Measures on spaces of Riemannian metrics CMS meeting December 6, 2015

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1 Measures on spaces of Riemannian metrics CMS meeting December 6, 2015 D. Jakobson (McGill), [CJW]: Y. Canzani, I. Wigman, DJ: arxiv: , Jour. of Geometric Analysis, 2013 [CCKJST]: B. Clarke, N. Kamran, L. Silberman, J. Taylor, Y. Canzani, DJ: arxiv: , Ann. Math. du Québec, 2015 December 6, 2015

2 Conjectures in Quantum Chaos: Random Wave conjecture, GOE level spacing conjecture, conjectures about nodal sets etc. Integration on manifolds of metrics: D. Ebin defined L 2 metric on manifolds of metrics. Restricted to the space of hyperbolic metrics on a compact surface, we get Weil-Petersson metric on moduli spaces of hyperbolic metrics (singular manifolds!). Many people studied differentiation on manifolds of metrics; such notions as Levi-Civita connection, geodesics, parallel transport etc. have been defined. Want to define integration. Our spaces of metrics are infinite-dimensional, so it is convenient to define Gaussian measures on those spaces.

3 Fix a compact smooth Riemannian manifold M n. We shall discuss several measures on manifolds of metrics on M. Measures on conformal classes of metrics: concentrated near a reference metric g 0, supported on regular (e.g. Sobolev, real-analytic) metrics a.s. Applications to the study of Gauss curvature. Measures on manifolds of metrics with the fixed volume form, applications to the study of L 2 (Ebin) distance function, and to integrability of the diameter, eigenvalue and volume entropy functionals. Remark: All measures are invariant by the action of diffeomorphisms.

4 Conformal class: g 0 - reference metric on M. Conformal class of g 0 : {g 1 = e f g 0 }; f is a random (suitably regular) function on M. 0 - Laplacian of g 0. Spectrum: 0 φ j + λ j φ j = 0, 0 = λ 0 < λ 1 λ Define f by f (x) = a j c j φ j (x), a j N (0, 1) are i.i.d standard Gaussians, c j = F(λ j ) 0 (damping): j=1

5 The covariance function r f (x, y) := E[f (x)f (y)] = j=1 c 2 j φ j (x)φ j (y), for x, y M. For x M, f (x) is mean zero Gaussian of variance r f (x, x) = cj 2 φ j (x) 2. j=1 Examples: Random Sobolev metric: c j = λ s j, = r f (x, y) = j φ j (x)φ j (y), spectral zeta function. λ 2s j Random real-analytic metric c j = e λ j t, = r f (x, y) = j φ j(x)φ j (y)e 2λ j t, heat kernel.

6 Sobolev regularity: If E f 2 H s = j c 2 j (1 + λ j ) s < then f H s (M) a.s. Weyl s law + Sobolev embedding imply Proposition: If c j = O(λ s j ), s > n/2, then f C 0 a.s; if c j = O(λ s j ), s > n/2 + 1, then f C 2 a.s.

7 [CJW]: Let n = 2, and let g 0 have non-vanishing Gauss curvature (M T 2 ). Can estimate the probability that after a random conformal perturbation, the Gauss curvature will change sign somewhere on M. Techniques: curvature transformation in 2d under conformal changes of metrics, large deviation estimates (Borell, Tsirelson-Ibragimov-Sudakov, Adler-Taylor). n 3: related results for scalar curvature and Q-curvature.

8 Random (Sobolev) embeddings into R k : 1-dimensional i.i.d. Gaussians k-dimensional i.i.d. Gaussians. F. Morgan (1979): M = S 1, k = 3: a.s. results about minimal surfaces spanned by random knots. F. Morgan (1982): general compact M, a.s. Whitney embedding theorems + applications. We shall use similar ideas to define Gaussian measures on manifolds of metrics with fixed volume form; transverse to conformal classes.

9 Random (Sobolev) embeddings into R k : 1-dimensional i.i.d. Gaussians k-dimensional i.i.d. Gaussians. F. Morgan (1979): M = S 1, k = 3: a.s. results about minimal surfaces spanned by random knots. F. Morgan (1982): general compact M, a.s. Whitney embedding theorems + applications. We shall use similar ideas to define Gaussian measures on manifolds of metrics with fixed volume form; transverse to conformal classes.

10 Random (Sobolev) embeddings into R k : 1-dimensional i.i.d. Gaussians k-dimensional i.i.d. Gaussians. F. Morgan (1979): M = S 1, k = 3: a.s. results about minimal surfaces spanned by random knots. F. Morgan (1982): general compact M, a.s. Whitney embedding theorems + applications. We shall use similar ideas to define Gaussian measures on manifolds of metrics with fixed volume form; transverse to conformal classes.

