geometric and analytic aspects of higgs bundle …swoboda/heidelberg lectures/… · geometric and...

GEOMETRIC AND ANALYTIC ASPECTS OF HIGGS BUNDLEMODULI SPACES

JAN SWOBODA

Abstract. The following are lecture notes prepared for a four-hours minicourse on Higgsbundles I presented at Heidelberg University in November and December 2014.

Contents

1. Introduction 12. Higgs bundles 22.1. Vector bundles over Riemann surfaces 23. The moduli space of Higgs bundles 53.1. The topology of the moduli space 63.2. The Higgs bundle moduli space as a hyperkahler manifold 74. The structure of ends of the Higgs bundle moduli space 104.1. The limiting equation 104.2. Model solutions 114.3. Limiting configurations 144.4. Conic operators 154.5. Analysis of the operator ∆A 174.6. Deformation theory of limiting configurations 184.7. Gluing theorem 19References 23

1. Introduction

The present lecture notes grew out of a four-hours minicourse on the subject of Higgsbundle moduli spaces the author had the chance to present at Heidelberg University in thefall 2014. These were mainly aimed at graduate and doctoral students in mathematics andtheoretical physics, and did not assume any prerequisites going beyond a standard courseon differential geometry. The notes therefore start with a resume of the basic theory ofholomorphic vector bundles over a Riemann surface and of some related facts from gaugetheory. After these prerequisites, the moduli space of Higgs bundles as the space of solu-tions of Hitchin’s self-duality equations is introduced. We discuss its relation to the purelyholomorphic object of the Dolbeault moduli space of poly- and semistable bundles as well asits main topological and geometric properties. As it turns out, the geometry of this modulispace is particularly rich. It admits a natural interpretation as a hyperkahler quotient, theinduced hyperkahler metric being in many cases complete. On the other hand, this mod-uli space is non-compact due to the existence of configurations with arbitrarily large (with

1

2 JAN SWOBODA

respect to the L2-norm) Higgs fields. It is therefore a natural and interesting question to un-derstand the asymptotic properties of the ends of this space. Lacking a satisfactory answerconcerning the large scale geometry of the moduli space so far, we rather give an overviewof the results recently obtained in [MSWW14] concerning the degeneration behaviour ofsolutions to the self-duality equations with large Higgs fields. We discuss a certain space ofsingular limiting configurations which gives rise to a geometric compactification of an (openand dense) subset of the Higgs bundle moduli space. In course of its construction we are ledto study elliptic differential operators with conic singularities. The by now far developedtheory of this class of operators is briefly reviewed. These notes end with a discussion of agluing theorem, which gives rise to a rather concrete description of a tubular neighbourhoodof (some large region of) the boundary of the compactified moduli space.

Acknowledgments. I would like to thank Anna Wienhard and Andreas Ott for invitingme to give these lectures as part of an interdisciplinary workshop within the PartnershipMathematics and Physics at Heidelberg University. I am grateful to Andreas Ott for theperfect organization of this activity. Finally I would like to thank the audience of my lecturesfor their interest in the subject as well as for many helpful questions and comments.

2. Higgs bundles

We present standard facts from the theory of holomorphic vector bundles and Higgsbundles. A nice introduction to these topics can be found in the appendix by Oscar Garcia-Prada of the book [WGP08].

2.1. Vector bundles over Riemann surfaces. A Riemann surface is a real two-dimensionalclosed manifold Σ with complex structure J . Topologically, closed surfaces are classified (upto homeomorphisms) by their genus

γ = 1− 12χ(Σ) ≥ 0.

Here we are mostly interested in surfaces of genus γ ≥ 2, for reasons to become clear later.Throughout, we fix a Riemannian metric g on Σ compatible with J . That is, we requirethat

g(JX, JY ) = g(X,Y )

is satisfied for all vector fields X and Y on Σ. Then the two-form

ω := g(J ·, ·) ∈ Ω2(Σ)

is closed, dω = 0. Hence in particular, (Σ, g) is a Kahler manifold. Note that if g hasconstant Gauß curvature Kg ≡ −1, then by the theorem of Gauß-Bonnet,∫

Σ

ω = 4π(γ − 1).

Vector bundles. Let E → Σ be a complex vector bundle of rank r ≥ 1. Thus by definition,there exist an open cover (Ui)i∈I of Σ such that each restriction E|Ui is diffeomorphic toUi × Cr and smooth transition functions

ϕji : Ui ∩ Uj → Cr×r

satisfying the cocycle conditions ϕii = id and ϕkjϕji = ϕki. The degree of E is the integer

d =∫

Σ

c1(E) ∈ Z.

GEOMETRIC AND ANALYTIC ASPECTS OF HIGGS BUNDLE MODULI SPACES 3

Here c1(E) denotes the first Chern class of the vector bundle E. It is a well-known fact that,up to vector bundle isomorphisms, complex vector bundles over the surface Σ are classifiedby their rank and degree. A complex vector bundle E is called holomorphic if in additionall transition functions ϕji are holomorphic. For any holomorphic vector bundle, one has anatural operator

∂E : Ω0(Σ, E) → Ω0,1(Σ, E),

which we define using local holomorphic trivializations by

∂E(s1, . . . , sr) = (∂s1, . . . , ∂sr).

It satisfies the Leibniz rule

∂E(fs) = ∂f ⊗ s+ f∂Es (2.1)

for every f ∈ C∞(Σ) and s ∈ Ω0(Σ, E), and extends to an operator

∂E : Ωp,q(Σ, E) → Ωp,q+1(Σ, E) (p, q ≥ 0). (2.2)

This extension has the property that

∂E ∂E = 0. (2.3)

Conversely, any C-linear operator ∂ on a complex vector bundle E satisfying (2.1) and(2.3) induces a holomorphic structure on E such that ∂E = ∂. We denote the set of theseoperators by

C(E) := ∂E : Ω0(Σ, E) → Ω0,1(Σ, E) | (2.1) and (2.3).

It is acted on by the group of complex gauge transformations Gc = Gl(E) via

g · ∂E = g−1 ∂E g for all g ∈ Gc and ∂E ∈ C(E).

Connections. Let E be a complex vector bundle. Recall that a C-linear map

dA : Ω0(Σ, E) → Ω1(Σ, E)

which satisfies the Leibniz rule is called a connection. Any connection dA canonicallyextends to a connection

dA : Ωk(Σ, E) → Ωk+1(Σ, E) (k ≥ 0),

and induces connections on the complex vector bundles E∗,∧∗

E∗, End(E) etc. The com-plex structure on Σ gives rise to the splitting

dA = ∂A + ∂A,

where ∂A = π1,0dA and ∂A = π0,1dA. Here π1,0 and π0,1 denote the canonical projectionsonto the (1, 0)-, respectively the (0, 1)-part, of the decomposition Ω1(Σ, E) = Ω1,0(Σ, E)⊗Ω0,1(Σ, E). The curvature of the connection is the two-form

FA = dA dA

which we may view as a two-form FA ∈ Ω2(Σ,End(E)). This two-form satisfies the Bianchiidentity dAFA = 0.

