Metric measure spaces with Riemannian Ricci curvature bounded from below Lecture I

Transcription

1 1 Metric measure spaces with Riemannian Ricci curvature bounded from below Lecture I Giuseppe Savaré savare Dipartimento di Matematica, Università di Pavia Analysis and Geometry on Singular Spaces, Pisa, June 9-13, 2014

2 2 Outline 1 Smooth setting: energy forms, diffusion semgroups 2 Bochner identity and the Bakry-Émery approach to lower curvature bounds 3 Ricci curvature and optimal transport

3 Smooth setting (M, g) smooth complete Riemannian manifold of dimension n. In a local chart U M, x : U Ω R n is a system of local coordinates: x = (x i ) i=1,...,n. i = x i. Tangent vector: V = V i i. V 2 g = g ijv i V j, g = g ijdx i dx j ; V, W g = g(v, W ) = g ijv i W j. Smooth curve: x : [a, b] M, x(t) = (x i (t)), V i := ẋ i, ẋ g = gijẋ i ẋ j Length[x] = b a ẋ g dt Cotangent vector - differential form: ω = ω idx i, with dual norm g ω 2 g = g ij ω iω j, g ij g jk = δ i k; ω, η g = g (ω, η) = g ij ω iη j. Differential of a function f : M R: Df = df = if dx i Df 2 g = g ij if jf Volume measure: Vol g = e G L n, G := 1 2 log(det g) = 1 2 log(det g ).

4 Energy forms and differential operators m = e V Vol g = e (V +G) L n is a reference Borel measure, V : M R smooth. Energy form: E(f, h) := E(f) = E(f, f) = M M Df, Dh g dm, Df 2 g dm = g ij if jf e V dvol g Sobolev space D(E) = W 1,2 (M, g, m): the completion of the space of smooth functions in L 2 (M, m) with E < endowed with the scalar product E 1(f, h) := f h dm + E(f, h). L is the associated second order drift-diffusion differential operator: f D(L) Lf L 2 (M, m) : E(f, h) = Lf h dm h D(E) M Lf = e V +G i ( e (V +G) g ij jf) = i ( g ij jf ) g ij i ( V + G ) jf

5 6 Examples in R n Euclidean Dirichlet energy: M = R n, V g = V, m = Vol g = L n. E(f) = Df 2 dx, Lf = f = i 2 f. Weighted energy and drift-diffusion: M = R n, V g = V, m = e V L n. E(f) = Df 2 e V dx, Lf = f DV, Df Gaussian and Ornstein-Uhelenbeck operator: V (x) := 1 2 x 2 n log(2π) 2 1 E(f) = Df 2 e 1 (2π) n/2 2 x 2 dx, Lf = f x, Df Elliptic operator in divergence form: M = R n, V g = g ijz i Z j, m = L n. E(f) = g ij if jf dx, Lf = ( i g ij ) jf.

6 Examples Laplace-Beltrami: m = Vol g E(f) = g ij if jf e G dx, Lf = gf = e G i ( e G g ij jf ) = i ( g ij jf ) DG, Df g Conformal geometry: g = g Id, Z 2 g = g Z 2, m = Vol g = g n/2 L n. E(f) = g n/2 1 Df 2 dx, Lf = 1 ( i g n/2 1 ) if = 1 ( ) f + (n/2 1) D log g, Df g n/2 g ( In particular, when n = 1 Lf = 1 g f 1 g 2 2 g f ). When n = 2 Lf = 1 g f. Weighted geometry: m = e V Vol g E(f) = Df 2 g e V dvol g, Lf = gf DV, Df g

7 Diffusion semigroup Diffusion semigroup in L 2 (M, m) generated by E: (P t) t 0. For every f L 2 (M, m) f t = P tf D(L), t > 0, is the unique solution of ft = Lft, t lim t 0 ft = f in L2 (M, m). Variational formulation: d f th dm + E(f t, h) = 0 dt (P t) t 0 is symmetric: Ptf h dm = f P th dm. h D(E). contractive in every L p, 1 p : P tf L p f L p analytic in L p, 1 < p < : LP tf L p Ct 1 f L p order preserving: f h P tf P th. In particular f 0 P tf 0 mass preserving, if m(b r( x)) Ae Br2 : P tc c, P tf dm = f dm

8 10 Γ tensor, Lf 2, commutation and Lebnitz rule Γ-tensor Γ(f, h) := 1 2 Leibnitz rule yields Γ(f, g) = Df, Dh g. Choosing f = h ( ) L(fh) f Lh h Lf Γ(f) = Γ(f, f) = 1 2 Lf 2 f Lf, Γ(f) = Df 2 g. From the Energy form E it is possibile to recover the energy density: ( Df 2 1 ) g h dm = Γ(f) h dm = 2 Lf 2 f Lf h dm = 1 2 E(f 2, h) + E(f, fh)

