On the mass of asymptotically hyperbolic manifolds

Transcription

1 On the mass of asymptotically hyperbolic manifolds Mattias Dahl Institutionen för Matematik, Kungl Tekniska Högskolan, Stockholm April 6, 2013, EMS/DMF Joint Mathematical Weekend Joint work with Romain Gicquaud and Anna Sakovich. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

2 Motivation: asymptotically Euclidean case Let (R n, δ) be Euclidean space. (M, g) is asymptotically Euclidean if There is a dieomorphism Φ : M \ K R n \ B R (0), e = Φ g δ 0 suciently fast at innity. m ADM = lim r 1 2(n 1)ω n 1 Theorem (Schoen&Yau, Witten) S r ( ) div δ e d(tr δ e) (ν r ) dµ δ. Suppose a complete AE manifold (M, g) has Scal g 0. If either (M, g) has dim < 8 or if (M, g) is spin then m 0. Moreover, m = 0 i (M, g) is isometric to (R n, δ). Example. The slice t = 0 of the Schwarzschild spacetime is ( ( R n \ {0}, 1 + m ) 4 ) n 2 δ. 2r n 2 Asymptotically Euclidean, Scal = 0 and m ADM = m. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

3 Asymptotically hyperbolic mass Let (H n, b) be the hyperbolic space. In polar coordinates b = dr 2 + sinh r σ. Roughly, (M, g) is asymptotically hyperbolic if There is a dieomorphism Φ : M \ K H n \ B R (0), e = Φ g b 0 suciently fast at innity. Consider N = {V C (H n ) Hess b V = Vb}. Vector space with a basis V (0) = cosh r, V (1) = x 1 sinh r,..., V (n) = x n sinh r, where x 1,..., x n are the coordinate functions on R n restricted to S n 1. The mass functional of (M, g) w.r.t. the chart Φ is H Φ (V ) = lim r S r ( ) V (div b e d tr b e) + (tr b e)dv e( b V, ) (ν r ) dµ b. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

4 Asymptotically hyperbolic mass (continued) Theorem (Wang; Chrusciel and Herzlich) If (M, g) is a complete spin asymptotically hyperbolic manifold with Scal g n(n 1) then H Φ is either timelike future directed or zero. H Φ is zero i (M, g) is isometric to (H n, b). Let η be the Lorentzian inner product on N such that {V (0), V (1),..., V (n) } is orthonormal with respect to η, η(v (0), V (0) ) = 1, η(v (i), V (i) ) = 1, i = 1,..., n. Then H Φ is timelike future directed if H Φ (V ) > 0 for every 0 < V N +, where N + is the interior of the future lightcone. In this case the mass is dened as m = 1 2(n 1)ω n 1 inf N 1 H Φ(V ), where N 1 is the unit hyperboloid in N +. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

5 Asymptotically hyperbolic mass (continued) In fact, we can always nd a chart Φ such that m = 1 2(n 1)ω n 1 H Φ (V (0) ), where V (0) = cosh r. Such coordinates are called balanced. Example. Let ρ = sinh r. The slice t = 0 of the Anti-de Sitter-Schwarzschild spacetime outside the horizon has the metric g AdS-Schw = dρ 2 + ρ 2 σ. 1 + ρ 2 2m ρ n 2 Asymptotically hyperbolic, Scal = n(n 1), mass equals m. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

6 Penrose conjecture for AE manifolds There is a conjectured lower bound for the mass of AE manifolds in terms of their geometry: Conjecture (Penrose 1973) Let (M, g) be an n-dimensional AE manifold. If Scal g 0 then m ADM 1 2 ( Σ ω n 1 ) n 2 n 1, where Σ is the outermost minimal hypersurface. Equality holds i (M, g) is isometric to the t = 0 slice of the Schwarzschild spacetime outside its minimal hypersurface. proven in dim = 3 (Huisken and Ilmanen by IMCF; Bray 2001 Conformal ow); proof by Bray extended to dim < 8 (Bray and Lee 2009); AE graphs (Lam 2010; Huang and Wu 2012). Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

7 Penrose conjecture for AH manifolds Conjecture Let (M, g) be n-dimensional AH manifold. If Scal g n(n 1) then m 1 2 [ ( Σ ω n 1 ) n 2 n 1 + ( Σ ω n 1 ) ] n n 1, where Σ is the outermost minimal hypersurface. Equality holds i (M, g) is isometric to the t = 0 slice of the Anti-de Sitter-Schwarzschild spacetime outside its minimal hypersurface. Neves 2010: IMCF is inconclusive in this case. Lee-Neves 2013: IMCF proof for cases of negative mass. Bray's conformal ow relies on the fact that on AE manifolds there are only two dimensional quantities m and Σ. On H n there is a third one: curvature. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