Definition 2.1. A hermitian vector bundle is a complex vector bundle E together witha fibrewise hermitian inner product h, which depends smoothly on the base point.

4 JAN SWOBODA

A connection dA on a hermitian vector bundle (E, h) is called unitary if

dh(s1, s2) = h(dAs1, s2) + h(s1, dAs2) for all s1, s2 ∈ Ω0(Σ, E).

It follows that the curvature FA of a unitary connection is a section of Ω2(Σ,U(E, h)), thebundle of two-forms with values in the unitary automorphisms of (E, h). Assuming that(E, ∂E) is a holomorphic vector bundle we may ask in addition for the connection dA to becompatible with the complex structure of E, i. e. to satisfy

∂A = ∂E .

Theorem 2.2. Let (E, ∂E) be a holomorphic vector bundle with hermitian inner producth. Then there exists a unique unitary connection dA compatible with ∂E, called Chernconnection.

For a complex vector bundle E of rank rand degree d we call the number

µ(E) =d

r

the slope of E. In the following it is assumed that the Kahler metric on Σ is normalizedsuch that

∫Σω = 2π. A unitary connection dA ∈ A(E, h) is said to have constant central

curvature if

FA = −iµ(E) idE ⊗ω. (2.4)

For µ(E) = 0 (or equivalently d = 0) this condition means that the connection dA isflat, FA = 0. Otherwise it expresses that dA is as close to flatness as possible. We notethat condition (2.4) is invariant under unitary gauge transformations g ∈ G(E, h) =U(E, h). We define the moduli space of central curvature connections to be the quotientA0(E, h)/G(E, h), where

A0(E, h) = dA ∈ A(E, h) | (2.4).

In order to endow it with a natural topology, we have to invoke Sobolev spaces of connectionsand gauge transformations, which we postpone to a later point. A unitary connectionis called irreducible if there does not exist any nontrivial splitting (E, h) = (E1, h1) ⊕(E2, h2) of hermitian vector bundles and corresponding splitting dA = dA1 + dA2 of unitaryconnections.

Theorem 2.3. The moduli space A∗0(E, h)/G(E, h) of irreducible central curvature connec-tion carries the structure of a smooth manifold of dimension 2 + 2r2(γ − 1).

We consider the projection map

pr: A(E, h) → C(E), dA 7→ ∂A.

Note that this map is well-defined since for dimension reasons the operator ∂A has theproperty that ∂2

A = 0 for any connection dA. Also, dA is the Chern connection associatedwith the holomorphic vector bundle (E, ∂A), endowed with the hermitian inner product h.

Definition 2.4 (Stability). A holomorphic vector bundle (E, ∂E) is called stable if µ(F ) <µ(E), and semistable if µ(F ) ≤ µ(E) for every proper holomorphic subbundle F ⊂ E. Itis called polystable if there exists a splitting

E = ⊕iEi

into stable holomorphic bundles such that µ(Ei) = µ(E) for all i.


Remark 2.5. There are inclusions

Cs(E) ⊆ Cps(E) ⊆ Css(E) ⊆ C(E)

of the sets of (semi-/poly)stable bundles. These sets are preserved under the action of thegroup Gc(E) of complex gauge transformations.

Concerning the quotient space C(E)/Gc(E) the problem arises that it is in general not ahausdorff topological space. However, one can show that the subspace of stable holomorphicbundles admits a good quotient, i. e. that this quotient is a smooth manifold. More preciselyone has the following result.

Theorem 2.6 (Narasimhan-Seshadri). The following diagram commutes

A∗0(E, h)/G(E, h)pr−−−−→∼= Cs(E)/Gc(E)

i1

y i2

yA0(E, h)/G(E, h)

pr−−−−→∼= Cps(E)/Gc(E)

where the maps pr are homeomorphisms, and i1, i2 denote inclusions.

Proof. For a proof we refer to [WGP08, Appendix, Theorem 4.3].

Remark 2.7. (i) The moduli space Ms(r, d) = Cs(E)/Gs(E) of stable holomorphicvector bundles of rank r and degree d is a complex manifold of dimension 1+r2(γ−1).It admits a compactification by the moduli space of semistable bundles.

(ii) Case r = 1 (holomorphic line bundles). The manifold Ms(1, d) = M(1, d) coincideswith a connected component of the Picard variety Pic(Σ) of holomorphic line bundlesover Σ. The particular case M(1, 0) is known as Jacobian torus.

3. The moduli space of Higgs bundles

Let (E, ∂E) → Σ be a holomorphic vector bundle.

Definition 3.1. A Higgs field is a section Φ ∈ Ω1,0(Σ,End(E)) satisfying ∂EΦ = 0. Thepair (∂E ,Φ) is called a Higgs bundle. Equivalently, Φ ∈ H0(Σ,End(E) ⊗ K), where Kdenotes the canonical bundle, i. e. the holomorphic cotangent bundle T ∗Σ.

Stability. A Higgs bundle (∂E ,Φ) is called stable if µ(F ) < µ(E) holds for every properholomorphic subbundle F ⊂ E which is Φ-invariant in the sense that Φ(F ) ⊆ F ⊗ K.Semistability and polystability are defined similarly.

Remark 3.2. (i) If E is stable as a holomorphic vector bundle then (∂E ,Φ) is clearlystable for any Higgs field Φ.

(ii) The group Gc(E) acts on the space of Higgs bundles through

g∗(∂E ,Φ) = (g−1 ∂E g, g−1Φg),

preserving (poly-/semi)stability.

Example 3.3. This example was communicated to us by Richard Wentworth. Let E → Σbe a holomorphic vector bundle of rank r = 2 and (∂E ,Φ) be a Higgs bundle. Suppose Φtakes values in the trace-free endomorphisms of E and its determinant detΦ has only simplezeroes on Σ (cf. more on this condition below). Let L ⊂ E be a Φ-invariant line bundle.

6 JAN SWOBODA

Then near a zero of det Φ we can find a local holomorphic trivialization of E with respectto which

Φ(z) = ϕ(z) dz =(a(z) b(z)0 −a(z)

)dz

for holomorphic functions a and b. Assume ϕ(0) = 0. Then a(0) = 0, and it follows thatdetΦ has a double zero in z = 0, in contradiction to our assumption. Thus a nontrivialΦ-invariant holomorphic subbundle L ⊂ E cannot exist and the Higgs bundle (∂E ,Φ) isstable.

Self-duality equations. Let (Σ, h) be a hermitian vector bundle. In analogy to the constantcentral curvature equation we consider the system of nonlinear PDEs, called Hitchin’s self-duality equations

∂AΦ = 0,FA + [Φ ∧ Φ∗] = −iµ(E) idE ⊗ω.