11 Random (Sobolev) embeddings into R k : 1-dimensional i.i.d. Gaussians k-dimensional i.i.d. Gaussians. F. Morgan (1979): M = S 1, k = 3: a.s. results about minimal surfaces spanned by random knots. F. Morgan (1982): general compact M, a.s. Whitney embedding theorems + applications. We shall use similar ideas to define Gaussian measures on manifolds of metrics with fixed volume form; transverse to conformal classes.

12 Metrics = sections of Pos(M) Sym(M) Hom(TM, T M) (positive-definite, symmetric maps); symmetric matrices in local coordinates. GL(T x M) acts on Pos x (M) with stabilizer isomorphic to O(n). Fix a volume form v, consider Met v (M). SL(T x M) acts on the fibre Pos v x(m) by h.g x = h T g x h; stabilizer isomorphic to SO(n). We have Pos v x(m) = SL n (R)/SO n Fix g 0 Met v ; dv(x) = det g 0 (x) dx 1... dx n. Let G x = SL(T x M), K x = SO(g 0 x ).

13 Metrics = sections of Pos(M) Sym(M) Hom(TM, T M) (positive-definite, symmetric maps); symmetric matrices in local coordinates. GL(T x M) acts on Pos x (M) with stabilizer isomorphic to O(n). Fix a volume form v, consider Met v (M). SL(T x M) acts on the fibre Pos v x(m) by h.g x = h T g x h; stabilizer isomorphic to SO(n). We have Pos v x(m) = SL n (R)/SO n Fix g 0 Met v ; dv(x) = det g 0 (x) dx 1... dx n. Let G x = SL(T x M), K x = SO(g 0 x ).

14 f x frame in T x M orthonormal w.r.to g 0 x, A x G x positive diagonal matrices of determinant 1 (w.r. to f x ). Every g 1 x Pos v x(m) can be represented as g 1 x = (k x a x )g 0 x, k x K x, a x A x ; unique up to S n acting on f x. Assumption: M is parallelizable ( global section of the frame bundle). Examples: All 3-manifolds; All Lie groups; The frame bundle of any manifold; The sphere S n iff n {1, 3, 7}. Necessary condition: vanishing of the 2nd Stiefel-Whitney class. For orientable spin.

15 f x frame in T x M orthonormal w.r.to g 0 x, A x G x positive diagonal matrices of determinant 1 (w.r. to f x ). Every g 1 x Pos v x(m) can be represented as g 1 x = (k x a x )g 0 x, k x K x, a x A x ; unique up to S n acting on f x. Assumption: M is parallelizable ( global section of the frame bundle). Examples: All 3-manifolds; All Lie groups; The frame bundle of any manifold; The sphere S n iff n {1, 3, 7}. Necessary condition: vanishing of the 2nd Stiefel-Whitney class. For orientable spin.

16 M parallelizable. Choose a global section f 0 of the frame bundle orthonormal w.r. to g 0. To define g 1 x, we apply to f 0 a composition of a rotation k x SO(T x M) and a diagonal unimodular transformation a x SL(T x M) which will define an orthonormal basis f 1 x for g 1 x. By construction, g 0 and g 1 will have the same volume form. We let a x = exp(h(x)), where H : M a = R n 1 is the Lie algebra of Diag 0 (n) SL n. Similarly, let k x = exp h(x), where h : M so n, the Lie algebra of SO n. Choice of g 0 + parallelizability makes the above construction well-defined.

17 We define Gaussian measures on {H : M a} and {h : M so n } as in Morgan. In the sequel, only need H; constructions are analogous.

18 Let H(x) = π n (A j )β j ψ j (x), (1) j=1 where ψ j + λ j ψ j = 0; A j are i.i.d standard Gaussians in R n ; π n : R n {x R n : x (1,..., 1) = 0} R n 1 - projection into the hyperplane n j=1 x j = 0; β j = F 2 (λ j ) > 0, where F 2 (t) is (eventually) monotone decreasing function of t, F(t) 0 as t.

19 Smoothness: Morgan showed Proposition 1: If β j = O(j r ) where r > (q + α)/n + 1/2, then H converges a.s. in C q,α (M, R n 1 ). Proposition 1 + Weyl s law = Proposition 2: If β j = O(λ s j ) where s > q/2 + n/4, then H converges a.s. in C q (M, R n 1 ).

20 Smoothness: Morgan showed Proposition 1: If β j = O(j r ) where r > (q + α)/n + 1/2, then H converges a.s. in C q,α (M, R n 1 ). Proposition 1 + Weyl s law = Proposition 2: If β j = O(λ s j ) where s > q/2 + n/4, then H converges a.s. in C q (M, R n 1 ).

