(Σ) + ‖∂sA− dAΦ‖L∞(Σ) ≤ c0, (14.15)

thensup

(t,s)∈S1×[s0,∞)

(‖Bs‖L∞(Σ) + ε‖C‖L∞(Σ)

)≤ ce−ρs0 , (14.16)

163

sup(t,s)∈S1×[s0,∞)

(N1 +N2 +N3 +N4) ≤ cε2e−ρs0 , (14.17)

sup(t,s)∈S1×[s0,∞)

(‖α‖L∞(Σ) + ε‖∇tα‖L∞(Σ)

)≤ ce−ρs0 , (14.18)

sup(t,s)∈S1×[s0,∞)

(‖d∗Aα‖L∞(Σ) + ‖dAα‖L∞(Σ)

)≤ ce−ρs0 , (14.19)

sup(t,s)∈S1×[s0,∞)

(ε‖∇tdAα‖Lp(Σ) + ε2‖∇sd

∗Aα‖Lp(Σ)

)≤ ce−ρs0 , (14.20)

sup(t,s)∈S1×[s0,∞)

(ε‖ψ‖L∞(Σ) + ε‖dAψ‖L∞(Σ) + ε2‖∇tψ‖L∞(Σ)

)≤ ce−ρs0 , (14.21)

sup(t,s)∈S1×[s0,∞)

(ε2‖∇tdAψ‖Lp(Σ) + ε3‖∇sdAφ‖Lp(Σ)

)≤ ce−ρs0 , (14.22)

supS1×[s0,∞)

(‖FA − FA+(s)‖L∞(Σ) + ε‖∇t(FA − FA+(s))‖Lp(Σ)

)≤ cε2− 1

p e−ρs0 ,

(14.23)

supS1×[s0,∞)

(ε1− 1

p‖∇sFA‖Lp(Σ + ε2‖∇t∇t(FA − FA+(s))‖Lp(Σ)

)≤ cε2− 1

p e−ρs0 ,

(14.24)sup

S1×[s0,∞)

(∥∥Bt −B+t

∥∥L∞

+ ε∥∥∇t(Bt −B+

t )∥∥Lp

)≤ ce−ρs0 , (14.25)

supS1×[s0,∞)

ε2∥∥∇s(Bt −B+

t )∥∥Lp≤ ce−ρs0 (14.26)

where s0 > s1 and, for g ∈ G2,20 (Σ× S1 × R) defined by g−1∂sg = Φ,

A+(s) + Ψ+(s)dt := g(s)∗ (A+ + Ψ+dt) ,

α(s) + φ(s)dt := (A(s) + Ψ(s)dt)− (A+(s) + Ψ+(s)dt),

B+t (s) := ∂tA+(s) + dA+(s)Ψ+(s).

Proof. By the estimate (13.12), the theorem 91 and the lemma 6 we know that theretwo constants ρ and c sucht that

sup(t,s)∈S1×[s0,∞)

(‖Bs‖L∞(Σ) + ε‖C‖L∞

)≤ ce−ρs0 ,

sup(t,s)∈S1×[s0,∞)

(N1 +N2 +N3 +N4) ≤ cε2e−ρs0 .(14.27)

Thus, if we integrate the first estimate of 14.27 we have

sup(t,s)∈S1×[s0,∞)

(‖α‖L∞(Σ) + ε‖ψ‖L∞(Σ)

)≤ ce−ρs0


and if we pick s2 ∈ [s0,∞), then the third estimate of the theorem follows from thecomputation

ε‖∇tα(s2)‖L∞(Σ) ≤∫ s2

∞ε‖∇s∇t(A(s)− A+(s)‖L∞(Σ)ds

≤∫ s2

∞ε‖C‖L∞‖A(s)− A+(s)‖L∞(Σ)ds

+

∫ s2

∞ε‖∇t∇s(A(s)− g(s)∗A+)‖L∞(Σ)ds

≤ce−ρs2 +

∫ s2

∞ε‖∇t(∂sA(s) + [Ψ(s), A(s)]

− [Φ(s), g(s)∗A+]− dg(s)∗A+Φ(s))‖L∞(Σ)ds

≤ce−ρs2 +

∫ s2

∞ε‖∇tBs‖L∞(Σ)ds

≤ce−ρs2 + c

∫ s2

∞

(ε‖∇tBs‖L2(Σ) + ε‖d∗AdA∇tBs‖L2(Σ)

)ds

+

∫ s2

∞ε‖dAd∗A∇tBs‖L2(Σ)ds

≤ce−ρs2

where the constant c does not depend on (t, s2) for (t, s2) ∈ S1× [s0,∞). The secondstep of the computation follows from the commutation formula (13.4), the third by thedefinition of g(s) and the previous estimates, the fifth by the lemma 6 and the last oneby 14.27. The estimates (14.19)-(14.22) follows in the same way. Next we prove thefirst part of the (14.23). By (13.11)

sup(t,s)∈S1×[s0,∞)

‖FA‖L∞(Σ) ≤ cε2

and by the Bianchi identity dABs = ∇sFA and by the lemma 6:

sup(t,s)∈S1×[s0,∞)

‖∇sFA‖L∞(Σ) =c sup(t,s)∈S1×[s0,∞)

‖dAd∗AdABs‖L2(Σ) ≤ cεe−ρs0 .

Thus, integrating the last estimate, sup(t,s)∈S1×[s0,∞) ‖FA − FA+‖L∞(Σ) ≤ cεe−ρs0 andhence

sup(t,s)∈S1×[s0,∞)

‖FA − FA+‖pL∞(Σ)

≤cεe−ρs0 sup(t,s)∈S1×[s0,∞)

(‖FA‖L∞(Σ) + ‖FA+‖L∞(Σ)

)p−1

≤cε2p−1e−ρs0

and finally we obtain

sup(t,s)∈S1×[s0,∞)

‖FA − FA+‖L∞(Σ) ≤ cε2− 1p e−

ρps0 .

The other estimates of (14.23)-(14.26) follow in the same way using the Bianchi iden-tity, the Yang-Mills equation (8.13) in order to commute the operators and the estimates(14.27).

L∞-bound for aYang-Mills flow 15

In this chapter we want to show that for any value b > 0, any Yang-Mills flow Ξε ∈Mε(Ξε

−,Ξε+), Ξε

± ∈ CritbYMε,H satisfies the L∞-estimate (15.1). This result allows usto apply the theorem 78 needed in the proof of the surjectivity in the chapter 17.

Theorem 94. We choose a regular value b > 0. Then there are two positive constants cand ε0 such that the following holds. For any ε, 0 < ε < ε0, any Ξε

−, Ξε+ ∈ CritbYMε,H

and any Ξε = Aε + Ψεdt+ Φεds ∈Mε(Ξε−,Ξ

ε+)

‖∂sAε − dAεΦε‖L∞(Σ) + ‖∂tAε − dAεΨε‖L∞(Σ) ≤ c. (15.1)

Proof. Since the two estimates (13.8) and (13.9) in theorem 78 hold also is we assume

sup(t,s)∈S1×R

(‖∂sAε − dAεΦε‖L∞(Σ) + ‖∂tAε − dAεΨε‖L4(Σ)

)≤ c (15.2)

then by the Sobolev estimate for 1-forms on Σ, (15.2) implies (15.1). In order toprove the theorem we assume therefore that there are sequences εν → 0 and Ξν :=Aν + Ψνdt+ Φνds ∈Mεν (Ξεν

− ,Ξεν+ ), Ξεν

± ∈ CritbYMεν ,H such that

mν := sup(t,s)∈S1×R

(‖∂sAν − dAνΦν‖

12

L∞(Σ) + ‖∂tAν − dAνΨν‖L4(Σ)

)→∞; (15.3)

furthermore we assume that there is a sequence (tν , sν) such that∥∥∂sAν(tν , sν)− dAν(tν ,sν)Φν(tν , sν)

∥∥ 12

L∞(Σ)

+∥∥∂tAν(tν , sν)− dAν(tν ,sν)Ψ

ν(tν , sν)∥∥L4(Σ)

= mν .(15.4)

We will consider the following three cases. We denote by ‖ · ‖ the L2-norm on Σ.

Case 1: limν→∞ ενmν = 0. We take the sequence of connections Ξν = Aν + Ψνdt +Φνds defined by

Aν(t, s) := Aν(

1

mν

t+ tν ,1

m2ν

s+ sν

),

Ψν(t, s) :=1

mν

Ψν

(1

mν

t+ tν ,1

m2ν

s+ sν

),

Φν(t, s) :=1

m2ν

Φν

(1

mν

t+ tν ,1

m2ν

s+ sν

);

(15.5)

165

166 15. L∞-bound for a Yang-Mills flow

then Ξν satisfies the Yang-Mills flow equations

∂sAν − dAν Φν +

1

ε2νm

2ν

d∗AνFAν −∇Ψν

t

(∂tA

ν − dAν Ψν)− ∗ 1

m2ν

X 1mν

t+tν(Aν) = 0,

∂sΨν −∇Ψν

t Φν − 1

ε2νm

2ν

d∗Aν(∂tA

ν − dAν Ψν)

= 0.

(15.6)

If we define Bνt := ∂tA

ν − dAν Ψν , Bνs := ∂sA

ν − dAν Φν and Cν := ∂sΨν − ∂tΦν −

[Ψν , Φν ], then we have the following estimates for the norms:

sup(t,s)∈S1×R

(∥∥Bνs

∥∥ 12

L∞(Σ)+∥∥Bν

t

∥∥L4(Σ)

)= 1, (15.7)

∥∥Bνs

∥∥2

L2 =

∫R

∫ mν2

−mν2

∥∥Bνs

∥∥2

L2(Σ)dt ds

=

∫R

∫ 12

− 12

1

m4ν

‖∂sAν − dAνΦν‖2L2(Σ)m

3ν dt ds

=1

mν

‖∂sAν − dAνΦν‖2L2 ≤

c

mν

,

(15.8)

ε2νm

2ν

∥∥Cν∥∥2

L2 =

∫R

∫ mν2

−mν2

∥∥Cν∥∥2

L2(Σ)dt ds

=

∫R

∫ 12

− 12

ε2νm

2ν

m6ν

‖∂sΨν − ∂tΦ− [Ψ,Φ]‖2L2(Σ)m

3ν dt ds

=ε2ν

mν

‖∂sΨν − ∂tΦ− [Ψ,Φ]‖2L2 ≤

c

mν

.

(15.9)

The theorem 78 tell us that for every open interval Ω ⊂ R, 0 ∈ Ω, and every compactset Q ∈ Ω there is a positive constat c such that

sup(t,s)∈S1×Q

(ε2νm

2ν‖Bν

t ‖2 + ‖dAν Bνt ‖2 + ‖d∗Aν B

νt ‖2)

+ sup(t,s)∈S1×Q

(ε2νm

2ν‖Bν

s ‖2 + ‖dAνd∗Aν Bνs ‖2 + ‖dAνd∗Aν B

νs ‖2)

≤c∫S1×Ω

(‖FAν‖2 + ε2

νm2ν‖Bν

t ‖2 + ε2νm

2νc 1

m2νX 1mν

t+tν(Aν)

)dt

+ c

∫S1×Ω

(ε2νm

2ν‖Bν

s ‖2 + ε4νm

4ν‖Cν‖2

)dt

≤cε2νm

2ν

(c 1

m2νX 1mν

t+tν(Aν) +

1

mν

)where for the last inequality we used that∫

S1×Ω

‖FAν‖2dt ≤ mνc sup s ∈ Ω

∫S1

‖FAν‖2dt ≤ cε2νmν .

167

Since∣∣∣∣∣∣∣∣ 1

m2ν

X 1mν

t+tν(Aν)

∣∣∣∣∣∣∣∣L∞

=1

m3ν

‖Xt(Aν)‖L∞ ≤

c

m3ν

=:

(c 1

m2νX 1mν

t+tν(Aν)

) 12

, (15.10)

it follows that

sup(t,s)∈S1×R

‖Bνt ‖L4(Σ) ≤c sup

(t,s)∈S1×R

(‖Bν

t ‖+ ‖dAν Bνt ‖+ ‖d∗Aν B

νt ‖)

≤ c√mν

→ 0 (ν →∞),

sup(t,s)∈S1×R

‖Bνs ‖L∞(Σ) ≤c sup

(t,s)∈S1×R

(‖Bν

s ‖+ ‖dAν Bνs ‖+ ‖d∗Aν B

νs ‖

+ ‖dAνd∗Aν Bνs ‖+ ‖d∗AνdAν B

νs ‖)

≤ c√mν

→ 0 (ν →∞),

and this is a contradiction.