9 11 Computation of L Df 2 g In R n 1 2 Df 2 = Df, D f + D 2 f 2 D 2 f 2 = ( 2 ij f )2. Drift part Z(f) = DV, Df : Lf = f Z(f), 1 2 Z( Df 2) = Df, DZ(f) D 2 V (Df, Df) 1 2 L Df 2 = Df, DLf + D 2 f 2 + D 2 V (Df, Df) Γ 2(f) := 1 2 LΓ(f) Γ(f, Lf) = 1 2 L Df 2 Df, DLf Γ 2(f) = D 2 f 2 + D 2 V (Df, Df) Gaussian: m = 1 e 1 (2πλ) n/2 2λ x 2 L n. Γ 2(f) = D 2 f λ Df 2

10 12 Computation of L Df 2 g: Laplacian and covariant derivative Riemannian connection ; i = i. Z X, Y g = ZX, Y g + X, ZY g, XY Y X = [X, Y ] i j = γ k ij k, ( iz) k = iz k + j γ k ijz j, ( iω) j = iω j k γ k ijω k Hessian: D 2 f = df = H ijdx i dx j, H ij = 2 ijf k γ k ij kf Laplacian: gf = trace(d 2 f) = ij g ij H ij = ij g ij( 2 ijf k ) γij k kf The variational and the covariant representation of g coincide!

11 Ricci curvature and the Bochner s formula Second order derivative: 2 X,Y := X Y X Y ; Riemann curvature tensor: 2 ij = i j γ k ij k Rm(X, Y ) = 2 X,Y 2 Y,X, Rm(X, Y ; Z, W ) := 2 X,Y Z 2 Y,XZ, W g Ricci curvature tensor: Ric(X, Y ) := i Rm(X, E i; Y, E i), E i orthonormal frame Bochner identity 1 2 g Df 2 g = Df, D gf g + D 2 f 2 g + Ric(Df, Df) Z = V, Lf = gf Z(f), 1 2 Z( Df 2 ) g = Df, DZ(f) g D 2 V (Df, Df). 1 2 L Df 2 g = Df, DLf g + D 2 f 2 g + Ric L(Df, Df) Ric L = Ric + D 2 V

12 Examples Sphere: S n R n+1 ; local chart x = (x 1,, x n, y), x < 1; y := 1 x 2 xi x j g ij(x) = δ ij + 1 x, 2 gij (x) = δ ij x i x j, Df 2 g = (δ ij x i x j ) if jf G = 1 2 log (1 x 2), Vol g = 1 1 x 2 L n. S nf = (δ ij x i x j ) 2 ijf n x i if Ric(Df, Df) = Ric L(Df, Df) = (n 1) Df 2 g Hyperbolic space: H n = {(x 1, x 2,, x n 1, x n ) R n : x n > 0}, z = x n ; g ij(x) = 1 z 2 δij, gij (x) = z 2 δ ij, Df 2 g = z 2 Df 2 G = n log z, Vol g = z n L n H nf = z 2 f (n 2) zf Ric(Df, Df) = Ric L(Df, Df) = (n 1) Df 2 g

13 Bakry-Émery Γ 2 conditions Γ 2(f) := 1 LΓ(f) Γ(f, Lf) 2 = 1 2 L Df 2 g Df, DLf g Γ 2(f) = D 2 f 2 g + Ric L(Df, Df) Bakry-Émery condition BE(K, N), K R, N n: ( ) Γ 2(f) K Γ(f) N Lf Ric L(Df, Df) K Df 2 g + 1 N n V, Df 2 g When N = Γ 2(f) K Γ(f) Ric L(Df, Df) K Df 2 g

14 Pointwise gradient bounds for the diffusion semigroup Fix t > 0, f smooth and define for 0 < s < t A s(f) := 1 2 Pt s ( Psf )2, B s(f) := 1 2 Pt s DPsf 2 g = 1 2 Pt sγ(psf). Setting f s := P sf d ( 1 ) ds As(f) = Pt s 2 Lf s 2 f slf s = P t sγ(f s) = 2B s(f) d ( 1 ) ds Bs(f) = Pt s 2 LΓ(fs) Γ(fs, Lfs) = P t sγ 2(f s) If BE(K, ) holds, i.e. Γ 2 KΓ d Bs(f) KPt sγ(fs) = 2KBs(f) ds B + 2KB 0, A 2KA 0 2B t(f) = DP tf 2 g 2e 2Kt B 0(f) = e 2Kt P t Df 2 g Γ(P tf) e 2Kt P tγ(f)

15 2I2K(t) Lip(f) f Lipschitz regularization B t e 2K(t s) B s, A s = 2B s 2e 2K(t s) B t Integrating w.r.t. s A t A 0 2B ti 2K(t), I 2K(t) := t 0 e 2Kr dr = f 2 t + 2I 2K(t)Γ(f t) f 2 { t if K = 0 (2K) 1( e 2Kt 1 ) if K 0