21 Lipschitz distance ρ: ρ(g 0, g 1 ) = sup x M sup 0 ξ T x M ln g 1(ξ, ξ) g 0 (ξ, ξ) (2) A related expression appeared in the paper by Bando-Urakawa. If g 1 (x) = (k(x)d(x))g 0 (x) then ρ only depends on the diagonal part d(x). Tail estimate for ρ: One can show that for large R, Prob{ρ(g 0, g 1 ) > R} 2 n (n + ɛ) Prob{sup d 1 (x) > R/2} (3) x M

22 Definition of d 1 : Recall from (1): H(x) = j=1 π n (A j )β j ψ j (x). Define D(x) = (d 1 (x),..., d n (x)) by D(x) = A j β j ψ j (x) j=1 ( don t project A j ). The covariance function for d 1 (x): r d1 (x, y) = βk 2 ψ k(x)ψ k (y), k=1 Define σ 2 by σ 2 := σ(d 1 ) 2 := sup r d1 (x, x). (4) x M

23 Borell-TIS theorem applied to the random field d 1 implies Proposition 3. Let σ be as in (4). Then ln Prob{ρ(g 0, g 1 ) > R} lim R R 2 1 8σ 2. (5) More precise result: Proposition 4. There exists α > 0 such that for a fixed ɛ > 0 and for large enough R, we have ( ) αr Prob{ρ(g 1, g 0 ) > R} 2 n (n + ɛ) exp 2 R2 8σ 2.

24 ρ controls diameter and eigenvalues: Proposition 5. Assume that dvol(g 0 ) = dvol(g 1 ) and ρ(g 0, g 1 ) < R. Then e R diam(m, g 1) diam(m, g 0 ) er (6) and e 2R λ k( (g 1 )) λ k ( (g 0 )) e2r. (7)

25 Propositions 4 and 5 imply Theorem 6. Let h : R + R + be a monotonically increasing function such that for some δ > 0 ( [ ]) h(e y ) = O exp y 2 (1/(8σ 2 ) δ). Then h(diam(g 1 )) is integrable with respect to the probability measure dω(g 1 ) constructed earlier. and Theorem 7. Let h : R + R + be a monotonically increasing function such that for some δ > 0 h(e 2y ) = O ( exp [ y 2 (1/(8σ 2 ) δ) ]). Then h(λ k ( (g 1 ))) is integrable with respect to the probability measure dω(g 1 ) constructed earlier.

26 Similar results can be established for volume entropy, h vol = lim s ln volb(x, s) s

27 L 2 or Ebin distance between can be computed as follows [Ebin, Clarke]: Ω 2 2 (g0, g 1 ) := d 2,x (g 0 (x), g 1 (x)) 2 dv(x) M where d 2,x (g 0 (x), g 1 (x)) is the distance in SL n /SO n ; = H(x), H(x) g 0 (x) dv(x). M For us Ω 2 2 is a random variable whose distribution we shall compute. Note: only depends on H, hence it suffices to consider the Gaussian measure on {H : M a}.

28 L 2 or Ebin distance between can be computed as follows [Ebin, Clarke]: Ω 2 2 (g0, g 1 ) := d 2,x (g 0 (x), g 1 (x)) 2 dv(x) M where d 2,x (g 0 (x), g 1 (x)) is the distance in SL n /SO n ; = H(x), H(x) g 0 (x) dv(x). M For us Ω 2 2 is a random variable whose distribution we shall compute. Note: only depends on H, hence it suffices to consider the Gaussian measure on {H : M a}.

29 In local coordinates, let a x = diag(exp(b 1 (x)), exp(b 2 (x)),..., exp(b n (x))), where n j=1 b j(x) = 0, x M. Then d x (g 0 x, g 1 x ) 2 = n b j (x) 2. j=1 Accordingly, Ω 2 (g 0, g 1 ) 2 = M n b j (x) 2 dv(x). j=1

30 π n : R n {x : x j = 0} standard projection. P n - matrix of π n (in the usual basis of R n ) with singular values Then in distribution (1,..., 1, 0) := µ i,n, 1 i n. Ω 2 2 D = j β 2 j n µ 2 i,n W i,j i=1 where W i,j χ 2 1 are i.i.d. We get Ω2 D 2 = j β2 j V j where V j χ 2 n 1 are i.i.d.

31 Theorem 8. Moment generating function of Ω 2 2 : M Ω 2(t) = E(exp(tΩ )) = j = n 2tµ j i=1(1 2 i,n β2 j ) 1/2 = j Characteristic function of Ω 2 2 : n M χ 2(tµ 2 1 i,n β2 j ) i=1 (1 2tβ 2 j ) (n 1)/2 n j i=1(1 2itµ 2 i,n β2 j ) 1/2 = j (1 2itβ 2 j ) (n 1)/2

32 Corollary 9. Tail estimates for Ω 2 2 : applying results of Laurent-Massart, one can show that Prob{Ω 2 2 s2 } exp( s 2 /(2β1 2 )).