Case 2: limν→∞ ενmν = c1 > 0. We consider the sequence of connections Ξν =Aν + Ψνdt+ Φνds defined by

Aν(t, s) := Aν(ενt+ tν , ε

2νs+ sν

), Ψν(t, s) := ενΨ

ν(ενt+ tν , ε

2νs+ sν

),

Φν(t, s) := ε2νΦ

ν(ενt+ tν , ε

2νs+ sν

);

then Ξν satisfies the Yang-Mills flow equations

∂sAν − dAνΦν + d∗AνFAν −∇

Ψν

t

(∂tA

ν − dAν Ψν)− ε2

ν ∗Xενt+tν (Aν) = 0,

∂sΨν −∇Ψν

t Φν − d∗Aν(∂tA

ν − dAν Ψν)

= 0,

In addition, if define Bνt := ∂tA

ν − dAν Ψν , Bνs := ∂sA

ν − dAν Φν and Cν := ∂sΨν −

∂tΦν − [Ψν , Φν ], for ν →∞, we have that

sup(t,s)∈S1×R

(∥∥Bνs

∥∥ 12

L2(Σ)+∥∥Bν

t

∥∥L2(Σ)

)= cν → c1,

∥∥∂sAν − dAν Φν∥∥2

L2 =

∫R

∫ 12εν

− 12εν

∥∥∂sAν − dAν Φν∥∥2

L2(Σ)dt ds

=

∫R

∫ 12

− 12

ε4ν ‖∂sAν − dAνΦν‖2

L2(Σ)

1

ε3ν

dt ds

=εν ‖∂sAν − dAνΦν‖2L2 ≤ cεν ,∥∥Cν

∥∥2

L2 =

∫R

∫ 12εν

− 12εν

∥∥Cν∥∥2

L2(Σ)dt ds

=

∫R

∫ 12

− 12

ε6ν ‖∂sΨν − ∂tΦν − [Ψν ,Φν ]‖2

L2(Σ)

1

ε3ν

dt ds

=ε3ν ‖∂sΨν − ∂tΦν − [Ψν ,Φν ]‖2

L2 ≤ cεν


We can compute the estimates (13.8) and (13.9) of the theorem 78 also for ε = 1exactly in the same way as we did in the proof of that theorem. Thus, for every openinterval Ω ⊂ R, 0 ∈ Ω, and every compact set Q ∈ Ω there is a positive constat c suchthat

sup(t,s)∈S1×Q

(‖Bν

t ‖2 + ‖dAν Bνt ‖2 + ‖d∗Aν B

νt ‖2)

+ sup(t,s)∈S1×Q

(‖Bν

s ‖2 + ‖dAνd∗Aν Bνs ‖2 + ‖d∗AνdAν B

νs ‖2)

≤c∫S1×Ω

(‖FAν‖2 + ‖Bν

t ‖2 + cε2νXενt+tν (Aν) + ‖Bνs ‖2 + ‖Cν‖2

)dt

≤cεν + c1

ενcε2νXενt+tν (Aν)

where for the last estimate we used that

∫S1×Ω

‖FAν‖2dt ≤ c

ενsups∈Ω

∫S1

‖FAν‖2dt ≤ cεν .

Next, we consider

∣∣∣∣∣∣ε2νXενt+tν

∣∣∣∣∣∣L∞

= ε3ν‖Xt(A)‖L∞ ≤ cε3

ν =:(cε2νXενt+tν (Aν)

) 12,

then

sup(t,s)∈S1×R

‖Bνt ‖L4(Σ) ≤c sup

(t,s)∈S1×R

(‖Bν

t ‖+ ‖dAν Bνt ‖+ ‖d∗Aν B

νt ‖)

≤cε12ν → 0 (ν →∞),

sup(t,s)∈S1×R

‖Bνs ‖L∞(Σ) ≤c sup

(t,s)∈S1×R

(‖Bν

s ‖+ ‖dAν Bνs ‖+ ‖d∗Aν B

νs ‖

+ ‖dAνd∗Aν Bνs ‖+ ‖d∗AνdAν B

νs ‖)

≤ε12ν → 0 (ν →∞),

and this is a contradiction.

Case 3: limν→∞ ενmν = ∞. We consider the substitution (15.5) as in the case 1 andhence the new connection satisfies the Yang-Mills equations (15.6) and the estimates(15.7)-(15.7). In addition, we denote Bν

s := ∂sAν − dAνΦ

ν , Bνt := ∂tA

ν − dAν Ψν ,

Bνs := ∂sA

ν − dAν Φν . We recall that by the computations in the first and in the secondstep of the proof of theorem 78 we have

169

1

2(∂2t − ∂s)‖Bν

t ‖2 =‖∇νt B

νt ‖2 +

1

ε2νm

2ν

‖dAν Bνt ‖2 +

1

ε2νm

2ν

‖d∗Aν Bνt ‖2

+1

ε2νm

2ν

〈∗[Bνt , ∗FAν ], Bν

t 〉 − 〈1

m2ν

d ∗X 1mν

t(Aν)Bν

t

+1

m2ν

X 1mν

t+tν(Aν), Bν

t 〉,

1

2

(∂2t − ∂s

)‖FAν‖2 =‖∇ν

tFAν‖2 +1

ε2νm

2ν

‖d∗AνFAν‖2

+ 〈FAν , [Bνt ∧ Bν

t ]〉 − 〈d∗AνFAν , ∗1

m2ν

X 1mν

t+tν(Aν)〉,

1

2

(∂2t − ∂s

)‖Bν

s ‖2 =‖∇νt Bs‖2 +

1

ε2νm

2ν

‖dAν Bνs ‖+

1

ε2νm

2ν

‖d∗Aν Bνs ‖2

+1

ε2νm

2ν

〈Bνs , [d

∗Aν Bt, B

νt ]〉 − 2

ε2νm

2ν

〈Bνs , ∗[Bν

s , ∗FAν ]〉

− 2〈Bνs ,∇ν

s ∗1

m2ν

X 1mν

t+tν(Aν)〉,

thus for a constant c0 > 0(∂2t − ∂s

)( c0

ε2νm

2ν

‖FAν‖2 + ‖Bνt ‖2 + ‖Bν

s ‖2

)≥− c 1

ε2νm

2ν

‖FAν‖2 − c‖Bνt ‖2 − c‖Bν

s ‖2 − c

m2ν

and hence by the lemma 80 there is, for any open set Ω ⊂ R, 0 ∈ Ω, and any compactinterval Q ⊂ Ω, a positive constant c such that

sup(t,s)∈S1×Q

(‖Bν

s ‖2 + ‖Bt‖2)

≤c∫S1×Ω

(‖Bν

s ‖2 + ‖Bνt ‖2 +

1

ε3νm

3ν

‖FAν‖2 +1

m2ν

)dt ds

≤ c

mν

+ c sups∈Ω

∫S1

‖Bνt ‖2dt ≤ c

mν

.

Analogously, using the computations of the proof of theorem 78 we can obtain thatthere is a constant c0 such that(

∂2t − ∂s

) (‖dAν Bν

t ‖2 + ‖d∗AνdAν Bνs ‖2 + ‖d∗Aν B

νt ‖2 + ‖dAνd∗Aν B

νs ‖2)

+(∂2t − ∂s

)( c0

ε2νm

2ν

‖FAν‖2 + ‖Bνt ‖2 + ‖Bν

s ‖2

)≥− c

ε2νm

2ν

‖FAν‖2 − c‖dAν Bνt ‖2 − c‖d∗AνdAν B

νs ‖2 − c‖Bν

t ‖2

− c‖d∗Aν Bνt ‖2 − c‖dAνd∗Aν B

νs ‖2 − c‖Bν

s ‖2

(15.11)


thus, by the Sobolev estimates for 1-forms on Σ and the lemma 80 we get, for Ω1 =Σ× S1 × Ω,

sup(t,s)∈S1×Q

(‖Bν

t ‖2L4(Σ) + ‖Bν

s ‖2L∞(Σ)

)≤ sup

(t,s)∈S1×Q

(‖Bν

s ‖2 + ‖d∗AνdAν Bνs ‖2 + ‖dAνd∗Aν B

νs ‖2)

+ sup(t,s)∈S1×Q

(‖Bν

t ‖2 + ‖dAν Bνt ‖2 + ‖d∗Aν B

νt ‖2)

≤c∫S1×Ω

(‖d∗AνdAν B

νs ‖2 + ‖dAνd∗Aν B

νs ‖2)dt ds

+ c

∫S1×Ω

(‖dAν Bν

t ‖2 + ‖d∗Aν Bνt ‖2)dt ds+

c

mν

≤c 1

mν

(‖d∗AνdAνBν

s ‖2L2(Ω1) + ‖dAνd∗AνBν

s ‖2L2(Ω1)

)+ cεν‖dAνBν

s ‖2L2(Ω1) +

cε2ν

mν

‖∇tBνs ‖2

L2(Ω1) +c

mν

+ cεν

≤ c

mν

+ cεν → 0

where the last estimate follows from the lemma 79 and the third by

‖dAν Bνt ‖2

L2(Ω1) + ‖d∗Aν Bνt ‖2

L2(Ω1)

≤ c

mν

+ cεν + cεν‖dAνBνs ‖2

L2(Ω1) + cε2ν

mν

‖∇tBνs ‖2

L2(Ω1)

(15.12)

and therefore we have a contradiction. In order to finish the proof of the theorem itremains only to prove (15.12). By the identities

ε2

∥∥∥∥Bνs +

1

ε2d∗AνFAν −∇tB

νt − ∗X 1

mνt+tν

(Aν)

∥∥∥∥2

L2(Σ×S1)

+ ‖∇tFAν − dAν Bνt ‖2

L2(Σ×S1)

=‖Bνs ‖2 +

1

ε2‖d∗AνFAν‖

2L2(Σ×S1)

+ ε2∥∥∇tB

νt

∥∥2

L2(Σ×S1)

+ ε2∥∥∥X 1

mνt+tν

(Aν)∥∥∥2

L2(Σ×S1)+ ‖∇tFAν‖

2L2(Σ×S1) +

∥∥dAν Bνt

∥∥2

L2(Σ×S1)

− 2ε2

⟨∗X 1

mνt+tν

(Aν),1

ε2d∗AνFAν −∇tB

νt

⟩+ 2

⟨Bνs , d

∗AνFAν − ε

2∇tBνt − ε2 ∗X 1

mνt+tν

(Aν)⟩

− 2⟨d∗AνFAν , ε

2∇tBνt

⟩−⟨∇tFAν , dAν B

νt

⟩,

−2⟨d∗AνFAν ,∇tB

νt

⟩− 2

⟨∇tFAν , dAν B

νt

⟩= 2

⟨FAν , [B

νt ∧ Bν

t ]⟩,

2⟨Bνs , d

∗AνFAν − ε

2∇tBνt

⟩= 2

⟨dAν B

νs , FAν

⟩− ε2

⟨∇tB

νs , B

νt

⟩,

171

by the Bianchi identity ∇tFAν − dABt = 0 and by the perturbed Yang-Mills equation(1.26) we have:

‖dAν Bνt ‖2

L2(Ω1) + ‖d∗Aν Bνt ‖2

L2(Ω1) + ε2νm

2ν‖∇tB

νt ‖2

L2(Ω1)

≤ c

m2ν

‖ ∗X 1mν

t+tν(Aν)‖L∞‖FAν‖2

L2(Σ×S1) + ε2ν |〈∗∇tX 1

mνt+tν

(Aν), Bνt 〉|

+ c

∫Ω

(ενmν‖dAν Bν

s ‖2Σ×S1 +

1

ενmν

‖FAν‖2Σ×S1

)ds

+ c

∫Ω

(ε2νm

2ν‖∇tB

νs ‖2

Σ×S1 + sups∈Ω‖Bν

t ‖2Σ×S1

)ds

+ c

∫Ω

(ε2ν‖Bν

s ‖2Σ×S1 + cε2

ν

)ds

+ c

∫Ω

‖FAν‖L2(Σ×S1)(ενmν)− 1

2

(‖Bν

t ‖2L2(Σ×S1) + ‖dAν Bν

t ‖2L2(Σ×S1)

)+ c

∫Ω

‖FAν‖L2(Σ×S1)(ενmν)− 1

2

(‖d∗Aν B

νt ‖2

L2(Σ×S1) + ε2νm

2ν‖∇tB

νt ‖2

L2(Σ×S1)

)≤c εν

m2ν

+ ε2ν + cεν‖dAνBν

s ‖2L2(Ω1) + cεν

+ cε2ν

mν

‖∇tBνs ‖2

L2(Ω1) +c

mν

+ cε2ν

+ cε12ν sups∈Ω‖Bν

t ‖2L2(Σ×S1)

+ cε12ν

(‖dAν Bν

t ‖2L2(Ω1) + ‖d∗Aν B

νt ‖2

L2(Ω1) + ε2νm

2ν‖∇tB

νt ‖2

L2(Ω1)

)where we use the Holder inequality and the Sobolev estimate in the first estimate. Thuschoosing εν small enough

‖dAν Bνt ‖2

L2(Ω1) + ‖d∗Aν Bνt ‖2

L2(Ω1)

≤ c

mν

+ cεν + cεν‖dAνBνs ‖2

L2(Ω1) + cε2ν

mν

‖∇tBνs ‖2

L2(Ω1).