(3.1)

Clearly, solutions (A,Φ) of (3.1) are invariant under unitary gauge transformations. Irre-ducible solutions are defined as before, the set of which we denote by

B∗0 := (A,Φ) | (3.1), irreducible.It is invariant under the action of G(E, h).

Theorem 3.4 (Hitchin). The moduli space B∗0/G(E, h) is a smooth manifold of dimension4 + 4r2(γ − 1).


Note that any solution A of the constant central curvature equation (2.4) gives rise to asolution of (3.1) with vanishing Higgs field. Furthermore, under

(A,Φ) 7→ (∂A,Φ)

solutions of (3.1) are mapped to Higgs bundles. This mapping gives rise to a precise analogof the Narasimhan-Seshadri correspondence.

Theorem 3.5 (Hitchin). The following diagram commutes

B∗0(E, h)/G(E, h)[(A,Φ)] 7→[(∂A,Φ)]−−−−−−−−−−−→∼=

MsHiggs(E)/Gc(E)

i1

y i2

yB0(E, h)/G(E, h)

[(A,Φ)] 7→[(∂A,Φ)]−−−−−−−−−−−→∼=Mps

Higgs(E)/Gc(E)

where the horizontal maps are homeomorphisms and i1, i2 denote inclusions.


3.1. The topology of the moduli space. From now on, we restrict our discussion to thecase of rank r = 2 Higgs bundles and set Md := MHiggs(2, d), and similarly for Ms

d,Mps

d and Mssd . Note that if the degree d of E is odd then

Msd = Mps

d = Mssd .

As noted above, the moduli space Msd contains the space A∗0/G(E, h) of irreducible constant

central curvature connections as a subset. Furthermore, one can show that the cotangent


bundle of A∗0/G(E, h) embeds into Msd as an open dense subset. In particular, the moduli

space of stable Higgs bundles is noncompact. This statement can be made more precise.

Theorem 3.6 (Hitchin). The map

f : Msd → R, [(A,Φ)] 7→ ‖Φ‖2L2(Σ)

is proper. As a consequence, a sequence ([(Aν ,Φν)])ν∈N in Msd does not have an accumula-

tion point if and only if the sequence (‖Φν‖L2(Σ))ν∈N of norms diverges.

Proof. For a proof we refer to [Hi87, Proposition 7.1].

The proof of this theorem is analytic and uses Uhlenbeck’s compactness theorem. Sta-bility enters crucially. We remark that f−1(0) is the compact moduli space of irreducibleconstant curvature connections. Furthermore, in [Hi87] the fact that the map f is a S1-equivariant Morse-Bott function is used to calculate topological invariants of the modulispace Ms

d such as its homology and homotopy groups. For instance, it turns out thatπ1(M s

d) = 0.

Holomorphic quadratic differentials. Let K2 denote the symmetric product of the canonicalbundle K with itself. Thus sections of K2 are locally, with respect to a holomorphic trivial-ization of K, of the form q = h dz2. A section q of K2 is called a holomorphic quadraticdifferential if ∂h = 0. We denote by QD(Σ) the C-vector space of holomorphic quadraticdifferentials on Σ. Its complex dimension is dim QD(Σ) = 3(γ − 1).

Now let (E, h) be a hermitian vector bundle and suppose (A,Φ) is a solution of the self-duality equations (3.1). Then with respect to a local holomorphic trivialization of the vectorbundle (E, ∂A) we can write Φ = ϕdz, where the map ϕ satisfies

∂ϕ =(∂ϕ11 ∂ϕ12

∂ϕ21 ∂ϕ22

)= 0.

In particular, the determinant detϕ = ϕ11ϕ22 − ϕ12ϕ21 is holomorphic. It follows that forany Higgs field Φ the determinant detΦ is a holomorphic quadratic differential. We henceobtain a fibration (called Hitchin fibration)

det : Msd → QD(Σ), [(A,Φ)] 7→ det Φ.

A basic property of this map is that it is surjective with fibres det−1(q) diffeomorphic tocompact tori. A generic fibre has dimension 1

2 dimMsd = 6(γ − 1). We again recover the

moduli space of constant central curvature connections as the fibre det−1(0).

3.2. The Higgs bundle moduli space as a hyperkahler manifold. Let (M,J) be acomplex manifold with compatible Riemannian metric g. Recall that M is called a Kahlermanifold if the two-form ω = g(J ·, ·) is closed. In particular, the pair (M,ω) is then asymplectic manifold. A hyperkahler manifold is a tuple (M, I, J,K, g) such that I, Jand K are complex structures satisfying the quaternion relations

IJ = −JI = K, (3.2)

and g is a Kahler metric for each I, J and K.

8 JAN SWOBODA

Kahler quotients. Let (M,J, g) be a Kahler manifold. Let G be a compact Lie group withLie algebra g. Suppose G acts on M preserving both the complex structure J and theRiemannian metric g. A map

µ : M → g∗

is called moment map for this action if for all ξ ∈ g

dµ(ξ) = ω(·, Xξ) ∈ Ω1(M)

and µ is G-equivariant (with respect to the coadjoint action of G on g∗). Here we let Xξ bethe fundamental vector field generated by ξ, i. e.

Xξ(p) =d

dt

∣∣∣∣t=0

exp(tξ)p (p ∈M).

It follows that the level set N := µ−1(0) is invariant under G. If 0 ∈ g∗ is a regular valueof µ then N is a submanifold of M of dimension dimM − dimG. We assume this to be thecase and in addition that the action of G on N is free. Then one can pass to the quotientmanifold M//G := N/G which is again Kahler by the following result.

Theorem 3.7 (cf. [HKLR87]). The manifold M G := N/G inherits from M a complexstructure J and a Riemannian metric g which turn this quotient into a Kahler manifold ofreal dimension dimM − 2 dimG. The induced Kahler form ω = g(J ·, ·) satisfies

ω(X, Y ) = ω(X,Y )

for all vector fields X and Y on N . (Here we denote by π : N → N/G the canonicalprojection).

Proof. For a proof we refer to [HKLR87, Theorem 3.1].

Remark 3.8. One can show that the Kahler quotient M G coincides with the quotientM ss/Gc in the sense of geometric invariant theory [Ki84].

Example 3.9. Consider M = sl(n,C) with complex structure J given by multiplicationwith i. We endow M with the Kahler metric g(X,Y ) = Re tr(AB∗). The special unitarygroup SU(n) acts on M by isometries through

g ·X = g−1Xg.

This action preserves the complex structure J . Identifying su(n)∗ with su(n) via the Killingform, a moment map is given by

µ : M → su(n), X 7→ i

2[X,X∗].

It follows that µ−1(0) comprises the subset of normal endomorphisms in M . Since anynormal X ∈ M is diagonalizable by a special unitary matrix it follows that the quotientM SU(n) is in bijection to the set X = diag(d1, . . . , dn) | detX = 1/Sn.