16 18 The abtract framework for Γ-calculus A (Polish) topological space (X, τ) A probability Borel measure m a strongly local Dirichlet form E in L 2 (X, m), i.e. a closed, symmetric, nonnegative bilinear form on D(E) L 2 (X, m) satisfying E(f +, f +) E(f, f), E(f, h) = 0 if f, h D(E), fh = 0. (P t) t 0 is the positivity and mass preserving Markov semigroup in L 2 (X, m) (in fact in any L p (X, m)) generated by E L : D(L) L 2 (X, m) is the selfadjoint accretive operator Lu ϕ dm = E(u, ϕ), Lu u dm = E(u, u) 0. Energy density: there exists a bilinear map Γ : D(E) L 1 (X, m): 1 2 E(f 2, h) + E(f, fh) = Γ(f) h dm for every f, h D(E) L E(f, h) = Γ(f, h) dm. Γ(f) plays the role of Df 2 g, Γ(f, h) corresponds to Df, Dh g.

17 19 Bakry-Émery condition BE(K, ) in energy-measure spaces Strong form: Γ 2 tensor Γ 2(f) = 1 LΓ(f) Γ(f, Lf) KΓ(f) 2 Pointwise gradient commutation estimate: Γ ( P tf ) e 2Kt P t ( Γ(f) ) Weak form: the quantity A s[f, h] := 1 2 Psf 2 P t sh dm satisfies d 2 ds 2 As[f, h] + 2K d ds As[f, h] 0 in D (0, t) for every f L 2 (X, m), h L (X, m), ϕ 0 Applications (Bakry, Ledoux, Lott, Gentil, Qian, Wang, Wei,... ): volume and geometric comparison in weighted Riemannian manifold, Log-Sobolev and spectral-gap inequalities, hypercontractivity of the Markov semigroup, Levy-Gromov isoperimetric inequality in infinite dimension, Li-Yau and Harnack inequalities (for the finite dimensional version BE(K, N)),...

18 21 Riemannian distance, minimal geodesics and exponential map } d g(x 0, x 1) = min {Length[x] : x smooth curve joining x 0 to x 1 (M, d g) is a complete metric space. { d 2 1 g(x 0, x 1) = min 0 } ẋ 2 dr : x : [0, 1] M, x(i) = x i, i = 0, 1 x : [0, 1] M is a minimal, constant speed geodesic if In local coordinates d g(x(s), x(t)) = t s d g(x(0), x(1)). ẍ k (t) = γ k ij(x(t))ẋ i (t)x j (t). ( ) Exponential map: if Z is a vector field and x M, exp x (tz) is the value x(t) of the solution of ( ) with initial conditions x k (0) = x k 0, ẋ k (0) = Z k (x). If Z is smooth, T t(x) := exp x (tz(x)) is smooth flow in M, with T 0(x) = x.

19 Push forward of measures and Ricci curvature X, Y are separable metric space, T : X Y is a Borel map, µ P(X) is a Borel probability measure. Y ν = T µ P(Y ), ν(b) = µ(t 1 (B)) B B(Y ), Borel. f dν(y) = X f(t(x)) dµ(x) f bounded or nonnegative, Borel. X = Y = M, T t smooth with invertible differential dt t, µ = ϱm, µ t = ϱ tm = (T t) µ. ϱ t(t t(x))e V (Tt(x)) V (x) det(dt t(x)) = ϱ(x)e ( ) J t(x) := log ϱ t(t tx)/ϱ(x) = V (T t(x)) V (x) log ( det dt ) t(x). When T t(x) := exp x (t Ψ(x)) then J t(x) 1 N ( J t(x)) 2 + Ric L(Ṫt(x), Ṫt(x))

20 23 Couplings and Wasserstein distance (X, d) is a metric space, µ 0, µ 1 P(X), π i : X X X are the projections π i (x 0, x 1) = x i. Coupling µ P(X X) between µ 0, µ 1: (π i ) µ = µ i, i.e. µ 0(A) = µ(a X), µ 1(B) = µ(x B). P p(x): space of Borel probability measures with finite p-moment: d p (x, x) dµ(x) < for some x X. X µ 1 µ 1 µ µ 0, µ 1 P p(x), W p p (µ 0, µ 1) := min { d p (x 0, x 1) dµ(x 0, x 1) : µ coupling for µ 0, µ 1 } µ 0 µ 0 X