With this last estimate we conclude the discussion of the third case and thus, also theproof of the theorem 94.

Relative Coulombgauge 16

Theorem 95. Assume q ≥ p > 2, q > 4 and qp/(q − p) > 4. We choose Ξ0 =A0 + Ψ0dt+ Φ0ds ∈ A1,p(Ξ−,Ξ+) such that FA0 = 0. Then for every constant c0 > 0there exist constants δ > 0 and c > 0 such that the following holds for 0 < ε ≤ 1. IfΞ ∈ A1,p(Ξ−,Ξ+) satisfies

ε2∥∥d∗εΞ0

(Ξ−Kε2(Ξ0))∥∥Lp≤ c0ε

3p , ‖Ξ−Kε2(Ξ0)‖0,q,ε ≤ δε

3q , (16.1)

then there exists a gauge transformation g ∈ G2,p0 (P × S1 × R) such that

d∗εΞ0(g∗Ξ−Kε2(Ξ0)) = 0

and

‖g∗Ξ− Ξ‖1,p,ε ≤ cε2(

1 + ε−3p ‖Ξ−Kε2(Ξ0)‖1,p,ε

)∥∥d∗εΞ0(Ξ−Kε2(Ξ0))

∥∥Lp. (16.2)

This last theorem is analogous to the proposition 6.2 in [7], but if we compare them,we will remark some differences. The first one is given by different rescaling in thes direction induced by the equations (we have an ε2 factor instead of ε) and this alsoinduces a difference in the Sobolev’s estimates and it causes the change in some expo-nents: 2

p, 2q

and −2p

become 3p, 3q

and −3p. The second difference is an extra ε2 factor in

the first estimate of (16.1) and in (16.2); this is given by the difference in the definitionsof d∗εΞ0

. In [7] it is defined by, with Ξ0 =: A+ Ψdt+ Φds,

d∗εΞ0(α + ψdt+ φds) = d∗Aα− ε2∇tψ − ε2∇sφ, (16.3)

our definition is instead

ε2d∗εΞ0(α + ψdt+ φds) = d∗Aα− ε2∇tψ − ε4∇sφ. (16.4)

The third difference is that we use the difference Ξ − Kε2(Ξ0) instead of Ξ − Ξ0 andthis is needed in order to have finite norms. For any σ ∈ R we define ρσ : R2 → R2

by ρσ(t, s) = (t, s+ σ).

Theorem 96. We choose p > 10 and b > 0. Let Ξ0 ∈ M0(Ξ−,Ξ+), Ξ± ∈ CritbEHwith index difference 1. Then there exist three positive constants ε0, δ and c such thatthe following holds. If 0 < ε < ε0 and Ξ ∈Mε(T ε,b(Ξ−), T ε,b(Ξ+)) such that

‖Ξ−Kε2(Ξ0)‖1,p,ε ≤ δε1− 4p , ε2 ‖∇sΞ‖0,p,ε ≤ cε1+ 7

p (16.5)

173

174 16. Relative Coulomb gauge

then there exist σ ∈ R and g ∈ G2,p0 (P × S1 × R) such that Ξε = g∗ (Ξ ρσ) satisfies

d∗εΞ0(Ξε −Kε2(Ξ0)) = 0, Ξε −Kε2(Ξ0) ∈ im (Dε(Kε2(Ξ0)))∗ (16.6)

and‖Ξε −Kε2(Ξ0)‖1,p,ε ≤ c ‖Ξ−Kε2(Ξ0)‖1,p,ε . (16.7)

Furthermore, for Ξε−Kε2(Ξ0) := αε+ψεdt+φεds and Ξ−Kε2(Ξ0) := α+ψdt+φds,then

‖∇tαε‖Lp ≤ ‖∇tα‖Lp + c ‖Ξ−Kε2(Ξ0)‖1,p,ε , (16.8)

‖∇sαε‖Lp ≤ ‖∇sα‖Lp + c ‖Ξ−Kε2(Ξ0)‖1,p,ε , (16.9)

‖∇tψε‖Lp ≤ ‖∇tψ‖Lp + c ‖Ξ−Kε2(Ξ0)‖1,p,ε , (16.10)

‖∇sψε‖Lp ≤ ‖∇sψ‖Lp + c ‖Ξ−Kε2(Ξ0)‖1,p,ε , (16.11)

‖∇tφε‖Lp ≤ ‖∇tφ‖Lp + c ‖Ξ−Kε2(Ξ0)‖1,p,ε , (16.12)

‖∇sφε‖Lp ≤ ‖∇sφ‖Lp + c ‖Ξ−Kε2(Ξ0)‖1,p,ε . (16.13)

In order to prove the theorem 95 we need the following lemma

Lemma 97. Assume q ≥ p > 2, q > 4, and pq/(q − p) > 4. Given c0 > 0 there existsa constant c > 0 such that, if ‖η‖L∞ ≤ c0 and g = exp(η), then

ε2∥∥d∗εΞ0

(g∗Ξ− Ξ− dΞη)∥∥Lp≤cε−

3q(‖η‖1,q,ε + ‖Ξ−Kε2(Ξ0)‖0,q,ε + ε2

)‖η‖2,p,ε

+ cε−3q(ε2∥∥d∗εΞ0

(Ξ−Kε2(Ξ0))∥∥Lp

+ ε2)‖η‖1,q,ε

(16.14)

and if ‖η‖1,q,ε + ‖Ξ− Ξ0‖0,q,ε ≤ c0ε3q , then

‖g∗Ξ− Ξ‖0,q,ε ≤ c‖η‖1,q,ε, (16.15)

‖g∗Ξ− Ξ‖1,p,ε ≤ c(‖η‖2,p,ε + ε−

2q ‖Ξ− Ξ0‖1,p,ε‖η‖1,q,ε

). (16.16)

Proof. The lemma follows exactly as the lemma 6.6 in [7] using the estimate (11.5).

Proof of theorem 95. We choose Ξ1 = Ξ and we define the sequence Ξν , for ν ≥ 2,by

Ξν+1 = g∗νΞν , gν = exp(ην), d∗εΞ0(dΞ0ην + Ξν −Kε2(Ξ0)) = 0

by the definition of ην and the lemma 6.4 in [7] and the Sobolev theorem 51 we havethat

‖ην‖2,p,ε + ε3p− 3q ‖ην‖1,q,ε ≤ c1ε

2∥∥d∗εΞ0

(Ξν −Kε2(Ξ0))∥∥Lp, (16.17)

‖ην‖1,q,ε ≤ c1‖Ξν −Kε2(Ξ0)‖0,q,ε. (16.18)

In order to conclude the proof of the theorem we need first to show by induction thatthere are three positive constants c2, c3 and c4 such that the following estimates hold.

‖Ξν −Kε2(Ξ0)‖0,q,ε ≤ c2 ‖Ξ−Kε2(Ξ0)‖0,q,ε + c2

∥∥Πim dΞ0(Ξ−Kε2(Ξ0))

∥∥0,p,ε

,

(16.19)

175

ε2∥∥d∗εΞ0

(Ξν −Kε2(Ξ0))∥∥Lp≤c3ε

2− 3q ‖Ξν−1 −Kε2(Ξ0)‖0,q,ε

∥∥d∗εΞ0(Ξν−1 −Kε2(Ξ0))

∥∥Lp

+ c3ε2‖ην−1‖2,p,ε,

(16.20)∥∥d∗εΞ0(Ξν −Kε2(Ξ0))

∥∥Lp≤ 21−ν ∥∥d∗εΞ0

(Ξ−Kε2(Ξ0))∥∥Lp, (16.21)

‖ην‖1,q,ε ≤ c42−ν(‖Ξ−Kε2(Ξ0)‖0,q,ε +


∥∥0,p,ε

). (16.22)

For ν = 1 (16.19) and (16.21) are satisfied by definition and with c2 ≥ 1, (16.18)implies (16.22) for c4 ≥ 2c1 and (16.20) is empty. Next, we consider ν ≥ 2. By theassumptions of the theorem and by (16.18), for δ small enough, we have that

‖ηj‖1,q,ε + ‖Ξj −Kε2(Ξ0)‖0,q,ε ≤ ε3q , j = 1, ..., ν − 1.

By lemma 97 and (16.22)

‖Ξj+1 − Ξj‖0,q,ε ≤c5‖ηj‖1,q,ε

≤c5c42−ν(‖Ξ−Kε2(Ξ)‖0,q,ε +


∥∥0,p,ε

)and thus we have (16.19). Next,

d∗εΞ0(Ξν+1 −Kε2(Ξ0)) = d∗εΞ0

(g∗νΞν − Ξν − dΞ0ην)

=d∗εΞ0(g∗νΞν − Ξν − dΞνην) +

[d∗εΞ0

(Ξν −Kε2(Ξ0)) ∧ ην]

+[d∗εΞ0

(Kε2(Ξ0)− Ξ0) ∧ ην]− ∗ε [∗ε (Ξν −Kε2(Ξ0)) ∧ dΞ0ην ]

− ∗ε [∗ε (Kε2(Ξ0)− Ξ0) ∧ dΞ0ην ]

and hence by the lemma (16.17) and by (16.18), (16.20), we can conclude (16.20). By(16.17) and (16.20) we get

‖ην‖2,p,ε + ε3p− 3q ‖ην‖1,q,ε

≤c1c3ε2− 3

q ‖Ξν−1 −Kε2(Ξ0)‖0,q,ε

∥∥d∗εΞ0(Ξν−1 −Kε2(Ξ0))

∥∥Lp

+ c2c3ε2− 3

p‖dΞ0ην‖0,p,ε

and hence

‖ην‖2,p,ε + ε3p− 3q ‖ην‖1,q,ε

≤c1c3ε− 3q ‖Ξν−1 −Kε2(Ξ0)‖0,q,ε

∥∥d∗εΞ0(Ξν−1 −Kε2(Ξ0))

∥∥Lp

+ c2c3ε2− 3

p‖ΠdΞ0(Ξν−1 −Kε2(Ξ0))‖0,p,ε

and

ε2∥∥d∗εΞ0

(Ξν −Kε2(Ξ0))∥∥Lp

≤2c3ε2− 3

q ‖Ξν−1 −Kε2(Ξ0)‖0,q,ε

∥∥d∗εΞ0(Ξν−1 −Kε2(Ξ0))

∥∥Lp

+ c2c3ε2− 3

p‖ΠdΞ0(Ξν−1 −Kε2(Ξ0))‖0,p,ε.

(16.23)


By (16.18) and (16.19)

‖η2‖1,q,ε ≤ c0c3 ‖Ξ−Kε2(Ξ0)‖0,q,ε

using (16.23)

‖ην‖1,q,ε ≤16c1c2c23δ2−ν ‖Ξν−1 −Kε2(Ξ0)‖0,q,ε

+ c2c3ε2− 3

p‖ΠdΞ0(Ξν−1 −Kε2(Ξ0))‖0,p,ε.

(16.22) therefore holds for c4 = 1 whenever δ and ε are small enough. The lemma 97with (16.17) and (16.18) implies

‖Ξν+1 − Ξν‖1,p,ε ≤ c6ε2(

1 + ε−3p ‖Ξν −Kε2(Ξ0)‖1,p,ε

)∥∥d∗εΞ0(Ξν − Ξ0)

∥∥Lp

and thus, for δ sufficiently small,

‖Ξν − Ξ0‖1,p,ε ≤ ε3p + 2 ‖Ξ−Kε2(Ξ0)‖1,p,ε .

The sequence converges therefore in W 1,p to a connection Ξε which satisfies the con-dition d∗εΞ0

(Ξε − Ξ0) = 0 and the estimate (16.2). In addition, the sequence hν :=

g1g2...gν satisfies h∗νΞ = Ξν and converges in G2,p0 (P × S1 × R) to a gauge transfor-

mation g which satisfies g∗Ξ = Ξε.