Coming back to Higgs bundles, we consider the affine space

X := A(E, h)× Ω1,0(Σ,End(E))

(respectively a suitable Sobolev completion of it). The map

A(E, h) → C(E), dA 7→ ∂A


identifies A(E, h) with the space of holomorphic structures on E. Under this identification,the tangent space of X at (A,Φ) is

T(A,Φ)X = Ω0,1(Σ,End(E))× Ω1,0(Σ,End(E))

with complex structure J given by

J(α, ϕ) = (iα, iϕ).

One may check that

Y := (A,Φ) ∈ X | ∂AΦ = 0

is a complex submanifold of (X, J). This submanifold is invariant under the action of thecomplex gauge group Gc(E). Furthermore, it carries a G(E, h) invariant Kahler metric givenby

G(A,Φ)((α, ϕ), (α′, ϕ′)) =∫

Σ

tr(α ∧ (α′)∗) +∫

Σ

tr(ϕ∗ ∧ ϕ′). (3.3)

As it turns out, the map

µ1 : Y → Ω0(Σ, u(E, h)), (A,Φ) 7→ FA + [Φ ∧ Φ∗]

has all the properties required of a moment map for the action by G(E, h). We set

λ := −idr

idE ⊗ω.

Notice then that the quotient µ−11 (λ)/G(E, h) is precisely the set of gauge equivalence

classes of solutions of (3.1). Consequently, by an adaption of Theorem 3.7 to this infinite-dimensional situation, the space

B∗0/G(E, h) ⊆ B0/G(E, h)

of gauge equivalence classes of irreducible solutions inherits from (Y, J,G) the structure ofa Kahler manifold. We may proceed similarly with two other complex structures on X,i. e. with

J2(α, ψ) = (iψ∗,−iα∗), J3(α, ψ) = (−ψ∗, α∗).

One easily checks that these are again complex structures compatible with the Riemann-ian metric G. Furthermore, with J1 = J it follows that these complex structures satisfythe quaternion relations (3.2), turning X into a (flat) hyperkahler manifold. Equivariantmoment maps with respect to the latter two complex structures are

µ2(A,Φ) = Re(∂AΦ), µ3(A,Φ) = Im(∂AΦ).

We set µ = (µ1, µ2, µ3). By the same quotient construction as before, it follows that themanifold

µ−1(λ, 0, 0)/G(E, h) = B∗0/G(E, h)

is a hyperkahler manifold.

Theorem 3.10 (Hitchin). Assume that the degree d of E is odd. Then the induced hy-perkahler metric G on B∗0/G(E, h) is complete.

10 JAN SWOBODA

Proof. We sketch the instructive proof and refer to [Hi87, Theorem 6.1] for details. Assumeby contradiction that there exists a maximal unit speed geodesic s 7→ γ(s) of finite length,i. e. the set s ∈ R | γ(s) ∈ Md is bounded above. Let smax denote the supremum ofthis set. We can lift γ to a horizontal curve γ in the space Md of all solutions to theself-duality equations, which we may regard as the total space of a principal G(E, h)-bundleover Md. Note that at this point we use the assumption d odd, hence all solutions in Md

are irreducible, and this implies that G(E, h) acts freely on Md. Note that by definition ofthe metric on Md, the curve γ is also parametrized by arc length. Hence for any s0 ≤ sn

within the finite time of existence of γ, the length sn− s0 of the segment of γ between γ(s0)and γ(sn) is bounded below by the straight line distance with respect to the metric G in(3.3). Now let (sn)n∈N be a sequence of points which converges from below to smax. Itfollows that there exists a constant M > 0 such that

‖A(sn)−A(s0)‖2L2(Σ) + ‖Φ(sn)− Φ(s0)‖2L2(Σ) ≤M

for all n ∈ N. In particular, the sequence of norms ‖Φ(sn)‖L2(Σ) is uniformly bounded. ByTheorem 3.6 it follows that, after modifying by unitary gauge transformations if necessary,the sequence ((A(sn),Φ(sn))n∈N has a (uniformly) convergent subsequence. Let (A∗,Φ∗)denote its limit. It is again a solution to the self-duality equations. Hence the geodesic γmay be continued up to time smax + ε for some ε > 0, a contradiction.

4. The structure of ends of the Higgs bundle moduli space

The basic reference for this section is the preprint [MSWW14]. Recall that a smoothmanifold has k ≥ 0 ends if there exists a compact subset K0 ⊆ M such that for anycompact subset K containing K0, the space M \ K consists of k unbounded connectedcomponents. A compact manifold is a manifold with k = 0 ends, while R or R × S1 eachhave k = 2 ends. Our aim in this section is to give an account on the degeneration profileof those solutions to Eq. (3.1) which represent points in the ends of the moduli space Ms

d.In view of Theorem 3.6 this amounts to understanding solutions (A,Φ) where ‖Φ‖L2(Σ)

becomes large.

4.1. The limiting equation. To motivate the kind of behaviour one should expect of suchlarge solutions, we consider for large parameter t > 0 the rescaled self-duality equations

∂AΦ = 0,F⊥A + t2[Φ ∧ Φ∗] = 0.

(4.1)

Clearly, any solution (A,Φ) of (4.1) amounts to a solution (A, tΦ) of the unrescaled equations(3.1). Now let (Aj ,Φj) be a sequence of the rescaled equations (4.1) to parameter tj suchthat tj →∞ as j →∞. Suppose this sequence converges suitably, e.g. in the C1-topology,to some limit (A∞,Φ∞). This limit would then have to satisfy the decoupled limitingequations

∂AΦ = 0,F⊥A = [Φ ∧ Φ∗] = 0.

(4.2)

With [Φ∞ ∧Φ∗∞] = 0 we conclude that the Higgs field is everywhere normal, meaning thatwith respect to a local unitary frame

Φ∞ = ϕ∞ dz,


where the endomorphism ϕ∞ satisfies [ϕ∞, ϕ∗∞] = 0. In particular, ϕ∞(z) is unitarily diag-onalizable for every z. On the other hand, consider the holomorphic quadratic differentialq∞ := det Φ∞ ∈ QD(Σ). Using the Riemann-Roch theorem one can show that q∞ has4(γ − 1) zeroes, counted with multiplicities. Let us assume that all these zeroes are simple,which generically is the case. Then at any such z ∈ q−1

∞ (0) the endomorphism ϕ∞(z) hasJordan type

ϕ∞(z) ∼(

0 10 0

).

Otherwise, ϕ∞ had to vanish at z = 0, and then detϕ vanishes to at least order 2 at z = 0,contradicting our assumption. Thus ϕ∞(z) is not diagonalizable. We are thus lead to theconclusion that

• the sequence (Aj ,Φj) cannot uniformly converge, at least not to a limiting solutiondefined globally on Σ. A limit, if exists, will be expected to have singularities in thezeroes q−1

∞ (0);• the limiting equation (4.2) does not admit an everywhere smooth solution (A,Φ);• the holomorphic quadratic differential q∞ = detΦ∞ corresponding to a solution of

(4.2) still might be nonsingular.