21 Metric properties of P p (X). Optimal coupling: µ Opt(µ 0, µ 1) such that d p (x 0, x 1) dµ = W p p (µ 0, µ 1) (P p(x), W p) is a metric space. If (X, d) is complete (resp. separable, compact, length, geodesic), then (P p(x), W p) is complete (resp. separable, compact, length, geodesic). A metric space is geodesic if every couple of points can be connected by a geodesic. W p(µ n, µ) 0 iff f dµ n f dµ for every f C(X), f(x) A + B d p (x, x). If d is bounded then P p(x) = P(X) and the topology induced by W p coincides with the usual weak topology in P(X), in duality with C b (X)

22 Dynamic properties of P p (X) Path space: C([0, 1]; X). Geo(X) subset of all minimal geodesics. Evaluation map e t : C([0, 1]; X) X, e t(x) := x(t). Dynamic plans: probability measures π on C([0, 1]; X). π is a geodesic plan if it is concentrated on Geo(X), i.e. π(geo(x)) = 1. If π is a dynamic plan, µ t = (e t) π is a continuous curve in P(X). If X is geodesic and µ Opt(µ 0, µ 1) then there exists π P(Geo(X)) such that µ = (e 0, e 1) π. In this case π GeoOpt(µ 0, µ 1) and µ t = (e t) π is a minimal, constant speed, geodesic in P p(x). Geodesic parametrization: conversely, if t µ t is a geodesic in P p(x) there exists an optimal geodesic plan π GeoOpt(µ 0, µ 1) such that µ t = (e t) π for every t [0, 1].

23 [Brenier, McCann, Otto-Villani, Cordero Erausquin-McCann-Schmuckenschlager, Von Renesse-Sturm...] 26 Optimal transport in Riemannian manifold Suppose X = M, m = e V Vol g and µ i = ϱ im P 2(X). There exists a unique geodesic (µ t) t [0,1] connecting µ 0 to µ 1 and a unique geodesic optimal plan π GeoOpt(µ 0, µ 1), µ t = (e t) π. T t(x) = exp(tz) such that x(t) = T t(x(0)) for π-a.e. x, µ t = (T t) µ 0, W2 2 (µ s, µ t) = d 2 (T s(x), T t(x)) dµ 0(x). ( ) µ t = ϱ tm and J t(x) := log ϱ t(t t(x))/ϱ 0(x) satisfies J t(x) 1 N ( J t(x)) 2 + Ric L(Ṫt(x), Ṫt(x)) If Ric L Kg the Relative entropy functional Ent m(µ t) := ϱ t log ϱ t dm satisfies d 2 dt 2 Entm(µt) KW 2 2 (µ 0, µ 1).

24 Second derivative of the entropy E(t) := Ent m(µ t) = = ϱ t log ϱ t dm = log ϱ t dµ t = log(ϱ t(t t(x))) dµ 0 = J t(x) dµ 0 + E(0). Ë(t) J t(x) dµ 0 K Ṫt(x) 2 g dµ 0 = K d 2 g(t 1(x), x) dµ 0 K W 2 2 (µ 1, µ 0) E(t) 1 2 KW 2 2 (µ 1, µ 0)t 2 is convex, E(t) (1 t)e(0) + te(1) K 2 t(1 t)w 2 2 (µ 1, µ 0)

25 Intrinsic metric approach, Bonnet-Myers diameter comparison, Bishop-Gromov volume comparison, stability w.r.t. Sturm-Gromov-Hausdorff convergence (Cheeger-Colding: limits of Riemannian manifold), nonsmooth calculus and Metric measure spaces satisfying a lower Ricci curvature bound: the approach by Lott, Sturm, Villani. The basic object is a metric measure space: (X, d, m) : CD(K, ) spaces (X, d) is a complete and separable metric space, m is a Borel probability measure in P(X) (X, d, m) satisfies the lower Ricci curvature bound CD(K, ) according to Lott-Sturm-Villani if for every µ 0, µ 1 P(X) with finite entropy there exists µ ϑ P(X) such that: Geodesic interpolation in the transport metric: W 2(µ ϑ, µ 0) = ϑw 2(µ 0, µ 1), W 2(µ ϑ, µ 1) = (1 ϑ)w 2(µ 0, µ 1), K-convexity of the Entropy: Ent m(µ ϑ ) (1 ϑ)ent m(µ 0) + ϑent m(µ 1) K 2 ϑ(1 ϑ)w 2 2 (µ 0, µ 1).

26 Main problem: how to connect BE to LSV? Bakry-Émery: gradient commutation along the Heat flow Γ ( P tu ) e 2Kt P tγ(u). Lott-Sturm-Villani: K-convexity of the entropy along geodesic interpolation of measures. Ent m(µ ϑ ) (1 ϑ)ent m(µ 0) + ϑent m(µ 1) K 2 ϑ(1 ϑ)w 2 2 (µ 0, µ 1). Bakry-Émery Time t Heat flow P tµ? µ Geodesic interpolation Space ϑ Lott-Sturm-Villani