Proof of theorem 96. We follows the proof of the proposition 6.3 in [7] adapting it forour needs. First, we consider the estimates

‖Ξ τσ −Kε2(Ξ0)‖1,p,ε ≤‖Ξ−Kε2(Ξ0)‖1,p,ε + ‖Kε2(Ξ0) τσ −Kε2(Ξ0)‖1,p,ε

≤‖Ξ−Kε2(Ξ0)‖1,p,ε + |σ| · ‖∂sKε2(Ξ0)‖1,p,ε ,

‖∇s (Ξ τσ −Kε2(Ξ0))‖0,p,ε ≤‖∇s (Ξ−Kε2(Ξ0))‖0,p,ε + c|σ| · ‖Ξ−Kε2(Ξ0)‖Lp+ ‖∇s (Kε2(Ξ0) τσ)−∇sKε2(Ξ0)‖0,p,ε

≤‖∇s (Ξ−Kε2(Ξ0))‖0,p,ε + c|σ| · ‖Ξ−Kε2(Ξ0)‖Lp+ c|σ| · ‖∂s∇sKε2(Ξ0)‖0,p,ε ,

‖∇t (Ξ τσ −Kε2(Ξ0))‖0,p,ε ≤‖∇t (Ξ−Kε2(Ξ0))‖0,p,ε + c|σ| · ‖Ξ−Kε2(Ξ0)‖Lp+ ‖∇t (Kε2(Ξ0) τσ)−∇tKε2(Ξ0)‖0,p,ε

≤‖∇t (Ξ−Kε2(Ξ0))‖0,p,ε + c|σ| · ‖Ξ−Kε2(Ξ0)‖Lp+ c|σ| · ‖∂s∇tKε2(Ξ0)‖0,p,ε ,

(16.24)

where, by the definitions of the chapter 11,

‖∂sKε2(Ξ0)‖1,p,ε ≤‖∂sΞ0‖1,p,ε + ‖∇s(Kε2(Ξ0)− Ξ0)‖1,p,ε

≤‖∂sΞ0‖1,p,ε + cε2,

‖∂s∇sKε2(Ξ0)‖0,p,ε ≤‖∂s∇sΞ0‖0,p,ε + ‖∇s∇s(Kε2(Ξ0)− Ξ0)‖Lp+ ‖∇s(Kε2(Ξ0)− Ξ0)‖Lp + cε2 ≤ c,

‖∂s∇tKε2(Ξ0)‖0,p,ε ≤‖∂s∇tΞ0‖0,p,ε + ‖∇s∇t(Kε2(Ξ0)− Ξ0)‖Lp+ ‖∇t(Kε2(Ξ0)− Ξ0)‖L∞ + cε2 ≤ c.

(16.25)

177

Therefore for |σ| ≤ δε3p , ‖Ξ τσ −Kε2(Ξ0)‖1,p,ε ≤ δε

3p for ε small enough, and thus

by theorem 95, there is a gauge transformation gσ ∈ G2,p0 (P × S1 × R) such that for

Ξσ = g∗σ (Ξ τσ), d∗εΞ0(Ξσ −Kε2(Ξ0)) = 0 and

‖Ξσ −Kε2(Ξ0)‖1,p,ε ≤ c1

(|σ|+ ‖Ξ−Kε2(Ξ0)‖1,p,ε

).

We assume that g0 = 1. We need to show that there is a σ such that Ξσ − Kε2(Ξ0) ∈imDε(Kε2(Ξ0))∗ and |σ| ≤ c2 ‖Ξ−Kε2(Ξ0)‖1,p,ε. The estimates (16.8)-(16.13) followsas (16.24) and (16.25) computing separately every component.

D0(Ξ0) is onto and has index 1; therefore its kernel is spanned by ξ0 = ∂sΞ0 ∈ W 1,p.Then Dε(Kε2(Ξ0)) has index 1 with the kernel spanned by

ξε = ξ0 −Dε(Kε2(Ξ0))∗ (Dε(Kε2(Ξ0))Dε(Kε2(Ξ0))∗)−1Dε(Kε2(Ξ0))ξ0.

Consider now the function θ(σ) = 〈ξε,Ξσ − Kε2(Ξ0)〉ε; thus, if θ(σ) = 0, then Ξσ −Kε2(Ξ0) ∈ imDε(Kε2(Ξ0))∗. We assume that there are positive constants δ0, ε0 and ρ0

such that, for 0 < ε < ε0,

|σ|+ ‖Ξ−Kε2(Ξ0)‖1,p,ε ≤ δ0ε1− 4

p ⇒ θ′(σ) ≥ ρ0. (16.26)

Then the existence of a zero for θ(σ) follows from

|θ(0)| = |〈ξε,Ξ−Kε2(Ξ0)〉ε| ≤ ‖ξε‖0,q,ε ‖Ξ−Kε2(Ξ0)‖0,p,ε ≤ c3δε

1− 4p

where q = pp−1

. In fact, if c3δ <12δ0ρ0, δ ≤ 1

2δ0, then ‖Ξ−Kε2(Ξ0)‖1,p,ε ≤

12δ0ε

1− 4p .

Therefore, by (16.26), there is a σ ∈ R with |σ| ≤ θ(0)ρ0≤ 1

2δ0ε

1− 4p such that θ(σ) = 0.

For this σ, we have

|σ| ≤ c ‖Ξ−Kε2(Ξ0)‖1,p,ε , Ξσ −Kε2(Ξ0) ∈ imDε(Kε2(Ξ0))∗.

Thus, in order to finish the proof of the theorem we need only to show (16.26).

Proof of (16.26). We define ησ = g−1σ (∂σgσ − ∂sgσ), then

θ′(σ) = 〈ξε, ∂sΞσ + dΞσησ〉ε, ∂σd∗εΞ0

(Ξσ −Kε2(Ξ0)) = 0. (16.27)

Thus,d∗εΞ0

∂sΞσ + d∗εΞ0dΞ0ησ + d∗εΞ0

[(Ξσ − Ξ0) ∧ ησ] = 0. (16.28)

If ε−3p ‖Ξσ −Kε2(Ξ0)‖1,p,ε and ‖Kε2(Ξ0)− Ξ0‖1,∞,ε are sufficiently small, then there is

a unique ησ which satisfies (16.28), furthermore

‖ησ‖1,p,ε ≤c ‖∂sΞσ‖0,p,ε ≤ c(

1 + ‖∂s (Ξσ − Ξ0)‖0,p,ε

)≤c(

1 + ‖∇s (Ξσ −Kε2(Ξ0))‖0,p,ε + ‖Ξσ −Kε2(Ξ0)‖0,p,ε

).

(16.29)


where the last step follows from the definition of Kε2 and the estimate (11.5). Sinced∗εΞ0

ξε = 0, we have

|〈ξε, dΞσησ〉ε| = |〈ξε, [(Ξσ − Ξ0) ∧ ησ]〉ε|≤c ‖Ξσ − Ξ0‖∞,ε ‖ησ‖0,p,ε

≤cε−3p ‖Ξσ − Ξ0‖1,p,ε

(1 + ‖∇s (Ξσ −Kε2(Ξ0))‖0,p,ε

)+ cε−

3p ‖Ξσ − Ξ0‖1,p,ε ‖Ξσ −Kε2(Ξ0)‖0,p,ε

≤cδ + cδε−3p ε1− 4

p ‖∇s (Ξσ −Kε2(Ξ0))‖0,p,ε .

where the second inequality follows from the Sobolev theorem 51 and from (16.29)and the last one is a consequence of the assumptions. Next we consider

〈ξε, ∂sΞσ〉ε = 〈∂sξε,Kε2(Ξ0)− Ξσ〉ε + 〈ξε, ∂sKε2(Ξ0)〉ε,

Since ‖∂sΞ0‖0,2,ε ≥ 3ρ0 > 0 for some ρ0, 〈ξε, ∂sKε2(Ξ0)〉ε ≥ 2ρ0 by the definitionof ξε and thus 〈ξε, ∂sΞσ + dΞσησ〉ε > ρ0 for δ0 small enough. Hence, by (16.27)θ′(σ) > ρ0.

Surjectivity of Rε,b 17In this chapter we will show that the mapRε,b defined in the chapter 12 is also surjec-tive.

Theorem 98. We assume that the energy functionalEH is Morse-Smale and we chooseb > 0 to be a regular value of EH . Then there is a constant ε0 > 0 such that thefollowing holds. For every ε ∈ (0, ε0), every pair Ξ± := A± + Ψ±dt ∈ CritbEH , themap

Rε,b :M0 (Ξ−,Ξ+)→Mε(T ε,b(Ξ−), T ε,b(Ξ+)

)(17.1)

is surjective.

Proof. We prove the theorem indirectly. We assume that there is a sequence Ξν ∈Mεν

(T εν ,b(Ξ−), T εν ,b(Ξ+)

), εν → 0, that is not in the image of Rεν ,b. Hence by the

theorems 78 and 94, for a positive constant c0,

‖∂sAν − dAνΦν‖L∞(Σ) + εν‖∂sΨ− ∂tΦ− [Ψ,Φ]‖L∞(Σ) ≤ c0,

ε2ν‖∂tAν − dAνΨν‖L∞(Σ) + ‖FAν‖L∞(Σ) + ‖dAνd∗AνFAν‖L2(Σ) ≤ c0ε

2ν

(17.2)

and all the estimates of the theorem 93. The proof is structured in the following way.In the first step, see figure 17.1, we will define a sequence Ξν

1 which converges to Ξ±for s→ ±∞ and in the following step we will project it on the space A0(P ) defininga new sequence Ξν

2; then by the implicit function theorem 100 there is a subsequenceof Ξν

2 which converges to a geodesic flow (step 3). Finally, after choosing appropri-ate gauge transformations and time shifts (steps 4-6) we can show that the sequencesatisfies the assumptions of the local uniqueness theorem 76 (step 7) and therefore asubsequence turns out to lie in the image ofRε,b. This yields to a contradiction.

FA = 0Ξ− Ξ+

T εν ,b(Ξ−)

T εν ,b(Ξ+)

Ξν2

αν− + ψν−dtαν+ + ψν+dt

Ξν1

Ξν

Figure 17.1: Idea of the proof of theorem 98.

179

180 17. Surjectivity ofRε,b

Furthermore, we assume that there is a positive S0 such that Φν = 0 for |s| ≥ S0. Inthe general case, the Φν converge to 0 for |s| → ∞ in an exponential way and hencewe can find a sequence gνν∈N of gauge transformations such that g∗νΞ

ν has the aboveproperty. First, we pick the sequence gνν∈N of gauge transformations defined byg−1ν ∂sgν := Φν and we define

Aν±(s) + Ψν±(s)dt := g(s)∗(Aν± + Ψν

±dt),

for Ξν± := Aν± + Ψν

±dt := lims→±∞ Ξν . Second, like in the chapter 12 we choosea smooth positive function θ(s) = 0 for s ≤ 1 and θ(s) = 1 for s ≥ 2, such that0 ≤ θ ≤ 1 and 0 ≤ ∂sθ ≤ c0 with c0 > 0 and we define a family of 1-forms αν0 +ψν0dtas

ξν0 = αν0 + ψν0dt :=θ(−s)(

(Aν− + Ψν−dt)−

(T εν ,b

)−1(Aν− + Ψν

−dt))

+ θ(s)(

(Aν+ + Ψν+dt)−

(T εν ,b

)−1(Aν+ + Ψν

+dt))

;(17.3)

in addition we denote

αν + ψνdt+ φνds := θ(−s)(Ξν − Ξν

−)

+ θ(s)(Ξν − Ξν

+

)which satisfies the uniformly exponential convergence estimates of the theorem 93.

Step 1. We define Ξν1 = Aν1 + Ψν

1dt+ Φν1ds := Ξν − ξν0 then there is a constant c > 0

such that

‖∂tAν1 − dAν1 Ψν1‖L∞(Σ) + ‖∂sAν1 − dAν1 Φν

1‖L∞(Σ) ≤ c (17.4)

‖FAν1‖L∞(Σ) + ε2ν

∥∥∥∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)∥∥∥Lp(Σ)

≤ cε2 (17.5)

and for s ≥ 0

∂sAν1 − dAν1 Φν

1 −∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)− ∗Xt(A

ν1)

=− 1

ε2ν

d∗Aν1dA−αν0,+ +

[ψν0 ,(

(∂tAν − dAνΨν)−

(∂tA

ν+ − dAν+Ψν

+

))]− 1

ε2ν

d∗Aν(FAν − FAν+

)+

1

ε2ν

∗[αν ,

(dA+

(αν0 − αν0,+

)+

1

2[αν0 ∧ αν0 ]

)]+ (∗Xt(A+)− ∗Xt(A

ν1)) +

(∗Xt(A

ν)− ∗Xt(Aν+))

+[ψν ,((∂tA

ν − dAνΨν)−(∂tA

ν1 − dAν1 Ψν

1

))]+∇Ψ+

t ([ψν , αν0 ]− [αν , ψν0 ])

where αν0,±(s) ∈ im d∗A±(s) are defined uniquely by

d∗A±dA±αν0,± = ε2

ν∇Ψ±t

(∂tA± − dA±Ψ±

).