4.2. Model solutions. To understand how degeneration might occur, consider first therescaled self-duality equations on the unit disc D ⊆ C. We look for solutions (A,Φ) of (4.1)such that

q = det Φ = −z,

i. e. where q has a simple zero at z = 0. Note that any holomorphic quadratic differential onD with a simple zero at z = 0 can be brought into this form by a change of the holomorphiccoordinate. The following ansatz appears at several places in the physics literature, cf. thearticles by Mason and Woodhouse [MasWood93], Ceccotti and Vafa [CeVa93], and Gaiotto,Moore and Neitzke [GMN10]. We are very grateful to Andy Neitzke for bringing this familyof special solutions to our attention. We let

At =(

18

+14|z| dht

d|z|

)(1 00 −1

)(dz

z− dz

z

),

Φt =

(0 |z| 12 eht

z

|z|12e−ht 0

)dz

(4.3)

for some function ht : [0,∞) → R. Inserting this ansatz into (4.1) and using that in polarcoordinates ∂ = eiθ

2 (∂r + ir∂θ) we obtain

∂AtΦt = ∂(|z| 12 eh

t )− 1z

(14

+12|z| dht

d|z|

)|z| 12 eht

=eiθ

4|z|− 1

2 eht +eiθ

2|z| 12 dht

d|z|eht − |z| 12

4zeiθ

−|z|32

2zdht

d|z|eht

= 0,

12 JAN SWOBODA

which shows that the first equation is automatically satisfied. Furthermore, using the iden-tities 2i dθ = dz

z − dzz and dr ∧ dθ = i

2|z|dz ∧ dz it follows that

FAt = dAt = − 14|z|

(|z|h′′t + h′)(

1 00 −1

)dz ∧ dz

and

[Φt ∧ Φ∗t ] =(

2|z| sinh(2ht) 00 −2|z| sinh(2ht)

)dz ∧ dz.

Whence the second equation in (4.1) is satisfied if ht solves the ODE

h′′t + |z|−1h′t = 8t2|z| sinh(2ht).

Substituting ρ = 8t3 |z|

32 and ht(|z|) = ψ(ρ) we obtain(

d2

dρ2+

1ρ

d

dρ

)ψ =

12

sinh(2ψ), (4.4)

which we recognize as a Painleve III equation for the function ψ : [0,∞) → R. It is a knownfact (cf. [Wi01]) that (4.4) has a unique positive solution with

• ψ(ρ) ∼ − log(ρ13∑∞

j=0 ajρ4j3 ) for ρ 0,

• ψ(ρ) ∼ K0(ρ) ∼ ρ−12 e−ρ, ρ ∞, with K0 denoting the so-called modified Bessel

function of the second kind (or Macdonald function),• ψ(ρ) is monotone decreasing.

t = ¥

t = 200

t = 100

0.2 0.4 0.6 0.8 1.0r

10

20

30

40

ÈAHrLÈ

Figure 1. Degeneration behaviour of the connection component Afidt (r, θ)

of the fiducial solution (Afidt ,Φfid

t ) as t→∞.

Consequently, the functions ht(|z|) = ψ( 8t3 |z|

32 ) and ft(z) = 1

8 + 14 |z|

dht

d|z| satisfy

ht(|z|) → 0 and ft(|z|) →18


as t→∞ for any fixed |z| 6= 0. It follows that the solutions (At,Φt) obtained through thisansatz converge uniformly and exponentially fast on D \ 0 to (A∞,Φ∞), where

A∞ =18

(1 00 −1

)(dz

z− dz

z

)is a flat connection and

Φ∞ =

(0 |z| 12z

|z|12

0

)dz (4.5)

is a normal Higgs field.

Definition 4.1 (Model solutions). The special solution (At,Φt) on D as in (4.3) withht(|z|) = ψ( 8t

3 |z|32 ) is called fiducial (or model) solution to parameter 0 < t < ∞. The

solution (A∞,Φ∞) on D× = D \ 0 is called limiting fiducial solution.

The complex gauge orbit of the fiducial solutions. We next show that all the fiducial solutions(Afid

t ,Φfidt ) are actually pairwise complex gauge equivalent. Towards that end, define with

respect to a fixed holomorphic frame the pair

A0 = 0, Φ0 =(

0 1z 0

)dz.

Proposition 4.2. (i) Over D, the fiducial solution (Afidt ,Φfid

t ) is complex gauge equivalentto (A0,Φ0). In particular, all fiducial solutions for 0 < t < ∞ are mutually complexgauge equivalent.

(ii) Over D×, the limiting fiducial solution (Afid∞ ,Φfid

∞ ) is complex gauge equivalent to(A0,Φ0) by the singular gauge transformation

g∞ =(|z|− 1

4 00 |z| 14

),

i. e., g∗∞(A0,Φ0) = (Afid∞ ,Φfid

∞ ).

Proof. The second assertion is a straightforward computation so we focus on the first. Forsimplicity, omit the superscript ‘fid’ from all quantities. We seek a complex gauge transfor-mation of the form

g =(eut 00 e−ut

)∈ Γ(D,SL(E)), ut = ut(r),

so that g∗(A0,Φ0) = (Afidt ,Φfid

t ). Since ∂z = 12e

iθ∂r on rotationally symmetric functionsand A0 = 0, we have

g−1 ∂A0 g = ∂ + g−1∂g = ∂ +12eiθ

(∂rut 0

0 −∂rut

)dz.

On the other hand,

∂At= ∂ − ft

(1 00 −1

)dz

z.

Thus g−1 ∂A0 g = ∂Atif and only if

∂rut = − 14r− 1

2∂rht,

14 JAN SWOBODA

which has the solutionut = −1

4log r − 1

2ht.

Hence g∗A0 = At; moreover

g−1Φ0g =(

0 e−2ut

ze2ut 0

)dz,

and e−2ut = r12 eht , so that g−1Φ0g = Φt.

4.3. Limiting configurations. Motivated by this local discussion we next seek to under-stand globally defined solutions on Σ of the limiting equation (4.2). We hence fix a simpleholomorphic quadratic differential q ∈ QD(Σ). Set Σ× = Σ \ q−1(0). We ask for solutions(A,Φ) of (4.2) with singularities in the set of zeroes of q and which are smooth on itscomplement Σ×.

Theorem 4.3 ([MSWW14], Theorem 4.1). Let (∂,Φ) be a Higgs bundle with simple Higgsfield Φ. Then there is a Hermitian metric H0 so that if A = A(H0, ∂) is the associatedChern connection then the pair (A,Φ) is complex gauge equivalent via some transformationg∞ ∈ Γ(Σ×,SL(E)) to a limiting configuration (A∞,Φ∞), i.e., (A∞,Φ∞) := g∗(A,Φ).

