Edge-cone Einstein metrics and Yamabe metrics

Transcription

1 joint 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 azuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 at Hokkaido University, July /

2 Goal of my talk (S n, h β ) : the n-sphere (n 3) with edge-cone Einstein metric h β S n (S n 2 S 1 ) = (0, π 2 ) S 1 S n 2 (r, θ, x) g S = dr 2 + sin 2 r dθ 2 + cos 2 r g S n 2=: h 1 : standard round metric h β := dr 2 + β 2 sin 2 r dθ 2 + cos 2 r g S n 2 = h1 : loc. isom. (β > 0) edge-cone Einstein metric of cone angle 2πβ on (S n, S n 2 ) Main Thm No edge-cone Yamabe metric on (S n, [h β ]) with 2πβ 4π at Hokkaido University, July /

3 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 S n at Hokkaido University, July /

4 References [Ak1] Computations of the orbifold Yamabe invariant, Math. Z. 271 (2012), [Ak2] Edge-cone Einstein metrics and the Yamabe invariant, 83th Geometry Symposium (2016), [ACM1] (with G. Carron and R. Mazzeo) The Yamabe problem on stratified spaces, GAFA 24 (2014), [ACM2] Hölder regularity of solutions for Schrödinger operators on stratified spaces, J. Funct. Anal. 269 (2015), [AM] [AN-34] [AL] [M] [V] (with I. Modello) Edge-cone Einstein metrics and the Yamabe invariant, in preparation. (with A. Neves), 3-manifolds with Yamabe invariant greater than that of RP 3, J. Diff. Geom. 75 (2007), M. Atiyah and C. LeBrun, Curvature, cones and charcteristic numbers, Math. Proc. Cambridge Phil. Soc. 155 (2013), I. Mondello, The local Yamabe constant of Einstein stratified spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), J. Viaclovsky, Monopole metrics and the orbifold Yamabe problem, Ann. Inst. Fourier (Grenoble) 60 (2010), at Hokkaido University, July /

5 0 On scalar curvature M n : cpt n mfd ( M =, n 2 n 3 later) g = (g ij ) : Riem. metric, M U x : local coor. system g U : U M(n, R), x (g ij (x)) : symm. & positive definite n n R g = (R i jkl ) = F (g, g, 2 g) : curv. tensor ( Kij := g ( R g (e i, e j )e j, e i ) : sect. curv. ) Ric g = (R ij := Σ k R k ikj ) : Ricci curv. tensor the same type as g = (g ij) averaging R g := Σ i,j g ij R ij ( = 2Σ i<j K ij ) C (M) : scalar curv. averaging Note (1) n = 2 R g = 2 Gauss curv. (2) Vol g (B r (p)) = ω n r n( 1 1 ) 6(n + 2) R g (p)r 2 + O(r 3 ) as r 0 at Hokkaido University, July /

6 conformal class/deformation/laplacian Note (3) E n K ij 0 R g 0 S n (1) K ij 1 R g n(n 1) H n ( 1) K ij 1 R g n(n 1) Assume n 3 from now on [g] := {e f g f C (M)} : conf. class of g Set g = u 4 n 2 g, u > 0 : conf. deform. & αn := 4(n 1) n 2 > 0 L g u := ( α n g + R g ) u = R g u n+2 n 2 L g : conf. Laplacian of g at Hokkaido University, July /

7 1 The Yamabe problem and Yamabe constants M n : cpt n mfd (n 3, M = ) E : M(M) R, E(g) := Rg dµg M V g (n 2)/n h Crit(E) h : Einstein metric : scale inv. E-H action (Energy of g) i.e., Ric h = const h = (R h /n) h Note inf g M(M) E(g) =, sup g M(M) E(g) = + Y (M, C) := inf E C = inf g C E(g) >, C C(M) Yamabe constant & g Crit(E C ) R g const at Hokkaido University, July /

8 Yamabe Problem Yamabe Problem C C(M), Find ǧ = u 4/(n 2) g C s.t. E(ǧ) = inf E C (minimizer!) ( Find u C >0 (M) s.t. Q g (u) = inf f >0 Q g (f ), L g f := 4(n 1) n 2 Q g (f ) := g f +R g f M ( 4(n 1) n 2 f 2 + R g f 2 )dµ g ( M M f = f L g f dµ ) g 2n/(n 2) dµ g ) (n 2)/n ( M f 2n/(n 2) dµ g ) (n 2)/n Resolution Thm (Yamabe, Trudinger, Aubin, Schoen) Such ǧ C always exists! Yamabe metric & Rǧ = Y (M, C) V 2/n ǧ const Note Yamabe metric constant scalar curvature metric at Hokkaido University, July / Kazuo AKUTAGAWA (Chuo University) (MSJ-SI Edge-cone Einstein 2018metricsThe and Yamabe Role metrics of Metrics in the Theory 1

9 Yamabe constant conformal invariant Definition Aubin s Inequality conf.inv. Y (M, [g]) := inf g [g] E( g) Y (S n, [g S ]) =: Y n Yamabe constant n(n 1)Vol(S n (1)) 2/n at Hokkaido University, July /

10 Resolution Thm (Yamabe, Trudinger, Aubin, Schoen) Such ǧ C always exists! Yamabe metric & Rǧ = Y (M, C) V 2/n ǧ const Note (Proof of Resolution) Y (M, [g]) < Y (S n, [g S ]) direct method Y (M, [g]) = Y (S n, [g S ]) (M, [g]) = (S n, [g S ]) : conf. PMT at Hokkaido University, July /

11 The Yamabe invariant and Fundamental Problems Definition Yamabe invariant (or σ invariant) O.Kobayashi, Schoen diff. top. inv. Y (M n ) := sup Y (M n, [g]) Y (S n, [g S ]) = Y (S n ) [g] (1) Find a (singular) supreme Einstein metric h with Y (M, [h]) = Y (M). A naive approach : Find a nice sequences {ǧ j } of Yamabe metrics or singular Einstein metrics on M satisfying Vǧj = 1 & Y (M, [ǧ j ]) Y (M) & Analizes the limit of ǧ j Not much progress! (2) Estimates Y (M) from below/above & Calculates Y (M) Nice progress, particularly n = 3, 4! at Hokkaido University, July / Kazuo AKUTAGAWA (Chuo University) (MSJ-SI Edge-cone Einstein 2018metricsThe and Yamabe Role metrics of Metrics in the Theory 1

12 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 } at Hokkaido University, July /

13 The Yamabe problem on cpt almost Riem. m-m spaces Setting (M = Ω 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]) = Q (Ω,g) (f ) := inf Q (Ω,g)(f ) [, Y n ] f Cc (Ω) {0} Ω ( 4(n 1) n 2 df 2 + R g f 2 )dµ g ( Ω f 2n/(n 2) dµ g ) (n 2)/n at Hokkaido University, July /

14 Local Yamabe constant Definition Y l (M, [g]) : local Yamabe constant depends only on U(S) Y l (M, [g]) := inf p M lim r 0 Y (B r (p) Ω, [g]) [, Y n ] (1) If p Ω lim r 0 Y (B r (p) Ω, [g]) = Y n (2) If p S lim r 0 Y (B r (p) Ω, [g]) Y n Note (M, Ω, g) : smooth manifold (i.e. M = Ω) Y l (M, [g]) = Y n (M, Ω, g) : orbifold with S = {(p 1, Γ 1 ),, (p l, Γ l )} / Y l (M, [g]) = min j {Y n Γj 2/n } < Y n at Hokkaido University, July /

15 azuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 at Hokkaido University, July /

16 Refined Aubin s inequality Definition Y l (M, [g]) : local Yamabe constant depends only on U(S) Y l (M, [g]) := inf p M lim r 0 Y (B r (p) Ω, [g]) [, Y n ] (X, d, µ) : cpt m-m space with appropriate cond.s (1) Y l (M, [g]) > 0 (2) Y (M, [g]) > (3) Y (M, [g]) Y l (M, [g]) Refined Aubin s inequality at Hokkaido University, July /