Proof of step 1. The estimates (17.4) and (17.5) follow from (17.2), from the inequal-ities of the theorem 24 and the Sobolev theorem 14:

‖∂sAν1 − dAν1 Φν1‖L∞(Σ) = ‖∂sAν − dAνΦν‖L∞(Σ) ≤ c,

181

‖∂tAν1 − dAν1 Ψν1‖L∞(Σ) ≤‖∂tAν − dAνΨν‖L∞(Σ)

+ ‖∇Ψν

t αν0‖L∞(Σ) + ‖dAν1ψν0‖L∞(Σ) ≤ c,

‖FAν1‖L∞(Σ) ≤ ‖FAν1‖L∞(Σ) + ‖dAν1αν0‖L∞(Σ) + c‖αν0‖L∞(Σ) ≤ c,

∥∥∥∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)∥∥∥Lp(Σ)

≤∥∥∇Ψν

t (∂tAν − dAνΨν)

∥∥Lp(Σ)

+ c

≤ 1

ε2ν

‖d∗AνFAν‖Lp(Σ) + ‖∂sAν − dAνΨν‖Lp(Σ) + c

≤c

where for the last estimate we use also the Yang-Mills flow equation (8.13). In orderto prove the identity, we first remark that


1 −∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)− ∗Xt(A

ν1)

=∂sAν − dAνΦν −∇Ψν

t (∂tAν − dAνΨν)− ∗Xt(A

ν)

+ ∗Xt(Aν)− ∗Xt(A

ν1) + [ψν0 , (∂tA

ν − dAνΨν)]

+∇Ψν1t

((∂tA


ν1 − dAν1 Ψν

1

));

(17.6)

next, in order to simplify the exposition, we consider s ≥ 0, for a negative s the proofis analogous. Since Ξν is a Yang-Mills flow, we have


1 −∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)− ∗Xt(A

ν1)

=− 1

ε2ν

d∗AνFAν +[ψν0 ,(∂tA

ν+ − dAν+Ψν

+

)]+[ψν0 ,(


(∂tA

ν+ − dAν+Ψν

+

))]+[ψν ,((∂tA


ν1 − dAν1 Ψν

1

))]+∇Ψ+

t

((∂tA


ν1 − dAν1 Ψν

1

))+ (− ∗Xt(A

ν1) + ∗Xt(A

ν)) .

Furthermore, since[ψν0 ,(∂tA

ν+ − dAν+Ψν

+

)]+∇Ψ+

t

((∂tA


ν1 − dAν1 Ψν

1

))= +∇Ψν+

t

(∂tA

ν+ − dAν+Ψν

+

)−∇Ψ+

t

(∂tA+ − dA+Ψ+

)+∇Ψ+

t ([ψν , αν0 ]− [αν , ψν0 ])

=1

ε2ν

d∗Aν+FAν+− ∗Xt(A

ν+)− 1

ε2ν

d∗A+dA+α

ν0,+

+ ∗Xt(A+) +∇Ψ+

t ([ψν , αν0 ]− [αν , ψν0 ])

where the last identity follows from the equations for the perturbed geodesics (1.16)


and for the perturbed Yang-Mills connections (1.26). Thus,


1 −∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)− ∗Xt(A

ν1)

=− 1

ε2ν

d∗AνFAν +1

ε2ν

d∗Aν+FAν+− 1

ε2ν

d∗A+dA+α

ν0,+

+∇Ψ+

t ([ψν , αν0 ]− [αν , ψν0 ])

+[ψν0 ,(


(∂tA

ν+ − dAν+Ψν

+

))]+ (− ∗Xt(A

ν1) + ∗Xt(A+)) +

(∗Xt(A

ν)− ∗Xt(Aν+))

+[ψν ,((∂tA


ν1 − dAν1 Ψν

1

))]Next, the first step then follows directely using the identities

d∗AνFAν = d∗Aν (FAν − FAν+)− ∗[αν , ∗

(dA+α

ν0 +

1

2[αν0 ∧ αν0 ]

)]+ d∗Aν+FA

ν+,

− 1

ε2ν

d∗A+dA+α

ν0,+ = − 1

ε2ν

d∗Aν1dA+αν0,+ −

1

ε2ν

∗ [αν , ∗dA+αν0,+].

Since FAν1 + dAν1αν0 + 1

2[αν0 ∧ αν0 ] = FAν , dA+α

ν0 + 1

2[αν0 ∧ αν0 ] = FAν+ for s ≥ 2, we

haveFAν1 = FAν − FAν+ − [αν ∧ αν0 ] , for s ≥ 2, (17.7)

and thus we can estimate the norm of FAν1 , for any q ≥ 2, by∥∥FAν1∥∥Lq(Σ)≤∥∥∥FAν − FAν+∥∥∥Lq(Σ)

+ ‖[αν ∧ αν0 ]‖Lq(Σ) (17.8)

for s ≥ 2.

Step 2. There are two positive constants c and δ such that the following holds. Forevery Ξν there is a connection Ξν

2 := Aν2 + Ψν2dt + Φν

2ds = Ξν1 + αν1 + ψν1dt + φν1ds,

αν1 ∈ im d∗Aν1 , which satisfies

i) FAν2 = 0, ii) d∗Aν2 (∂tAν2 − dAν2 Ψν

2) = 0,

iii) d∗Aν2 (∂sAν2 − dAν2 Φν

2) = 0, iv) ‖αν1(s)‖L∞(Σ) ≤ ce(θ(−s)−θ(s))δsε2− 1

pν

v)∥∥πAν2 (F0 (Ξν

2))∥∥Lp≤ cε

1− 1p

ν , vi)∥∥πAν2 (F0 (Ξν

2))∥∥Lp(Σ)

≤ cε1− 1

pν ,

vii) lims→±∞ Ξν2 = Ξ±.

Proof of step 2. By the the first step we know that ‖FAν1‖L∞ ≤ cε2 and thus, for εsmall enough, the lemma 7 allows us to find a positive constant c such that for any Aν1there is a unique 0-form γν such that

FAν1+∗dAν1 γν = 0,

∥∥dAν1γν∥∥L∞(Σ)≤ c

∥∥FAν1∥∥L∞(Σ)≤ cε2

ν . (17.9)

183

and Ψν2 , Φν

2 are then uniquely given by

d∗Aν2

(∂tA

ν2 − dAν2 Ψν

2

)= 0, d∗Aν2

(∂sA

ν2 − dAν2 Φν

2

)= 0. (17.10)

Therefore i), ii) and iii) are satisfied. By (17.8), (17.9) one can remark that

‖αν1(s)‖L∞(Σ) ≤ ce(θ(−s)−θ(s))δsε2− 1

pν

using the a priori estimate of the theorem 93. In order to show the inequalities v) andvi) of the second step, we consider


2 −∇Ψν2t

(∂tA

ν2 − dAν2 Ψν

2

)− ∗Xt(A

ν2)

=∂sAν1 − dAν1 Φν

1 −∇Ψν1t

(∂tA

ν1 − dAν1 Ψν

1

)− ∗Xt(A

ν1)

+∇Φν1s αν1 − dAν1φ

ν1 − [αν1 , φ

ν1]−

[ψν1 ,(∂tA

ν1 − dAν1 Ψν

1

)]−∇Ψν1

t

(∇Ψν1t αν1 − dAν1ψ

ν1 − [αν1 , ψ

ν1 ])− (∗Xt(A

ν2)− ∗Xt(A

ν1))

and we remark that

∇Ψν1t ∇

Ψν1t αν1 =∇Ψν1

t ∇Ψν1t ∗ dAν1γ

ν

= ∗ dAν2∇Ψν1t ∇

Ψν1t γν + ∗∇Ψν1

t

[(∂tA

ν1 − dAν1 Ψν

1

), γν]

+ ∗[(∂tA

ν1 − dAν1 Ψν

1

),∇Ψν1

t γν]− ∗

[αν1 ,∇

Ψν1t ∇

Ψν1t γν

],

∇Ψν1t dAν1ψ

ν1 = dAν2∇

Ψν1t ψν1 −

[αν1 ,∇

Ψν1t ψν1

]+[(∂tA

ν1 − dAν1 Ψν

1

), ψν1],

∇Φν1s αν1 = ∗dAν2∇

Φν1s γν + ∗

[(∂sA

ν1 − dAn1uΦ

ν1

), γν]

+ ∗[αν1 ,∇Φν1

s γν].

Using the first step, the next lemma and the the uniformly exponential convergenceestimates of the theorem 93, we can conclude that∥∥∥πAν2 (∂sAν2 − dAν2 Ψν

2 −∇Ψν2t

(∂tA

ν2 − dAν2 Ψν

2

)− ∗Xt(A

ν2))∥∥∥

Lp≤ cε

1− 1p

ν ,∥∥∥πAν2 (∂sAν2 − dAν2 Ψν2 −∇

Ψν2t

(∂tA

ν2 − dAν2 Ψν

2

)− ∗Xt(A

ν2))∥∥∥

Lp(Σ)≤ cε

1− 1p

ν .

Lemma 99. There are two positive constants c and ε0 such that

εν

∥∥∥∇Ψν2t αν1

∥∥∥Lp

+ εν

∥∥∥∇Ψν2t γν

∥∥∥Lp

+ ε2ν

∥∥∥∇Ψν2t ∇

Ψν2t γν

∥∥∥Lp≤ cε

2− 1p

ν ,

εν


∥∥∥Lp(Σ)

+ εν

∥∥∥∇Ψν2t γν

∥∥∥Lp(Σ)

+ ε2ν

∥∥∥∇Ψν2t ∇

Ψν2t γν

∥∥∥Lp(Σ)

≤ cε2− 1

pν ,∥∥dAν2φν1∥∥Lp +

∥∥∇Φν2s αν1

∥∥Lp

+∥∥∇Φν2

s γν∥∥Lp≤ cεν ,∥∥dAν2φν1∥∥Lp(Σ)

+∥∥∇Φν2

s αν1∥∥Lp(Σ)

+∥∥∇Φν2

s γν∥∥Lp(Σ)

≤ cεν ,

‖ψν1‖L∞(Σ) + ‖dAν2ψν1‖L∞(Σ) + εν

∥∥∥∇Ψν2t ψν1

∥∥∥Lp(Σ)

+ εν

∥∥∥∇Ψν2t ψν1

∥∥∥Lp≤ cε

1− 1p

ν ,∥∥∂sAν2 − dAν2 Φν2

∥∥Lp(Σ)

≤ ceδ(θ(−s)−θ(s))s,∥∥∂tAν2 − dAν2 Ψν

2

∥∥Lp(Σ)

≤ c,

for any 0 < εν < ε0.


Proof of lemma 99. The first estimate of the lemma can be obtained analogously as(6.7) in fact in exactly the same way we have, by the theorem 78∥∥∥dAν1 ∗ dAν1∇Ψν1

t γν∥∥∥L2(Σ)

≤ c

(∥∥∥∇Ψν1t FAν1

∥∥∥L2(Σ)

+ ‖αν1‖L∞(Σ)

)≤ cε

1− 1p

ν (17.11)

or by the theorem 93 for |s| ≥ 2∥∥∥dAν1 ∗ dAν1∇Ψν1t γν

∥∥∥L2(Σ)

≤c(∥∥∥∇Ψν1

t FAν1

∥∥∥L2(Σ)

+ ‖αν1‖L∞(Σ)

)≤c∥∥∥∇Ψν1

t

(FAν − FAν+

)∥∥∥L2(Σ)

+ c

(∥∥∥∇Ψν1t [α ∧ αν0 ]

∥∥∥L2(Σ)

+ ‖αν1‖L∞(Σ)

)≤ce(θ(−s)−θ(s))δsε

1− 1p

ν ,

(17.12)

where for the second estimate we used (17.7) and for the last one from the theorem 93.Analogously, if we derive twice by∇Ψν1

t the formula

dAν1 ∗ dAν1γν = FAν1 −

1

2

[∗dAν1γ

ν ∧ ∗dAν1γν],

then we obtain ∥∥∥dAν1 ∗ dAν1∇Ψν1t ∇

Ψν1t γν

∥∥∥L2(Σ)

≤ cε− 1p

ν , (17.13)∥∥∥dAν1 ∗ dAν1∇Ψν1t ∇

Ψν1t γν

∥∥∥L2(Σ)

≤ ce(θ(−s)−θ(s))δsε− 1p

ν . (17.14)

In fact since∥∥∥∇Ψν1t ∇

Ψν1t FAν1

∥∥∥L2(Σ)

=∥∥∥∇Ψν1

t ∇Ψν1t

(FAν − FAν+ − [αν ∧ αν0 ]

)∥∥∥L2(Σ)

,

∥∥∥∇Ψν1t ∇

Ψν1t FAν1

∥∥∥L2(Σ)

≤ c

by theorem 78 and by the estimates on the 1-forms α0 +ψ0dt and α1 and if we considerthe estimates of the theorem 93 we get∥∥∥∇Ψν1

t ∇Ψν1t FAν1

∥∥∥L2(Σ)

≤ ce(θ(−s)−θ(s))δsε− 1p

ν .