8 Penrose type inequalities for AH graphs H n+1 = (H n R, b + V 2 ds ds), where V = V (0) = cosh r. Let f : H n \ Ω R be a continuous function which is smooth on H n \ Ω, where Ω is relatively compact and open. Assume that df (x) when x Ω and that f Ω = const. Consider M = {(x, s) H n R f (x) = s} as a hypersurface in H n+1 with the induced metric. Consider Φ : M H n \ Ω : (p, f (p)) p. The pushforward of the induced metric is g = b + V 2 df df. Let M be AH w.r.t. the chart Φ, i.e. e = g b = V 2 df df 0 suciently fast and let Φ be balanced. Σ = M is a minimal surface with Σ = Ω. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

9 Penrose type inequalities for AH graphs (continued) Main observation: [ ] V (Scal + n(n 1)) = div b V div b e Vd tr b e e( V, ) + (tr b e)dv, 1 + V 2 df 2 so that H Φ (V ) = H n \Ω V [Scal + n(n 1)] dµ g + HV dµ b HV dµ b. 1 + V 2 df 2 Ω Ω Assuming that B r0 (0) Ω and that H 0 we can estimate m from below. By Homan-Spruck inequality: ( ) n 2 n 2 Ω n 1 m V (r 2 n (n 1)n n 0 ), n 1 ω n 1 By Minkowski formula: m 1 2 V (r 0) Ω ω n 1. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

10 Penrose type inequalities for AH graphs (continued) Theorem (de Lima, Girão, 2012) Assume that Ω is star-shaped w.r.t. the origin, then the following inequality holds: HV dµ b Σ 1 (n 1)ω n 1 [ ( Σ ω n 1 ) n 2 n 1 + ( Σ ω n 1 with equality i Ω is a round sphere centered at the origin. ) ] n n 1, AH Penrose inequality holds for AH graphs (D.-Gicquaud-Sakovich-de Lima-Girão) Rigidity case from mean convex graph and maximum principles for scalar curvature (D.-Gicquaud-Sakovich following Huang-Wu for AE). Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

11 AE manifolds with small mass The only AE manifold with zero mass is the Euclidean space. Question: must an AE manifold (M, g) with Scal g 0 and small mass be close to Euclidean space? Spinor methods (Finster et al): L 2 -estimates for curvature tensor. Lee 2009: let (M, g) be harmonically at down to radius R i.e. Scal g = 0 and g = U 4 n 2 δ on R n \ B R. In this case: if m 0 then g δ uniformly on R n \ B αr for any α > 1. Lee's result used in Bray and Lee's proof of the Penrose conjecture. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

12 AH manifolds with small mass Consider the class A(R 0 ) of 4-tuples (M, g, Φ, U) such that (M, g) is AH w.r.t. Φ : M \ K H n \ B R0, Scal g n(n 1) on M, and Scal g = n(n 1) on M \ K. U > 0, Φ g = U 4 n 2 b on H n \ B R0, and U 1 at innity, the coordinates Φ at innity are balanced, the positive mass theorem holds for any asymptotically hyperbolic metric on M. Theorem (D., Gicquaud, Sakovich 2012) Let R 1 > R 0 and ε > 0. Then there is a constant δ > 0 such that U 1 εe nr on H n \ B R1 for all (M, g, Φ, U) A(R 0 ) with m g < δ. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

13 AH manifolds with small mass (continued) Idea of proof: Suppose that for a one parameter family of metrics {g s }, s [ s 0, s 0 ] the function H(s) = H g s Φ (V (0)) is analytic with bounded coecients. If Ḣ(0) > 0 is too large w.r.t. small H(0) we get a contradiction with the PMT. Ḣ(0) must be an integral of some geometric quantity. In AE case Lee 4 n 2 uses g s = λs (g sχric) where χ is a compactly supported function, λ s is such that Scal g s = 0. In this case Ḣ(0) = M χ Ric 2 dµ g. 4 n 2 For AH the obvious try is g s = λs (g sχ(ric + (n 1)g)) where χ is a compactly supported function, λ s is such that Scal g s = n(n 1). Ḣ(0) does not have a nice expression unless χ 1. Reason: in the proof we use functions V g V (0) such that g V g = nv g. They do not appear in the AE case. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

14 AH manifolds with small mass (continued) Second try g s = λ 4 n 2 s ( ( g sχ Ric g Hess(V )) g ). V g Then Ḣ(0) = M χv Ric g g Hess(V g ) V g Recall that small mass H(0) implies small Ḣ(0). And if Ḣ(0) = 0 then Ric g = Hess(V g ) V g 2 g dµ g. on H n \ B R1. In this case (V g ) 2 dt 2 + g is static. Kobayashi-Obata: U is the conformal factor for the Anti-de Sitter-Schwarzschild with m = 0 on H n \ B R1. Hence U = 1. Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15

15 Thank you for your attention! Mattias Dahl (KTH - Stockholm) On the mass of AH manifolds April 6, / 15