17 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 l (M, [g]) u C >0 (Ω) W 1,2 (M; dµ) L (M) s.t. u 2n/(n 2) = 1 4(n 1) Q (Ω,g) (u) = Y (M, [g]), n 2 g u +R g u = Y (M, [g]) u n+2 n 2 on Ω Moreover inf M u > 0 Note (M, Ω, ǧ := u 4 n 2 g) : cpt m-m space with the same sing. S Rǧ = Y (M, [g]) const on Ω at Hokkaido University, July /

18 Counterexample Thm (Ak [Ak1]) If { Y ( ˇM, [g] orb ) < min j Yn / Γ j 2/n} ǧ [g] orb : orbifold Yamabe metric (minimizer) Counterexample (Viaclovsky [V]) ( X 4, ĥ) : conf. cpt. of hyperkähler ALE (X 4, h) arising from C 2 /Γ, Γ < SU(2) : finite subgroup Y ( X 4, [ĥ] orb) = Y (S 4, [g S ])/ Γ 1/2 (= Y 4 / Γ 1/2 < Y 4 ) & ǧ [ĥ] orb : No orbifold Yamabe metric (Modifying the technique in Proof of Obata Uniqueness Thm!) at Hokkaido University, July / Kazuo AKUTAGAWA (Chuo University) (MSJ-SI Edge-cone Einstein 2018metricsThe and Yamabe Role metrics of Metrics in the Theory 1

19 azuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 at Hokkaido University, July /

20 3 The standard edge-cone Einstein metrics on S n (S n, h β ) : the n-sphere (n 3) with edge-cone Einstein metric h β S n (S n 2 S 1 ) = (0, π 2 ) S 1 S n 2 (r, θ, x) g S = dr 2 + sin 2 r dθ 2 + cos 2 r g S n 2=: h 1 : standard round metric h β := dr 2 + β 2 sin 2 r dθ 2 + cos 2 r g S n 2 = h1 : loc. isom. (β > 0) edge-cone Einstein metric of cone angle 2πβ on (S n, S n 2 ) Main Thm No edge-cone Yamabe metric on (S n, [h β ]) with 2πβ 4π at Hokkaido University, July /

21 azuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 at Hokkaido University, July /

22 azuo AKUTAGAWA (Chuo University) (MSJ-SI 2018 at Hokkaido University, July /

23 Local Yamabe constant of the edge-cone sphere (S n, h β ) Thm (Mondello [M]) Y (S n, [h β ]) = Y l (S n, [h β ]) = Y (R n, [δ β ]) = β 2/n Y n = E(h β ) if 0 < β 1 = Y n if β 1 In particular, h β : Yamabe metric on (S n, [h β ]) if 0 < β 1, not for β > 1 Main Thm (Non-exiatence) (Ak-Mondello [AM]) If β 2 (2πβ 4π), No Yamabe metric on (S n, [h β ])! Conjecture No Yamabe metric even if 1 < β < 2? at Hokkaido University, July /

24 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 proof! Existence of Yamabe metric on ( M, [ g]) is necessary for the Note Aubin s Lemma still holds for branced conf. double covering of (S n, [h β ]) (S n, [h β/2 ])! provided that (1) Existence of an edge-cone Yamabe metric u 4/(n 2) h β (u > 0) on (S n, [h β ]) (2) The following inequality holds : u hβ u dµ hβ S n = u 2 dµ hβ S n at Hokkaido University, July /

25 Proof of Main Thm (continuation) Suppose that (1) holds : ȟ β (β 2) : Yamabe metric on (S n, [h β ]) Set r(p) := dist hβ (p, S n 2 ). Then, by elliptic regularity result [ACM1, 2], on ε-open tubular neighborhood U ε = U ε (S n 2 ) u C 0,1/β (U ε ) & r u(r, θ, x) = O(r 1/β 1 ) as r 0 u hβ u dµ hβ + S n U ε u 2 dµ hβ = S n U ε u r udσ hβ = O(r 1/β ) 0 U ε as ε 0 Hence, we get (2), and thus Y (S n, [h β ]) > Y (S n, [h β/2 ]) On the other hand, by β > β/2 1, Modello s Thm implies that Y (S n, [h β ]) = Y n = Y (S n, [h β/2 ]) Contradiction! QED at Hokkaido University, July /

26 Thank you very much for your kind attention! at Hokkaido University, July /