Thus , we can also conclude that by (17.11) and (17.13) and the commutation formula(1.20)

εν


∥∥∥Lp(Σ)

+ ε2ν

∥∥∥∇Ψν1t ∇

Ψν1t αν1

∥∥∥Lp(Σ)

≤ cε2− 1

pν (17.15)

and by (17.12) and (17.14) and the commutation formula (1.20)

εν


∥∥∥Lp(Σ)

+ ε2ν


∥∥∥Lp(Σ)

≤ ce(θ(−s)−θ(s))δsε2− 1

pν .

185

Next, we can derive

FAν + dAν(∗dAν1γ

ν − αν0)

+1

2

[(∗dAν1γ

ν − αν0)∧(∗dAν1γ

ν − αν0)]

= 0

by ∇s and we obtain

dAν ∗ dAν∇Ψν

s γν =−∇Ψν

s FAν − dAν∇Ψν

s ∗ [αν0 , γν ]

+ [(∂sAν − dAνΦν) , αν0 ]

− [(∂sAν − dAνΦν) ∧ ∗dAνγν ]− dAν ∗ [(∂sA

ν − dAνΦν) , γν ]

−[∗(∇Ψν

s dAν1γν)∧(∗dAν1γ

ν − αν0)]

and thus we can conclude that∥∥dAν ∗ dAν∇Ψν

s γν∥∥L2(Σ)

≤ c ‖∇sFAν‖L2(Σ) + cε2− 1

pν eδs ≤ cενe

θ(−s)δse−θ(s)δs

by theorem 93 and hence ‖∇sαν1‖Lp ≤ cεν . In order to prove the other estimates we

need to use the definitions of Ψ2 and Φ2 and to expand the identity. On the one hand,

0 =d∗Aν2

(∂tA

ν2 − dAν2 Ψν

2

)=− d∗Aν2dAν2ψ

ν1 − ∗

[αν1 ∧ ∗

(∂tA

ν1 − dAν1 Ψν

1

)]+ d∗Aν2∇

Ψν1t αν1 + d∗Aν1

(∂tA

ν1 − dAν1 Ψν

1

);

(17.16)

where the last term can be written in the following way:

d∗Aν1

(∂tA

ν1 − dAν1 Ψν

1

)=d∗Aν (∂tA

ν − dAνΨν) + d∗Aν1

(∇Ψν

t αν0 − dAνψν0 − [αν0 , ψν0 ])

− ∗ [αν0 ∧ ∗ (∂tAν − dAνΨν)]

=d∗Aν (∂tAν − dAνΨν)− ∗

[αν ∧ ∗

(∇Ψν−t αν0 − dAν−ψ

ν0 − [αν0 , ψ

ν0 ])]

+ d∗Aν−

(−∇Ψν−

t αν0 + dAν−ψν0 − [αν0 , ψ

ν0 ])

+ ∗[αν0 ∧ ∗

(∂tA

ν− − dAν−Ψν

−

)]+ ∗

[αν0 ∧ ∗

(−∇Ψν−

t αν0 + dAν−ψν0 − [αν0 , ψ

ν0 ])]

+ d∗Aν1 (−[ψν , αν0 ] + [αν , ψν0 ])

+ ∗[αν0 ∧ ∗

((∂tA



−

))]where the second and the third line of the last expression vanish because they can bewritten as

d∗Aν−

(∂tA


−

)− d∗A−

(∂tA− − dA−Ψ−

)= 0

by the equations for the perturbed geodesics (1.17) and for the perturbed Yang-Millsconnections (1.27). Thus

d∗Aν1

(∂tA

ν1 − dAν1 Ψν

1

)=d∗Aν (∂tA

ν − dAνΨν)− ∗[αν ∧ ∗

(∇Ψν−t αν0 − dAν−ψ

ν0 − [αν0 , ψ

ν0 ])]

+ d∗Aν1 (−[ψν , αν0 ] + [αν , ψν0 ])

+ ∗[αν0 ∧ ∗

((∂tA



−

))]=:d∗Aν (∂tA

ν − dAνΨν) +Dν


and ∥∥∥d∗Aν1 (∂tAν1 − dAν1 Ψν1

)∥∥∥Lp≤ cε

1− 1p

ν . (17.17)

Furthermore for the term d∗Aν2∇Ψν1t αν1 we have

d∗Aν2∇Ψν1t αν1 =∇Ψν

t d∗Aν ∗ dAνγν − ∗[αν1 ∧ ∗∇

Ψν1t αν1

]− ∗

[(∂tA

ν1 − dAν1 Ψν

1

)∧ ∗αν1

]= ∗ [FAν ,∇Ψν

t γν ] + ∗[∇Ψν

t FAν , γν ]− ∗

[αν1 ∧ ∗∇

Ψν1t αν1

]− ∗

[(∂tA

ν1 − dAν1 Ψν

1

)∧ ∗αν1

].

Therefore, estimating term by term (17.16) we obtain∥∥∥d∗Aν1dAν1ψν1∥∥∥Lp(Σ)≤ cε

1− 1p

ν or∥∥∥d∗Aν1dAν1ψν1∥∥∥Lp(Σ)

≤ ce(θ(−s)−θ(s))δsε1− 1

pν (17.18)

and hence by (17.4), (17.9), (17.15) and (17.18)

‖∂tAν2 − dAν2 Ψν2‖Lp(Σ) ≤ c.

Analogously, deriving d∗Aν2(∂tA

ν2 − dAν2 Ψν

2

)by ∇Ψν2

t we can obtain ε2ν

∥∥∥∇Ψν2t ψν

∥∥∥Lp≤

ε2− 1

pν using∥∥∇Ψν

t d∗Aν (∂tAν − dAνΨν)

∥∥Lp(Σ)

≤∥∥d∗Aν∇Ψν

t (∂tAν − dAνΨν)

∥∥Lp(Σ)

= ‖d∗A (∂sAν − dAνΦν)‖Lp(Σ)

≤ce(θ(−s)−θ(s))δsεν

by the commutation formula (1.21),

[(∂sAν − dAνΦν) ∧ ∗ (∂sA

ν − dAνΦν)] = 0, [FAν , ∗FAν ] = 0,

the Yang-Mills flow equation (8.13) and the theorem 93. On the other hand, since Ξν

is a Yang-Mills flow and d∗A2ν

(∂sA

ν2 − dAν2 Φν

2

)= 0, we can estimate∥∥∥d∗Aν (∂sA

ν − dAνΦν)− d∗A2ν

(∂sA

ν2 − dAν2 Φν

2

)∥∥∥Lp(Σ)

≤ ce(θ(−s)−θ(s))δsεν

by the theorem 93 and the estimates computed so far; thus

d∗Aν2dAν2φν1 =− d∗Aν (∂sA

ν − dAνΦν) + d∗A2ν

(∂sA

ν2 − dAν2 Φν

2

)− ∗ [(αν0 + αν1) ∧ ∗ (∂sA

ν − dAνΦν)] + d∗Aν2∇Φν

s αν1

=− d∗Aν (∂sAν − dAνΦν)

− ∗ [(αν0 + αν1) ∧ ∗ (∂sAν − dAνΦν)]

+ d∗Aν1 [ψ0, α1] +[αν1 ∧ ∗∇Φν

s αν1]

+∇Φν1s ∗

[FAν1 , γ

ν]

+ ∗[(∂sA

ν1 − dAν1 Φν

1

)∧ ∗αν1

]and this implies that ‖dAφν1‖Lp ≤ cε1− 1

p , ‖dAφν1‖Lp ≤ ce(θ(−s)−θ(s))δsε1− 1p for |s| ≥ 2

and ‖∂sAν2 − dAν2 Φν2‖Lp(Σ) ≤ ce(θ(−s)−θ(s))δs.

187

Weber proved the following theorem (cf. [24], theorem 1.12).

Theorem 100 (Implicit function theorem). Fix a perturbation H : LM → R thatsatisfies (8.5). Assume EH is Morse and thatD0

u is onto for every u ∈M0(x−, x+;H)and every pair x± ∈ CritbEH . Fix two critical points x± ∈ CritbEH with Morse indexdifference one. Then, for all c0 > 0 and p > 2, there exist positive constants δ0 andc such that the following holds. If u : R × S1 → M is a smooth map such thatlims→±∞ u(s, ·) = x±(·) exists, uniformly in t, and such that

|∂su(s, t)| ≤ c0

1 + s2, |∂tu(s, t)| ≤ c0, |∇t∂tu(s, t)| ≤ c0 (17.19)

for all (s, t) ∈ R× S1 and

‖∂su−∇t∂tu− gradH(u)‖Lp ≤ δ0. (17.20)

Then there exist elements u0 ∈M0(x−, x+;H) and ξ ∈ im (D0u0

)∗ ∩Wu0 satisfying1

u = expu0(ξ), ‖ξ‖Wu0

≤ c‖∂su−∇t∂tu− gradH(u)‖Lp . (17.21)

Remark. The third condition of (17.19) follows from the first one and, for a positiveconstant c1,

|∂su−∇t∂tu− gradH(u)| ≤ c1;

therefore, in our case all the assumptions are satisfied by the second step and by thelemma 99.

Step 3. We choose p > 4. There are ε0, c > 0 such that the following holds. If εν < ε0,then there is a smooth map Aν3 : R2 → A0(P ) such that [Aν3] ∈M0(Ξ−,Ξ+),

d∗Aν3 (Aν3 − Aν2) = 0, (17.22)

‖Aν3 − Aν2‖Lp + ‖Aν3 − Aν2‖L∞ ≤ cε1− 1

pν (17.23)∥∥(∂tAν3 − dAν3 Ψν

3

)−(∂tA

ν2 − dAν2 Ψν

2

)∥∥Lp≤ cε

1− 1p

ν (17.24)∥∥(∂sAν3 − dAν3 Φν3

)−(∂sA

ν2 − dAν2 Φν

2

)∥∥Lp≤ cε

1− 1p

ν (17.25)

where Ψν3 and Φν

3 are defined uniquely by

d∗Aν3

(∂tA

ν3 − dAν3 Ψν

3

)= 0 and d∗Aν3

(∂sA

ν3 − dAν3 Φν

3

)= 0.

Proof of step 3. The third step follows directly from the theorem 100. The condition(17.22) can be reached using the lemma 5 and the local slice theorem (theorem 8.1 in[25]).

1The norm ‖ξ‖Wu0sums the the norms ‖ξ‖Lp , ‖∇tξ‖Lp ,‖∇t∇tξ‖Lp ,‖∇sξ‖Lp and the spaceWu0

is completion of the space of smooth, compactly supported, vector fields along u0 respect to the norm‖ · ‖Wu0

.


Step 4. We choose p > 4. There are ε0, c > 0 such that the following holds. If εν < ε0,then there is a smooth map Aν4 : R2 → A0(P ) such that [Aν4] ∈M0(Ξ−,Ξ+),

d∗Aν4 (Aν4 − Aν1) = 0, (17.26)

‖Aν4 − Aν1‖Lp + ‖Aν4 − Aν1‖L∞ ≤ cε1− 1

pν (17.27)∥∥(∂tAν4 − dAν4 Ψν

4

)−(∂tA

ν1 − dAν1 Ψν

1

)∥∥Lp≤ cε

1− 1p

ν (17.28)∥∥(∂sAν4 − dAν4 Φν4

)−(∂sA

ν1 − dAν1 Φν

1

)∥∥Lp≤ cε

1− 1p

ν (17.29)

where Ψν4 and Φν

4 are defined uniquely by

d∗Aν4

(∂tA

ν4 − dAν4 Ψν

4

)= 0 and d∗Aν4

(∂sA

ν4 − dAν4 Φν

4

)= 0.

Proof of step 4. By the previous two steps and the lemma 99 we can conclude that

‖Aν3 − Aν1‖Lp + ‖Aν3 − Aν1‖L∞ ≤ cε1− 1

pν ,

∥∥(∂sAν3 − dAν3 Φν3

)−(∂sA

ν1 − dAν1 Φν

1

)∥∥Lp≤ cε

1− 1p

ν .

Since

d∗Aν3 (Aν3 − Aν1) =d∗Aν3 (Aν3 − Aν2) + d∗Aν3αν1

= ∗ dAνdAνγν − ∗[(Aν3 − Aν2) ∧ αν1 ],

dAν (Aν1 − Aν2) = FAν1 −1

2[(Aν1 − Aν3) ∧ (Aν1 − Aν3)]

hold, we obtain∥∥∥d∗Aν3 (Aν3 − Aν1)∥∥∥Lp(Σ)

+ εν∥∥dAν3 (Aν3 − Aν1)

∥∥Lp(Σ)

≤ cε3− 2

pν .

Thus, by the local gauge theorem there are maps gν : R2 → G2,p0 (P ) such that

d∗Aν1 (g∗νAν3 − Aν1) = 0, ‖g∗νAν3 − Aν1‖W 1,p(Σ) ≤ c ‖Aν3 − Aν1‖W 1,p(Σ) ;

then we conclude the proof of the fourth step defining Aν4 := g∗νAν3 .

Step 5. For two positive constants c, ε0, 0 < ε < ε0, Ξν4 := Aν4 + Ψν

4dt + Φν4ds ∈

M0 (Ξ−,Ξ+) satisfies

∥∥(1− πAν4) (Ξν1 − Ξν

4)∥∥

Ξν4 ,1,p,εν+ εν‖dAν4∇

Ψν4t (Ψν

1 −Ψν4)‖ ≤ cε

2− 2p

ν , (17.30)

∥∥πAν4 (Aν1 − Aν4)∥∥

Ξν4 ,1,p,1≤ cε

1− 1p

ν . (17.31)

189

Proof of step 5. Since d∗Aν4 (Aν1 − Aν4) = 0 and

dAν4 (Aν1 − Aν4) = FAν1 −1

2[(Aν1 − Aν4) ∧ (Aν1 − Aν4)] ,

by lemma 6 ∣∣∣∣(1− πAν4 ) (Aν1 − Aν4)∣∣∣∣Lp

+∣∣∣∣dAν4 (Aν1 − Aν4)

∣∣∣∣Lp≤ cε

2− 2p

ν .

By (17.26) we have

d∗Aν4∇Ψν4t (Aν1 − Aν4) = ∗

[(∂tA

ν4 − dAν4 Ψν

4

)∧ ∗ (Aν1 − Aν4)

]d∗Aν4∇

Φν4s (Aν1 − Aν4) = ∗

[(∂sA

ν4 − dAν4 Ψν

4

)∧ ∗ (Aν1 − Aν4)

],

and by the properties and definitions of the connections

d∗Aν4

(∂tA

ν4 − dAν4 Ψν

4

)= 0, d∗Aν4

(∂sA

ν4 − dAν4 Φν

4

)= 0,

d∗Aν1

(∂tA

ν1 − dAν1 Ψν

1

)= d∗Aν (∂tA

ν + dAνΨν) +Dν ,

d∗Aν1

(∂sA

ν1 − dAν1 Φν

1

)= d∗Aν (∂sA

ν + dAνΦν)− [αν0 , (∂sA

ν − dAνΦν)] ,

where Dν is defined in the proof of the lemma 99. Hence we have

d∗Aν4dAν4

(Ψν4 −Ψν

1) =d∗Aν1

(∂tA

ν1 − dAν1 Ψν

1

)+ ∗

[(Aν1 − Aν4) ∧ ∗

(∂tA

ν1 − dAν1 Ψν

1

)]− d∗Aν4

(∇Ψν4t (Aν1 − Aν4)− [(Aν1 − Aν4) , (Ψν

1 −Ψν4)])

=d∗Aν1

(∂tA

ν1 − dAν1 Ψν

1

)+ ∗

[(Aν1 − Aν4) ∧ ∗

(∂tA

ν1 − dAν1 Ψν

1

)]− ∗

[(∂tA

ν4 − dAν4 Ψν

4

)∧ ∗ (Aν1 − Aν4)

]+ ∗

[∗ (Aν1 − Aν4) ∧ dAν4 (Ψν

1 −Ψν4)],

(17.32)

d∗Aν4dAν4

(Φν4 − Φν

1) =d∗Aν1

(∂sA

ν1 − dAν1 Φν

1

)+ ∗ [(Aν1 − Aν4) ∧ ∗ (∂sA

ν − dAνΦν)]

− ∗[(∂sA

ν4 − dAν4 Φν

4

)∧ ∗ (Aν1 − Aν4)

]+ ∗

[∗ (Aν1 − Aν4) ∧ dAν4 (Φν

1 − Φν4)] (17.33)

and thus by the first step, (17.17), the a priori estimates for a geodesics flow (8.7)-(8.7)and the a priori estimates of the theorem 93, we obtain∥∥∥d∗Aν4dAν4 (Ψν

4 −Ψν1)∥∥∥Lp

+∥∥∥d∗Aν4dAν4 (Φν

4 − Φν1)∥∥∥Lp≤ cε

1− 1p

ν .

By the lemma 6, the estimates (17.28), (17.29) and the triangular inequality, we havealso that

εν ‖Ψν4 −Ψν

1‖Lp + εν ‖Φν4 − Φν

1‖Lp ≤ cε2− 1

pν ,


εν∥∥dAν4 (Ψν

4 −Ψν1)∥∥Lp

+ εν∥∥dAν4 (Φν

4 − Φν1)∥∥Lp≤ cε

2− 1p

ν ,

εν

∥∥∥∇Ψν4t (Aν4 − Aν1)

∥∥∥Lp

+ εν∥∥∇Ψν4

s (Aν4 − Aν1)∥∥Lp≤ cε

2− 1p

ν .

Furthermore, deriving by ∇Φν4s and by ∇Ψν4

t the identities (17.32) and (17.33), we canobtain the other estimates needed for (17.30).

Step 6. We choose p > 10. Then there are ε0, c > 0 such that the following holds.There are two sequences gν ∈ G2,p

0 (P×S1×R) and sν ∈ R such that Ξν5 := g∗νΞ

ν4(t, s+

sν) = Aν5 + Ψν5dt+ Φν

5ds satisfy

d∗ενΞν5

(Ξν −Kεν2 (Ξν5)) = 0, Ξε −Kεν2 (Ξν

5) ∈ imDεν (Kεν2 (Ξν5))∗ (17.34)

∥∥(1− πAν5) (Ξν1 − Ξν

5)∥∥

Ξν5 ,1,p,εν≤ cε

2− 2p

ν , (17.35)∥∥πAν5 (Aν1 − Aν5)∥∥

Ξν5 ,1,p,1≤ cε

1− 1p

ν . (17.36)

Remark. In the sixth step we use the connectionK2(Ξν4) introduced in the chapter 11;

the definition of the 1-form αε0 + ψε0dt in that chapter is not the same as (17.3) even iswe consider that this holds. In fact, one can replace the definition (11.1) by

αε0(s) + ψε0(s)dt :=θ(−s)(h(s)g(s))−1(T ε,b(A− + Ψ−dt)− (A− + Ψ−dt))h(s)g(s)

+ θ(s)(h(s)g(s))−1(T ε,b(A+ + Ψ+dt)− (A+ + Ψ+dt))h(s)g(s),

where g(s) is defined as in (11.2) and h by h−1∂sh = g(Φν − Φν4)g−1. In this case,

(hg)−1∂s(hg) = g−1(h−1∂sh)g + g−1∂sg = Ψν . With this change all the theoremsproved for Kε2 continues to hold.

Proof of step 6. Step 5 and the theorem 73 tells us that

‖Ξν −K2(Ξν4)‖Ξν4 ,1,p,εν

≤ cε1− 3

pν ≤ δε

1− 4p

ν ,

ε2∥∥∇Φν4

s (Ξν −K2(Ξν4))∥∥

0,p,εν≤ cε

2− 3p

ν ≤ cε1+ 7

pν

for cε1p ≤ δ where δ is given by the theorem 96. Then by theorem 96, there is a

sequence gν ∈ G2,p0 (P × S1 × R), σν ∈ R such that

d∗εΞν4(g∗ν(Ξ

ν ρσν )−K2(Ξν4)) = 0,

g∗ν(Ξν ρσν )−K2(Ξν

4) ∈ im (Dε(K2(Ξν4)))∗ .

We define Ξν5 by (g−1

ν )∗Ξν4 ρ−σν . By the step 5, the theorem 96 and the triangular

inequality we have

‖Ξν1 − Ξν

5‖Ξν5 ,1,p,εν≤ cε

1− 2p

ν ,

εν

∥∥∥∇Ψν5t (Aν1 − Aν5)

∥∥∥Lp

+ ε2ν

∥∥∥∇Ψν5t (Ψν

1 −Ψν5)∥∥∥Lp

+ ε3ν

∥∥∥∇Ψν5t (Φν

1 − Φν5)∥∥∥Lp≤ cε

2− 2p

ν ,

191

ε3ν

∥∥∥∇Ψν5t (Φν

1 − Φν5)∥∥∥Lp

+ ε2ν

∥∥∇Φν5s (Aν1 − Aν5)

∥∥Lp≤ cε

2− 2p

ν ,

ε3ν

∥∥∇Φν5s (Ψν

1 −Ψν5)∣∣ ‖Lp + ε4

ν

∥∥∇Φν5s (Φν

1 − Φν5)∥∥Lp≤ cε

2− 2p

ν ,

∥∥πAν5 (Ξν1 − Ξν

5)∥∥

Ξν5 ,1,p,1≤ cε

1− 1p

ν ;

in order to improve the estimates for the non-harmonic part, we use the identity

d∗ενΞν5

(Ξν1 −Kεν2 (Ξν

5)) = 0,

i.e. if we define Ξν1−Ξν

5 =: αν+ψνdt+ψνds and Ξν−Kεν2 (Ξν5) =: αν+ψνdt+ψνds,

by the definition of Kεν2

‖d∗Aν5 αν‖Lp = ‖d∗Aν5α

ν‖Lp ≤ ε2ν‖∇tψ

ν‖Lp + ε4ν‖∇sφ

ν‖Lp ≤ cε2− 1p ;

and since FAν1 = dAν5 αν + 1

2[αν ∧ αν ] and

‖dAν5 α‖Lp ≤ cε2− 1p + ‖αν‖2

L2p ≤ cε2− 2p .

Step 7. We choose p > 13. There are three positive constants δ1, ε0, c such that forany εν < ε0 ∥∥πAν5 (Aν1 − Aν5)

∥∥Lp

+∥∥πAν5 (Aν1 − Aν5)

∥∥L∞≤ cε1+δ1

ν . (17.37)

End of the proof: Finally we can apply the theorem 76 choosing ε0 such that cεδ10 < δfor the δ needed in the theorem 76; thus, we can conclude that for ν big enough Ξν =Rεν ,b(Ξν

5) which is a contradiction to the fact that the Ξν are not in the image ofRεν ,b.Therefore the proof of theorem 98 is concluded.

Proof of step 7. The idea is to consider the situation in the figure 17.2 in order to im-prove the norm of πAν5 (Aν −Kεν2 (Ξν

5)). In particular, we use that αν1 ∈ im d∗Aν2 and thefact that the norm of Πim d∗

Aν5

(αν) can be estimate using the identity dAν5 αν = −1

2[αν ∧

αν ] deduced from FAν2 = FA0 + dAν5 αν + 1

2[αν ∧ αν ]. We denote by Aνk + Ψν

kdt+ Φνkds

the connection Kεν2 (Ξν5).

Let ∇t := ∇Ψν5t and ∇s := ∇Φν5

s . By the lemmas 58, 57 and the estimates (11.11) and(11.12) we have∥∥∥πAν5 (Aν −Kεν2 (Ξν

5))∥∥∥Lp

+∥∥∇tπAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

+∥∥∇sπAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

+∥∥∇t∇tπAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

≤c∥∥πAν5Dεν1 (Kεν2 (Ξν

5))(Ξν −Kεν2 (Ξν5))∥∥Lp

+ c∥∥(1− πAν5) (Aν − Aνk)

∥∥1,p,εν

+ cε3− 2p

+ c∥∥∇t

(1− πAν5

)(Aν − Aνk)

∥∥Lp

+ cε2ν ‖Dεν2 (Kεν2 (Ξν

5))(Ξν −Kεν2 (Ξν5))‖Lp .

(17.38)


FA = 0

αν(t)

Aν2(t, s) Aν5(t, s)

Aνk(t, s)Aν(t, s)

αν(t, s)

αν0(t, s)αν1(t, s)

Figure 17.2: The splitting of the seventh step.

and thus our task is to estimate all the norms on the right hand side of the inequality;

the second one can be estimate by cε2− 2

pν by the previous step and the lemma 73. The

last term of (17.38) can be estimate by

ε2ν

∥∥Dεν2 (Kεν2 (Ξν5))(Ξν −Kεν2 (Ξν

5))∥∥Lp

≤ε2ν ‖F εν2 (Kεν2 (Ξν

5))‖Lp+ ε2

ν ‖Cεν2 (Kεν2 (Ξν5))(Ξν −Kεν2 (Ξν

5))‖Lp ≤ cε2− 5p

(17.39)

where the first inequality follows from

Dεν2 (Kεν2 (Ξν5))(Ξν −Kεν2 (Ξν

5)) = −F εν2 (Kεν2 (Ξν5))− Cεν2 (Kεν2 (Ξν

5))(Ξν −Kεν2 (Ξν5))

because of F εν2 (Ξν) = 0 and the second estimating Cεν2 (Kεν2 (Ξν5))(Ξν −Kεν2 (Ξν

5)) termby term using the formula (10.4). Next, we define αν + ψνdt+ φνds := Ξν −Kεν2 (Ξν

5)and

Aν − Aνk = (Aν − Aν2) + (Aν2 − Aν5) + (Aν5 − Aνk) = αν1 + αν − αν0 (17.40)

where αν0 := Kεν2 (Ξν5) − Ξν

5 . We remark that 0 = FAν5+αν = dAν5 αν + 1

2[αν ∧ αν ].