To find a gauge transformation g∞ ∈ Γ(Σ×,SL(E)) as asserted in the theorem we proceedin two steps. The first step is to construct g ∈ Gc(Σ×) such that g−1Φg is normal everywhereon Σ×. This is certainly possible fibrewise over each point z ∈ Σ× as the endomorphismΦ(z) has the two nonzero eigenvalues λ(z) and −λ(z). One can check that there are notopological obstructions against finding a gauge transformation g ∈ Gc(Σ×,SL(E)) whichnormalizes Φ everywhere on Σ×. Hence from now on we may assume that (A,Φ) satisfieson Σ×

∂AΦ = 0, [Φ ∧ Φ∗] = 0.

Note that (A,Φ) does not yet satisfy the assertions of Theorem 4.3 since in general F⊥A 6= 0.However, one observes that the Higgs field Φ has a one-dimensional stabilizer, namely thecomplex line bundle

LcΦ := γ | [Φ, γ] = 0 ⊆ Ω0(Σ,End0(E).

The term ∗FA is a section of LcΦ. In fact, we can locally diagonalize Φ and write Φ = ϕdz,

where

ϕ =(λ 00 −λ

).

With respect to the same local unitary frame, the connection A is of the form A = αdz −α∗ dz with

α =(α0 α1

α2 −α0

).

Then the condition ∂AΦ = 0 reads

0 =(∂λ 00 −∂λ

)dz ∧ dz −

[(α0 α2

α1 −α0

),

(λ 00 −λ

)]dz ∧ dz =

(∂λ 2λα2

−2λα1 −λ

)dz ∧ dz.

It follows that α1 = α2 = 0 and hence the curvature of A satisfies

F⊥A =(−∂α0 − ∂α0 0

0 ∂α0 + ∂α0

)dz ∧ dz.


This shows that in fact ∗F⊥A ∈ Γ(LcΦ). The idea is now to choose a further complex gauge

transformation within the stabilizer subgroup of Φ in order to transform to zero the curvatureterm F⊥A . Such gauge transformations can be written as g = exp(γ) for a section γ ∈ Γ(Lc

Φ).We may assume in addition that γ is hermitian, i. e. that γ∗ = γ. A short calculation yields

Fg∗A = g−1FAg + g−1(∂A(gg∗∂A(gg∗)−1)

)g

= g−1FAg + g−1(∂A(exp(2γ)∂A(exp(−2γ))

)g

= g−1FAg − 2g−1(∂A∂Aγ)g.

Suppose g = exp(γ) satisfies F⊥g∗A = 0. From the identity

2i ∗ ∂A∂Aγ = ∆Aγ − 2i ∗ [FA, γ]

and the fact that, as seen above, [FA, γ] = 0 we infer the Poisson equation

∆Aγ = i ∗ F⊥A . (4.6)

Assuming a solution γ of this equation exists, we may apply the gauge transformationg = exp(γ) to obtain with (A∞,Φ∞) = g∗(A,Φ) the desired limiting configuration. Sincealready the connection A has a pole of order 1 in the points q−1(0) the induced Laplacian ∆A

is an elliptic operator with singular coefficients. In fact, as we shall see, these singularitiesare of conical type and we can invoke basic facts from the b-calculus to show existence anduniqueness of a solution.

4.4. Conic operators. As a simple model case, consider the scalar Laplacian in polarcoordinates

∆ =∂2

∂r2+

1r

∂

∂r+

1r2

∂2

∂ϕ2

on the punctured disc D× = D \ 0. We impose Dirichlet boundary conditions, i. e. weallow only for solutions u with u|∂D = 0. Let us formally calculate the elements of thenullspace of ∆. Assume u ∈ L2(D, rdr ∧ dθ). Let

u =∑k∈Z

fkeikθ

be the Fourier decomposition of the function u. For k ∈ Z we set

Dkf = f ′′ +1rf ′ − k2

r2fk.

Then it follows that

∆u =∑k∈Z

Dkfkeikϕ.

Thus Dkfk = 0 if and only if

fk =

Akr

k +Bkr−k, k 6= 0,

C +A0 log(r), k = 0.

Therefore the formal solutions of ∆u = 0 under Dirichlet boundary conditions are

u = A0 log(r) +∑

k∈Z\0

Ak(rk + r−k)eikϕ

for constants Ak ∈ R. We next impose the decay condition

u = O(rβ) (β ∈ R)

16 JAN SWOBODA

for solutions u at r = 0. The nullspace of ∆ is now finite-dimensional, its dimensiondepending on the choice of β. For instance, with −1 < β < 0 one obtains that dim ker ∆ = 1,this space being spanned by the function log(r). Moreover, we see that the operator ∆becomes injective for the choice β > 0. A similar consideration yields the dimension of thecokernel

coker∆ =v =

∑k∈Z

gkeikϕ∣∣∣ v|∂D=0, 〈v,∆u〉L2(r dr∧dϕ) = 0 for all u

under the above decay condition, cf. the table below. One observes that the dimensions ofker ∆ and coker ∆ jump precisely at integer values of β, the so-called indicial roots of theoperator ∆. These are characterized as those values of β for which there exists a functionu ∈ O(rβ) such that ∆u ∈ O(rβ−1) (rather than the expected decay rate O(rβ−2)). Denoteby Γ(∆) = Z the set of indicial roots. Let us introduce the b-Sobolev space

H2b :=

u ∈ L2(r dr ∧ dϕ)

∣∣ ViVj ∈ L2(r dr ∧ dϕ), Vi, Vj ∈ r∂r, ∂ϕ,

and analogously for H`b with ` ≥ 0. One may directly check that for β /∈ Γ(∆) the operator

∆: rβ+2H2b → rβL2(r dr ∧ dϕ)

is Fredholm and that its index is constant on each connected component of the set R\Γ(∆).We can summarize its properties in the following table.

β -3 -2 -1 0 1 2 3ker ∆ 5 3 1 0 0 0

coker∆ 0 0 0 0 1 3ind ∆ 5 3 1 0 -1 -3

Let us point out that ∆ is a bijective Fredholm operator precisely if the weight β is containedin the interval (0, 1).

This example generalizes to operators

L = −∂2r −

n− 1r

∂r +1r2LN

on a smooth manifold Mn with isolated cone points. I. e. we assume that about each conepoint there exists a neighbourhood U diffeomorphic to the cone C0,1(N) over a closed man-ifold N (called the link) of dimension n − 1. It is furthermore assumed that the operatorLN is elliptic with smooth coefficients. In the above example, N = S1 and LN = ∆S1 .The theory of conic operators readily extends to generalized Laplacians on vector bundles.So-called iterated edge operators provide a far-reaching extension of this theory to gen-eral stratified spaces, cf. [MaMo11] and the reference therein. The main result concerningmapping properties of conic differential operators is the following one.

Theorem 4.4. The operator L is Fredholm as a map

L : rδH`+2b (M) → rδ−2H`

b(M),

provided δ is not contained in the set

δ(γ) =γ +

n

2| γ indicial root for L

.

