Spaces with Ricci curvature bounded from below

Spaces with Ricci curvature bounded from below Nicola Gigli February 23, 2015

Topics 1) On the definition of spaces with Ricci curvature bounded from below 2) Analytic properties of RCD(K, N) spaces 3) Geometric properties of RCD(K, N) spaces 4) More on the differential structure of metric measure spaces

On the definition of spaces with Ricci curvature bounded from below Introduction The gradient flow of the relative entropy w.r.t. W 2 The gradient flow of the Dirichlet energy w.r.t. L 2 The heat flow as gradient flow

Completion / Compactification A common practice in various fields of mathematic is to start studying a certain class of smooth or nice objects, and to close it w.r.t. some relevant topology. In general, the study of the limit objects turns out to be useful to understand the properties of the original ones.

Gromov s plan When the original class of objects are Riemannian manifolds with some curvature bounds, this program has been proposed by Gromov.

Gromov s plan When the original class of objects are Riemannian manifolds with some curvature bounds, this program has been proposed by Gromov. Topology Synthetic notion Bounds from above/below on sectional curvature Gromov-Hausdorff convergence Bounds from below on the Ricci curvature

Gromov s plan When the original class of objects are Riemannian manifolds with some curvature bounds, this program has been proposed by Gromov. Topology Synthetic notion Bounds from above/below on Gromov-Hausdorff Alexandrov spaces sectional curvature convergence Bounds from below on the Ricci curvature

Gromov s plan When the original class of objects are Riemannian manifolds with some curvature bounds, this program has been proposed by Gromov. Topology Synthetic notion Bounds from above/below on Gromov-Hausdorff Alexandrov spaces sectional curvature convergence Bounds from below on the Ricci curvature measured Gromov-Hausdorff convergence

Gromov s plan When the original class of objects are Riemannian manifolds with some curvature bounds, this program has been proposed by Gromov. Topology Synthetic notion Bounds from above/below on Gromov-Hausdorff Alexandrov spaces sectional curvature convergence Bounds from measured CD(K, N) spaces / below on the Gromov-Hausdorff RCD(K, N) spaces Ricci curvature convergence

Aim of the game (1) To understand what it means for a metric measure space to have Ricci curvature bounded from below (2) To prove in the non-smooth setting all the theorems valid for manifolds with Ric K, dim N and their limits (3) To better understand the geometry of smooth manifolds via the study of non-smooth objects

The curvature condition Theorem (Sturm-VonRenesse 05) - see also Otto-Villani and Cordero Erausquin-McCann-Schmuckenschlager Let M be a smooth Riemannian manifold. Then the following are equivalent: i) The Ricci curvature of M is uniformly bounded from below by K ii) The relative entropy functional is K -convex on the space (P 2 (M), W 2 ) Definition (Lott-Villani and Sturm 06) (X, d, m) has Ricci curvature bounded from below by K if the relative entropy is K -convex on (P 2 (X), W 2 ). Called CD(K, ) spaces, in short.

Finsler structures are included Cordero-Erausquin, Villani, Sturm proved that (R d,, L d ) is a CD(0, ) space (in fact CD(0, d)) for any norm.

Finsler structures are included Cordero-Erausquin, Villani, Sturm proved that (R d,, L d ) is a CD(0, ) space (in fact CD(0, d)) for any norm. