joint work with ilaria mondello kazuo akutagawa (chuo ... · contents x0 on scalar curvature x1 the...

Edge-cone Einstein metrics and Yamabe metricsjoint work with Ilaria Mondello

Kazuo AKUTAGAWA (Chuo University)

MSJ-SI 2018

The Role of Metrics in the Theory of PDEs

at Hokkaido University, July 2018

Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metricsat Hokkaido University, July 2018 1 /

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Goal of my talk

• (Sn, hβ) : the n-sphere (n ≥ 3) with edge-cone Einstein metric hβ Sn − (Sn−2 t S1) = (0, π2 )× S1 × Sn−2 3 (r , θ, x)

gS = dr2 + sin2 r dθ2 + cos2 r · gSn−2 =: h1 : standard round metric

hβ := dr2 + β2 sin2 r dθ2 + cos2 r · gSn−2∼= h1 : loc. isom. (β > 0)

edge-cone Einstein metric of cone angle 2πβ on (Sn,Sn−2) Main Thm No edge-cone Yamabe metric on (Sn, [hβ]) with 2πβ ≥ 4π


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Contents

§ 0 On scalar curvature

§ 1 The Yamabe problem and Yamabe constants

§ 2 The Yamabe problem on singular spaces

§ 3 The standard edge-cone Einstein metrics on Sn


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References

[Ak1] Computations of the orbifold Yamabe invariant, Math. Z. 271 (2012),611–625.

[Ak2] Edge-cone Einstein metrics and the Yamabe invariant, 83th GeometrySymposium (2016), 101–113.

[ACM1] (with G. Carron and R. Mazzeo) The Yamabe problem on stratifiedspaces, GAFA 24 (2014), 1039–1079.

[ACM2] Holder regularity of solutions for Schrodinger operators on stratifiedspaces, J. Funct. Anal. 269 (2015), 815–840.

[AM] (with I. Modello) Edge-cone Einstein metrics and the Yamabe invariant,in preparation.

[AN-34] (with A.Neves), 3-manifolds with Yamabe invariant greater than thatof RP3, J. Diff. Geom. 75 (2007), 359–386.

[AL] M. Atiyah and C. LeBrun, Curvature, cones and charcteristic numbers,Math. Proc. Cambridge Phil. Soc. 155 (2013), 13–37.

[M] I. Mondello, The local Yamabe constant of Einstein stratified spaces,Ann. Inst. H. Poincare Anal. Non Lineaire 34 (2017), 249–275.

[V] J. Viaclovsky, Monopole metrics and the orbifold Yamabe problem,Ann. Inst. Fourier (Grenoble) 60 (2010), 2503–2543.


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§ 0 On scalar curvature

• Mn : cpt n−mfd (∂M = ∅, n ≥ 2 n ≥ 3 later)

• g = (gij) : Riem. metric, M ⊃ U 3 x : local coor. system

g |U : U → M(n,R), x 7→ (gij(x)) : symm. & positive definite n × n

Rg = (R ijk`) = ∃F (g , ∂g , ∂2g) : curv. tensor(

Kij := g(Rg (ei , ej)ej , ei

): sect. curv.

) Ricg = (Rij := ΣkR

kikj) : Ricci curv. tensor ← the same type as g = (gij)

averaging

Rg := Σi,jgijRij ( = 2Σi<jKij ) ∈ C∞(M) : scalar curv.

averaging

Note

(1) n = 2 Rg = 2× Gauss curv.

(2) Volg (Br (p)) = ωnrn(

1− 1

6(n + 2)Rg (p)r2 + O(r3)

)as r 0


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conformal class/deformation/Laplacian

Note

(3) En Kij ≡ 0 Rg ≡ 0

Sn(1) Kij ≡ 1 Rg ≡ n(n − 1)

Hn(−1) Kij ≡ −1 Rg ≡ −n(n − 1)

Assume n ≥ 3 from now on

[g ] := ef · g | f ∈ C∞(M) : conf. class of g

Set g = u4

n−2 · g , u > 0 : conf. deform. & αn := 4(n−1)n−2 > 0

Lg u := (−αn∆g + Rg ) u = Rgun+2n−2 Lg : conf. Laplacian of g


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§ 1 The Yamabe problem and Yamabe constants

• Mn : cpt n−mfd (n ≥ 3, ∂M = ∅) E :M(M) → R, E (g) :=

∫MRgdµg

V(n−2)/ng

: scale inv. E-H action (Energy of g)

h ∈ Crit(E ) ⇔ h : Einstein metric

i.e., Rich = const · h = (Rh/n) · h

Note

• infg∈M(M) E (g) = −∞, supg∈M(M) E (g) = +∞

• Y (M,C ) := inf E |C = infg∈C E (g) > −∞, ∀C ∈ C(M)Yamabe constant

& g ∈ Crit(E |C ) ⇔ Rg ≡ const


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Yamabe Problem Yamabe Problem∀C ∈ C(M), Find g = u4/(n−2) · g ∈ C s.t. E (g) = inf E |C (minimizer !)(Find u ∈ C∞>0(M) s.t. Qg (u) = inf f>0 Qg (f ), Lg f := − 4(n−1)

n−2 ∆g f +Rg f

Qg (f ) :=

∫M

( 4(n−1)n−2 |∇f |

2 + Rg f2)dµg

(∫Mf 2n/(n−2)dµg )(n−2)/n

=

∫Mf · Lg f dµg

(∫Mf 2n/(n−2)dµg )(n−2)/n

)