Furthermore, since by F εν1 (Ξν) = 0

Dεν1 (Kεν2 (Ξν5))(Ξν −Kεν2 (Ξν

5)) = −F εν1 (Kεν2 (Ξν5))− Cεν1 (Kεν2 (Ξν

5))(Ξν −Kεν2 (Ξν5))

and by (11.6)∥∥πAν5F εν1 (Kεν2 (Ξν

5))∥∥Lp≤ cε2,∥∥∥πAν5Dεν1 (Kεν2 (Ξν

5))(αν + ψνdt+ φνds)∥∥∥Lp

≤∥∥πAν5F εν1 (Kεν2 (Ξν

5))∥∥Lp

+∥∥∥πAν5Cεν1 (Kεν2 (Ξν

5))(αν + ψνdt+ φνds)∥∥∥Lp

≤cε2− 7

pν +

1

ε2ν

∥∥∥∥πAν5 ([αν ∧ ∗(dAν5 αν +1

2[αν ∧ αν ]

)])∥∥∥∥Lp

≤cε2− 7

pν +

1

ε2ν

∥∥πAν5 ([αν ∧ ∗ (dAν5 (αν − αν))])∥∥

Lp+ cε

1− 7p

ν

∥∥πAν5 (αν)∥∥Lp

(17.41)

where the second step follows estimating (10.3) term by term. Next, we consider thefollowing operator

Qεν (Ξν5) (Ξν −Kεν2 (Ξν

5)) :=Dεν (Ξν5) (Ξν −Kεν2 (Ξν

5)) +1

2ε2ν

d∗Aν5 [αν ∧ αν ] (17.42)

193

whose first component can be written as

Qεν1 (Ξν

5) (αν + ψνdt+ φνds) = ∇s

(αν − Πim d∗

Aν5

(αν))− dAν5 φ

ν

+1

ε2ν

d∗Aν5dAν5

(αν − αν)− d ∗Xt(Aν5)(αν − Πim d∗

Aν5

(αν))

−∇t∇t

(αν − Πim d∗

Aν5

(αν))− 2

[ψν ,(∂tA

ν5 − dAν5 Ψν

5

)]+∇s

(Πim d∗

Aν5

(αν))− d ∗Xt(A

ν5)Πim d∗

Aν5

(αν)−∇t∇t

(Πim d∗

Aν5

(αν)).

(17.43)

By theorem 54 we obtain that∥∥∥d∗Aν5dAν5 (αν − αν)∥∥∥Lp

+ ε2ν

∥∥∥∇t∇t(1− πAν5 )(αν − Πim d∗

Aν5

(αν))∥∥∥

Lp

+ εν

∥∥∥∇t(1− πAν5 )(αν − Πim d∗

Aν5

(αν))∥∥∥

Lp

≤cε2ν

∥∥∥Dεν1 (Ξν5)(αν − Πim d∗

Aν5

(αν), ψν , φν)∥∥∥

Lp+ ε2

∥∥∇sπAν5 (αν − αν)∥∥Lp

+ ε2∥∥∇t∇tπAν5 (αν − αν)

∥∥Lp

+ cε3− 1

pν

and by (17.43), step 2, the lemma 99 and the theorem 75

≤cε2ν

∥∥∥Qεν1 (Ξν

5)(αν , ψν , φν)∥∥∥Lp

+ cε3− 2

pν

+ ε2ν

∥∥∥∇t∇t

(Πim d∗

Aν5

(αν))∥∥∥

Lp+ ε2

ν

∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp

since F εν1 (Kεν2 (Ξν5)) + Qεν

1 (Kεν2 (Ξν

5)) (αν , ψν , φν) + Cεν1 (Kεν2 (Ξν

5)) (αν , ψν , φν) −1

2ε2νd∗Aν5 [αν ∧ αν ] = 0,

≤cε2ν ‖F εν1 (Kεν2 (Ξν

5))‖Lp + cε3− 2

pν

+ cε2ν

∥∥∥∥Cεν1 (Kεν2 (Ξν5))(αν , ψν , φν)− 1

2ε2ν

d∗Aν5 [αν ∧ αν ]∥∥∥∥Lp

+ ε2ν

∥∥∥∇t∇t

(Πim d∗

Aν5

(αν))∥∥∥

Lp+ ε2

ν

∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp

≤cε3− 7

pν + ε2

ν

∥∥∥∇t∇t

(Πim d∗

Aν5

(αν))∥∥∥

Lp+ ε2

ν

∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp

.

where the last step follows estimating (10.3) term by term. Hence, by the last estimate,(17.38), (17.39) (17.41) and the next claim:∥∥πAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

+∥∥∇tπAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

+∥∥∇t∇tπAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

+∥∥∇sπAν5 (Aν −Kεν2 (Ξν

5))∥∥Lp

≤cε2− 10

pν + ε

1− 1p

ν

∥∥∥∇t∇t

(Πim d∗

Aν5

(αν))∥∥∥

Lp

+ ε1pν

∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp≤ cε

2− 10p

ν .


Therefore, for p > 10 and by the Sobolev’s theorem 51 for ε = 1, there is a δ1 > 0,such that ∥∥πAν5 (Aν1 − Aν5)

∥∥Lp

+∥∥πAν5 (Aν1 − Aν5)

∥∥L∞≤ cε1+δ1

ν

holds for εν small enough. Thus we concluded the proof of the seventh step.

Claim 101. There are two positive constants c and ε0 such that, for 0 < ε < ε0,∥∥∥∇t∇t

(Πim d∗

Aν5

(αν))∥∥∥

Lp+∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp

≤cε1− 2p(1 + ‖∇t∇tπAν5 (αν)‖Lp + ‖∇sπAν5 (αν)‖Lp

)Proof of the claim. We write

(Πim d∗

Aν5

(αν))

= d∗Aν5ων for 2-form ων and hence

‖∇t∇td∗Aν5ων‖Lp ≤‖dAν5∇t∇td

∗Aν5ων‖Lp +

∥∥∥(1− Πim d∗Aν5

)∇t∇td

∗Aν5ων∥∥∥Lp

using the commutation formulas, the L∞-bound on the curvature terms and the lemma6, we obtain

≤‖∇t∇tdAν5 αν‖Lp + c‖d∗Aν5ω

ν‖Lp + c‖∇td∗Aν5ων‖Lp

and by the identity dAν5 αν + 1

2[αν ∧ αν ]

≤1

2‖∇t∇t[α

ν ∧ αν ]‖Lp + c‖αν‖Lp + c‖∇tαν‖Lp

≤c‖αν‖L∞‖∇t∇tαν‖Lp + c‖∇tα

ν‖L2p‖∇tαν‖L2p

+ c‖αν‖Lp + c‖∇tαν‖Lp

≤cε1− 2p

(1 +

∥∥∥∇t∇t

(Πim d∗

Aν5

(αν))∥∥∥

Lp

)+ cε1− 2

p

∥∥∇t∇t

(πAν5 (αν)

)∥∥Lp.

In the same way, we can show that∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp≤cε1− 2

p

(1 +

∥∥∥∇s

(Πim d∗

Aν5

(αν))∥∥∥

Lp

)+ cε1− 2

p

∥∥∇s

(πAν5 (αν)

)∥∥Lp

and thus the claim holds for ε sufficiently small.

With the last claim we concluded also the proof of the theorem 98.

The main theorem 18The definition 77 of the map Rε,b and the theorem 98 which assure its surjectivityallows us to state the following theorem.

Theorem 102. We assume that the energy functional EH is Morse-Smale and wechoose p > 2 and a regular value b > 0 of EH . There are constants ε0, c > 0 such thatthe following holds. For every ε ∈ (0, ε0), every pair Ξ0

± := A0± + Ψ0

±dt ∈ CritbEHthere exists a unique map

Rε,b :M0(Ξ0−,Ξ

0+

)→Mε

(T ε,b(Ξ0

−), T ε,b(Ξ0+))

satisfying for each Ξ0 ∈M0(Ξ0−,Ξ

0+

)d∗εΞ0

(Rε,b(Ξ0)−Kε2(Ξ0)

)= 0, Rε,b(Ξ0)−Kε2(Ξ0) ∈ im

(Dε(K2(Ξ0))

)∗, (18.1)∥∥Rε,b(Ξ0)−Kε2(Ξ0)

∥∥1,2;p,1

≤ cε2. (18.2)

Furthermore,Rε,b is bijective.

In this chapter, using this last theorem and the theorem 49 proved in the first part, wewill prove the following result.

Theorem 103. We assume that the energy functional EH is Morse-Smale. For everyregular value b > 0 of EH there is a positive constant ε0 such that, for 0 < ε < ε0, theinclusion LbMg(P )→ Aε,b (P × S1) /G0 (P × S1) induces an isomorphism

HM∗(LbMg(P ),Z2

) ∼= HM∗(Aε,b

(P × S1

)/G0

(P × S1

),Z2

).

Here LbMg(P ) ⊂ LMg(P ) denotes the subset where EH ≤ b and Aε,b (P × S1) ⊂A (P × S1) denotes the subset where YMε,H ≤ b.

Proof. If we fix a regular value b of EH , then by the theorem 49 there is a positiveconstant ε0 such that the map T ε,b is a bijection for 0 < ε < ε0. In addition, T ε,b,defined as in (4.7), maps perturbed closed geodesics to orbits of perturbed Yang-Millsconnections with the same Morse index and therefore we can see it as chain complexhomeomorphism

T ε,b : CEH ,b∗ → CYM

ε,H ,b∗ .

For any two perturbed geodesics γ± ∈ [CritbEH ] with index difference 1, the map Rε,b

is, for two lifts Ξ± ∈ CritbEH with [Ξ±] = γ±, by theorem 102, bijective and thus

]Z2M0 (Ξ−,Ξ+) /R = ]Z2Mε(T ε,b(Ξ−), T ε,b(Ξ+)

)/R

195

196 18. The main theorem

which yields that the following diagram commutes

. . . −−−→ CEH ,bk+1

∂EH

k−−−→ CEH ,bk

∂EH

k−1−−−→ CEH ,bk−1 −−−→ . . .yT ε,b yT ε,b yT ε,b

. . . −−−→ CYMε,H ,b

k+1

∂YMε,H,b

k−−−−−→ CYMε,H ,b

k

∂YMε,H,b

k−1−−−−−→ CYMε,H ,b

k−1 −−−→ . . .y(T ε,b)−1

y(T ε,b)−1

y(T ε,b)−1

. . . −−−→ CEH ,bk+1

∂EH

k−−−→ CEH ,bk

∂EH

k−1−−−→ CEH ,bk−1 −−−→ . . .

and hence(T ε,b

)∗ : HM∗

(LbMg(P ),Z2

)→ HM∗

(Aε,b

(P × S1

)/G0

(P × S1

),Z2

)is an isomorphism.

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Curriculum Vitae

Personal information

Name Remi JannerDate of Birth the 23rd December 1981Place of Birth Locarno, SwitzerlandNative of Bosco Gurin, SwitzerlandCitizenship swiss

Education

2006-2010 Graduate studies in Mathematics at the Swiss Federal Institute ofTechnology (ETH) Zurich, Switzerland

Advisor: Prof. Dr. D.A. Salamon

PhD thesis: Morse homology of the loop space on the moduli spaceof flat connections and Yang-Mills theory

2000-2005 Studies in Mathematics at the Swiss Federal Institute of Technology(ETH) Zurich, Switzerland

Advisor: Prof. Dr. D.A. Salamon

Thesis: Closed geodesics and harmonic maps on the moduli space offlat connections

1996-2000 High School Bellinzona, Switzerland Type C (scientific type)

1987-1996 Primary and Secondary School Arbedo-Castione, Switzerland

Employment history

2005-2010 Assistant at the ETH Zurich for the Department of Mathematics

Partial support from the ETH-Grant for the projekt TH-0106-1 duringthe period May 2007-August 2009.

2002-2005 Teaching aid at the ETH Zurich for the Department of Mathematics

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