Moreover, any solution u ∈ rδL2(M) of the equation

Lu = f (f ∈ rδ−2L2(M), f phg)


lies actually in rδH`b(M) for any ` ≥ 0. It has a polyhomogeneous expansion with exponents

γ + `, where γ is either an indicial root of L, or γ = γ′ + 2 and γ′ appears as exponent inthe polyhomogeneous expansion of f , and in either case with δ(γ) > δ.

Proof. For a proof we refer to [MaMo11, Proposition 6].

4.5. Analysis of the operator ∆A. We again take up the discussion of the proof ofTheorem 4.3. Recall that it remains to show solvability of the Poisson equation (4.6) in thespace of hermitian sections γ of the line bundle Lc

Φ. By appealing to Proposition 4.2 we maysimplify the analysis of this problem slightly. Namely, after modifying the pair (A,Φ) witha suitable complex gauge transformation, we may assume that (A,Φ) coincides with thelimiting fiducial solution (Afid

∞ ,Φfid∞ ) on discs about the punctures, while Φ remains normal

on the whole of Σ×. This modification has the advantage that the curvature component F⊥Avanishes on these discs, hence the right-hand side of the Poisson equation is now supportedaway from the punctures.

Let us consider the twisted Laplacian

∆A : Ω0(Σ, isu(E)) → Ω0(Σ, isu(E)),

where near the set q−1(0) of punctures the connection is of the form

A =18

(1 00 −1

)(dz

z− dz

z

)= αdθ.

We compute the indicial roots of ∆A on small discs about the punctures. In local coordi-nates,

∆A = −(∂2r +

1r∂r +

1r2T )γ,

where

Tγ = ∂2θγ + 2[α, ∂θγ] + [α, [α, γ]]

is the r-independent tangential operator. Note that r2∆A is a conic operator in the abovesense. The operator ∆A leaves invariant the rank-1 subbundles

L0 =(

u 00 −u

)| u ∈ R

and

iLΦ =(

0(i sin( θ

2 ) + cos( θ2 ))u(

−i sin( θ2 ) + cos( θ

2 ))u 0

)| u ∈ R

,

(iLΦ)⊥ =(

0(sin( θ

2 )− i cos( θ2 ))u(

sin( θ2 ) + i cos( θ

2 ))u 0

)| u ∈ R

of Ω0(D, i su0(E)). On L0, its set of indicial roots is Z, since ∆A|L0 = ∆std, cf. the com-putations in §4.4. Restricted to iLΦ and (iLΦ)⊥, the set of indicial roots of ∆A is Z + 1

2 .Furthermore, by symmetry of ∆A and an integration by parts argument,

∆A : rδH`+2b (Σ, iLΦ) → rδ−2H`

b(Σ, iLΦ)

is an isomorphism for 12 < δ < 3

2 . Because F⊥A = 0 near the punctures it follows that thePoisson equation (cf. (4.6))

∆Aγ = ∗F⊥A

18 JAN SWOBODA

admits a unique solution γ ∈ Γ(iLΦ), which in view of Theorem 4.4 decays as r12 as r

0. This discussion completes the proof of the existence theorem for limiting connections,cf. §4.3.

Figure 2. Hitchin fibration over the ray t2q, t → ∞, for a simple holo-morphic quadratic differential q ∈ QD(Σ).

4.6. Deformation theory of limiting configurations. Let (A∞,Φ∞) be a limiting con-figuration associated with a simple holomorphic quadratic differential q = det Φ∞, theexistence of which is guaranteed by Theorem 4.3. We study the solution space of pairs(A,Φ) such that det Φ = q and

∂AΦ = 0,F⊥A = [Φ ∧ Φ∗] = 0

(4.7)

on Σ×, modulo unitary gauge transformations. Because any two normal Higgs fields whosedeterminants coincide are related by a unitary conjugation, we may assume without loss of


generality that Φ = Φ∞. Set A = A∞ +α for some 1-form α. Then equations (4.7) become[Φ∞ ∧ α] = 0,dA∞α = 0.

(4.8)

Hence we are looking for 1-forms α ∈ Ω1(iL∞) which satisfy dA∞α = 0. Imposing thegauge fixing condition d∗A∞α = 0 it follows by Hodge theory that this space is isomorphic toH1(Σ×, iLΦ∞). Since there do not exists any nontrivial parallel sections of iL∞ it followsthat H0(Σ×, iLΦ∞) = 0. Thus by Poncare duality, H2(Σ×, iLΦ∞) = 0. Set M = Σ \Bε(q−1(0)) for some small ε > 0. It follows that the map

H1(M,∂M ; iL∞) → H1(Σ×, iL∞)

is an isomorphism. Hence we conclude that

dimR H1(Σ×, iLΦ∞) = 6γ − 6.

Having thus clarified the infinitesimal deformation theory of limiting configurations, we turnto a description of all such solutions. Note that with (A∞,Φ∞) as above and any solution αof the linear system (4.8), also (A∞+α,Φ∞) is a limiting equation. However, it might againbe contained in the same unitary gauge orbit as (A∞,Φ∞), and thus represent the samepoint in moduli space. As it turns out, the fibre above Φ∞ in this moduli space is actuallythe quotient of the de Rham cohomology space H1(Σ×, iL∞) by the lattice of classes withinteger periods.

Theorem 4.5. The moduli space of solutions of the limiting equations (4.7) with fixedsimple determinant q is a real torus of dimension 6γ − 6.

Proof. For more details on the proof we refer to [MSWW14, Corollary 4.11].

4.7. Gluing theorem. We fix a limiting configuration (A∞,Φ∞) such that the associatedholomorphic quadratic differential q = det Φ∞ is simple. The aim of this section is to provefor large parameter t the existence of a solution of the rescaled self-duality equations (4.1)of the form (At,Φt) = g∗t (A∞,Φ∞). The strategy we will follow is to first define a family ofapproximate solutions (Aappr

t ,Φapprt ) which will afterwards be perturbed into exact solutions

using the contraction mapping principle.

Choose a smooth cutoff-function χ : Σ → R with support in the union of discs B1(p), p ∈ p,and satisfying χ ≡ 1 on B 1

2(p). Let the complex gauge transformation gt be defined by

gt = exp(χγt),

where

γt =(− 1

2ht 00 1

2ht

).

The section γt is hereby chosen such that gt transforms (A∞,Φ∞) on each disc B 12(p)

to the fiducial solution (Afidt ,Φfid

t ). Set (Aapprt ,Φappr

t ) := g∗t (A∞,Φ∞). We may regard(Aappr

t ,Φapprt ) as a good approximation to a solution of (4.1) since the error satisfies the

exponential bound

‖FAapprt

‖L2(Σ) + ‖[Φapprt ∧ (Φappr

t )∗]‖L2(Σ) ≤ Ce−δt. (4.9)

Using the Banach fixed point theorem, this approximate solution will now be perturbed intoan exact one. We make the ansatz

20 JAN SWOBODA

(At,Φt) = exp(ut)∗(Aapprt ,Φappr

t )

for some hermitian ut ∈ Ω0(Σ, isu(E)). Inserting it into Eq. (4.1) yields the implicit equation

ut = −L−1t (Ht(A

apprt ,Φappr

t ) +Qt(ut)) ,

where Qt(ut) denotes quadratic and higher order terms in ut. Here we let Lt denote thelinearization of the map Ht,

Lt : Ω0(Σ, isu(E)) → Ω0(Σ, isu(E)), u 7→ ∆Aapprt

u− i ∗ t2MΦapprt

u,

where

MΦapprt

u = [Φapprt ∧ [Φappr

t ∧ u]]− [Φapprt ∧ [(Φappr

t )∗ ∧ u]].

We note that the linear operator Lt is strictly positive, hence invertible, and denote itsinverse by Gt. By elliptic regularity, the family of operators Gt : L2(Σ) → H2(Σ) is bounded,however not uniformly in t. To make the contraction argument work, we need a preciseestimate of its norm.

Figure 3. Gluing construction: a smooth model solution (Afidt ,Φfid

t ) isglued to a limiting configuration (A∞t ,Φ

∞t ) on small discs about the set

q−1(0). (A∞t ,Φ∞t ) remains unchanged on the exterior region Σext.


Global linear estimates. In the sequel, we write X int =⋃

p∈pD×1 (p) for the union of the

punctured unit discs, and assume that (A∞,Φ∞) is in fiducial form in each of these. Wealso set Xext = X \ X int.

Let Lt be computed at the pair (Aapprt ,Φappr

t ). Let λt(X) > 0 be the first eigenvalue ofLt = ∆Aappr

t+ t2MΦappr

ton X, and λt(X int), resp. λt(Xext) the first Neumann eigenvalues

of Lt on X int and Xext, respectively. The domain of the Neumann extension on either ofthese regions is

u ∈ H2(isu(E)|Xint / ext) | (dAtu)ν = 0where ν is the unit normal. The key result which allows us to extend the estimates aboveto the whole of X is the domain decomposition principle, see for instance [Ba, Proposition3], which states that

λt(X) ≥ minλt(X int), λt(Xext).Using this principle, it is not hard to obtain a uniform lower bound for λt(X). Since theoperator Lt over Xext does not depend on t at all, this clearly holds for λt(X int). On theother hand, considering the restriction of Lt to each disc in X int allows to estimate λt(X int)by Fourier analysis. One therefore arrives at the following result.

Lemma 4.6. For t ≥ 1, there is a uniform lower bound

λt(X) ≥ λ > 0.

Proof. For details of the proof we refer to [MSWW14, Lemma 6.3].

We need to strengthen this L2-estimate to an estimate for the norm of the operatorGt : L2(Σ) → H2(Σ). Here again a Fourier decomposition on the unit discs about thepunctures allows one to obtain such an estimate by studying an ordinary differential op-erator. Interestingly, the operator Gt is not diagonal with respect to the Hilbert spacebasis (exp(i`φ))`∈Z. However, the Hilbert space L2(r dr) decomposes into a sum of invari-ant 2-dimensional subspaces, the action of Gt on each subspace being not hard to analyze.Omitting the details, we state the result.

Lemma 4.7. For all u ∈ H2(D) ∩H10 (D), we have ‖u‖H2 ≤ Ct2‖u‖Lt , where ‖u‖Lt is the

graph norm of the operator Lt.

Proof. For a proof we refer to [MSWW14, Lemma 6.5].

We now use the t-dependent Sobolev space H2t := domLt, endowed with the graph norm

‖u‖2Lt= ‖u‖2L2 + ‖Ltu‖2L2 .

Clearly, ‖Gtv‖Lt≤ C‖v‖L2 for all t ≥ 1 and some C independent of t. Note that H2

t = H2

for all t, but the norms are not uniformly equivalent as t∞.

Lemma 4.8. If u ∈ H2(isu(E)), then ‖u‖H2 ≤ Ct2‖u‖Lt .

Proof. Using cut-off functions, write u = uint + uext with suppuint ⊂ X int and suppuext ⊂X \

⋃p∈pD1/2(p). Then by Lemma 4.6 we have

‖uint‖H2 ≤ C(1 + t2)‖u‖Lt.

22 JAN SWOBODA

On X \⋃

p∈pD1/2(p), consider the linear operator

Lt := ∆A∞ + t2MΦ∞

with Dirichlet boundary conditions. Then Lt is invertible and we write Gt := L−1t . Now

‖Ltu‖L2 ≤ ‖Ltu‖L2 + ‖(Lt − Lt)u‖L2

and since At converges to A∞ and Φt converges to Φ∞ exponentially in t,

‖(Lt − Lt)u‖L2 ≤ Ce−δt‖u‖L2 .

In addition,

‖∆A∞u‖L2 = ‖∆A∞u+ t2MΦ∞u− t2MΦ∞u‖

≤ ‖Ltu‖L2 + t2‖MΦ∞u‖L2

≤ ‖Ltu‖L2 + t2 sup |MΦ∞ |‖u‖L2 ,

which leads to the estimate

‖∆A∞u‖L2 ≤ ‖Ltu‖L2 + Ct‖u‖L2 + Ct2‖u‖L2 .

This gives the claim since the graph norm of ∆A∞ is equivalent to the standardH2-norm.

Summarizing we proved the following global linear estimate.

Theorem 4.9. Let (Aapprt ,Φappr

t ) be the initially defined approximate solution. Then theinverse Gt to Lt = ∆Aappr

t+ t2MΦappr

tsatisfies

‖Gtv‖H2 ≤ Ct2‖v‖L2 .

Proof of the main result. We can finally state and prove the main result of [MSWW14].

Theorem 4.10. There exists a constant m > 0 and for every sufficiently large t > 1 aunique map ut ∈ Bt−m(0) ⊆ H2(Σ) such that

(At,Φt) = exp(ut)∗(Aapprt ,Φappr

t )

satiesfies Eq. (4.1).

Proof. For some ρ > 0 consider the map

T : Bρ(H2) → H2, u 7→ −Gt (Ht(Aapprt ,Φappr

t ) +Qt(ut)) .

We claim it is a contraction if ρ = ρt = t−4−ε, where ε > 0 and t > 0 is sufficiently large.Assuming this claim, the assertion of the theorem follows immediately from the Banachfixed point theorem. To check the claim we note that by the error estimate (4.9) and since‖Qt(ut)‖L2 ≤ Ct2‖ut‖2H2 , the image of the ball Bρt

(H2) under the map

u 7→ Ht(Aapprt ,Φappr

t ) +Qt(u)

is contained in some ball BCt2ρ2t(L2). Applying Theorem 4.9 it follows that T maps Bρt

(H2)into itself and is contractive. The claim thus follows.


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