Resolution Thm (Yamabe, Trudinger, Aubin, Schoen)

Such g ∈ C always exists ! · · · Yamabe metric

& Rg = Y (M,C ) · V−2/ng ≡ const

Note Yamabe metric ⇒ constant scalar curvature metric

:Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metrics

at Hokkaido University, July 2018 8 /1

Yamabe constant ← conformal invariant

Definition

Aubin’s Inequality

conf.inv. Y (M, [g ]) := infg∈[g ]

E (g) ≤ Y (Sn, [gS]) =: Yn

Yamabe constant ↑ ||n(n − 1)Vol(Sn(1))2/n


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Resolution Thm (Yamabe, Trudinger, Aubin, Schoen)

Such g ∈ C always exists ! · · · Yamabe metric

& Rg = Y (M,C ) · V−2/ng ≡ const

Note

(Proof of Resolution)• Y (M, [g ]) < Y (Sn, [gS]) direct method

• Y (M, [g ]) = Y (Sn, [gS]) (M, [g ]) ∼= (Sn, [gS]) : conf.PMT


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The Yamabe invariant and Fundamental Problems

Definition

Yamabe invariant (or σ invariant) · · · O.Kobayashi, Schoen

diff.− top. inv. Y (Mn) := sup[g ]

Y (Mn, [g ]) ≤ Y (Sn, [gS]) = Y (Sn)

(1) Find a (singular) supreme Einstein metric h with Y (M, [h]) = Y (M) .

A naive approach : Find a nice sequences gj of Yamabe metrics orsingular Einstein metrics on M satisfying Vgj = 1 & Y (M, [gj ]) Y (M)

& Analizes the limit of gj ← Not much progress ! (2) Estimates Y (M) from below/above

& Calculates Y (M) ← Nice progress, particularly n = 3, 4 ! Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metrics


§ 2 The Yamabe problem on singular spaces

For the study of the smooth Yamabe invariant& supreme Einstein metrics,

we will study of the Yamabe problem on singular spaces !

smooth Riem. manifolds ⊂ orbifolds ⊂ conic manifolds

⊂ simple edge spaces (⊃ edge-cone manifolds)

⊂ iterated edge spaces

⊂ almost Riem. metric-measured spaces ⊂ Dirichlet spaces Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metrics


The Yamabe problem on cpt almost Riem. m-m spaces

Setting (M = Ω t S, d , µ) : compact metric-measure space (M, d , µ)

Ω ⊂ M : open dense & ∃n-mfd str. : regular part, S : singular part

∃g : C∞ Riem. metric on Ω compatible with d & µ

Definition

• Y (M, d , µ) = Y (M, [g ]) : Yamabe constant

Y (M, [g ]) := Y (Ω, [g ]) = inff∈C∞c (Ω)−0

Q(Ω,g)(f ) ∈ [−∞,Yn]

Q(Ω,g)(f ) :=

∫Ω

( 4(n−1)n−2 |df |

2 + Rg f2)dµg

(∫

Ω|f |2n/(n−2)dµg )(n−2)/n


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Local Yamabe constant

Definition

• Y`(M, [g ]) : local Yamabe constant · · · depends only on U(S)

Y`(M, [g ]) := infp∈M

limr→0

Y (Br (p) ∩ Ω, [g ]) ∈ [−∞,Yn]

(1) If p ∈ Ω limr→0 Y (Br (p) ∩ Ω, [g ]) = Yn

(2) If p ∈ S limr→0 Y (Br (p) ∩ Ω, [g ]) ≤ Yn

Note • (M,Ω, g) : smooth manifold (i.e. M = Ω) Y`(M, [g ]) = Yn

• (M,Ω, g) : orbifold with S = (p1, Γ1), · · · , (p`, Γ`)

Y`(M, [g ]) = minj Yn

/|Γj |2/n < Yn


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Refined Aubin’s inequality

Definition

• Y`(M, [g ]) : local Yamabe constant · · · depends only on U(S)

Y`(M, [g ]) := infp∈M

limr→0

Y (Br (p) ∩ Ω, [g ]) ∈ [−∞,Yn]

(X , d , µ) : cpt m-m space with appropriate cond.s

(1) Y`(M, [g ]) > 0

(2) Y (M, [g ]) > −∞

(3) Y (M, [g ]) ≤ Y`(M, [g ]) · · · Refined Aubin’s inequality


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Existence of Yamabe metrics Thm ACM (Ak-Carron-Mazzeo [ACM1])

•(M, d , µ) = (M,Ω, g) : cpt almost Riem. m-m space with appropriate cond.s

Assume : Y (M, [g ]) < Y`(M, [g ])

∃u ∈ C∞>0(Ω) ∩W 1,2(M; dµ) ∩ L∞(M) s.t. ||u||2n/(n−2) = 1

Q(Ω,g)(u) = Y (M, [g ]), −4(n − 1)

n − 2∆gu+Rgu = Y (M, [g ]) ·u

n+2n−2 on Ω

Moreover infM u > 0 Note • (M,Ω, g := u

4n−2 · g) : cpt m-m space with the same sing. S

Rg = Y (M, [g ]) ≡ const on Ω Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metrics


Counterexample Thm (Ak [Ak1]) If Y (M, [g ]orb) < minj

Yn/|Γj |2/n

∃g ∈ [g ]orb : orbifold Yamabe metric (minimizer) Counterexample (Viaclovsky [V])

(X 4, h) : conf. cpt. of hyperkahler ALE (X 4, h)

arising from C2/Γ, Γ < SU(2) : finite subgroup

Y (X 4, [h]orb) = Y (S4, [gS])/|Γ|1/2 (= Y4/|Γ|1/2 < Y4)

& @g ∈ [h]orb : No orbifold Yamabe metric

(Modifying the technique in Proof of Obata Uniqueness Thm !) Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metricsat Hokkaido University, July 2018 18 /

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§ 3 The standard edge-cone Einstein metrics on Sn

• (Sn, hβ) : the n-sphere (n ≥ 3) with edge-cone Einstein metric hβ Sn − (Sn−2 t S1) = (0, π2 )× S1 × Sn−2 3 (r , θ, x)

gS = dr2 + sin2 r dθ2 + cos2 r · gSn−2 =: h1 : standard round metric

hβ := dr2 + β2 sin2 r dθ2 + cos2 r · gSn−2∼= h1 : loc. isom. (β > 0)

edge-cone Einstein metric of cone angle 2πβ on (Sn,Sn−2) Main Thm No edge-cone Yamabe metric on (Sn, [hβ]) with 2πβ ≥ 4π


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Local Yamabe constant of the edge-cone sphere (Sn, hβ) Thm (Mondello [M]) Y (Sn, [hβ]) = Y`(S

n, [hβ]) = Y (Rn, [δβ])

= β2/n · Yn = E (hβ) · · · if 0 < β ≤ 1

= Yn · · · if β ≥ 1

In particular, hβ : Yamabe metric on (Sn, [hβ]) if 0 < β ≤ 1, not for β > 1 Main Thm (Non-exiatence) (Ak-Mondello [AM])

If β ≥ 2 (2πβ ≥ 4π), No Yamabe metric on (Sn, [hβ]) !

Conjecture

No Yamabe metric even if 1 < β < 2 ? Kazuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 The Role of Metrics in the Theory of PDEs)Edge-cone Einstein metrics and Yamabe metrics


Proof of Main Thm Aubin’s Lemma (M, [g ])→ (M, [g ]) : finite conf. covering & Y (M, [g ]) > 0

⇒ Y (M, [g ]) > Y (M, [g ]) (Here, (M, g) is C∞)

Note Existence of Yamabe metric on (M, [g ]) is necessary for theproof !

Note Aubin’s Lemma still holds for branced conf. double covering of(Sn, [hβ]) → (Sn, [hβ/2]) !

provided that

(1) Existence of an edge-cone Yamabe metric u4/(n−2) · hβ (u > 0) on (Sn, [hβ])

(2) The following inequality holds :∫Sn

u ·∆hβu dµhβ = −∫Sn

|∇u|2dµhβ


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Proof of Main Thm (continuation)

Suppose that (1) holds : ∃hβ (β ≥ 2) : Yamabe metric on (Sn, [hβ])

Set r(p) := disthβ (p,Sn−2). Then, by elliptic regularity result [ACM1, 2],

on ε-open tubular neighborhood Uε = Uε(Sn−2)

u ∈ C 0,1/β(Uε) & ∂ru(r , θ, x) = O(r1/β−1) as r 0

∫Sn−Uε

u∆hβu dµhβ +

∫Sn−Uε

|∇u|2dµhβ =

∫∂Uε

u ∂rudσhβ = O(r1/β) 0

as ε 0

Hence, we get (2), and thus Y (Sn, [hβ]) > Y (Sn, [hβ/2])

On the other hand, by β > β/2 ≥ 1, Modello’s Thm implies that

Y (Sn, [hβ]) = Yn = Y (Sn, [hβ/2]) Contradiction ! QED


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Thank you very much for your kind attention !


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joint work with ilaria mondello kazuo akutagawa (chuo ... · contents x0 on scalar curvature x1 the...

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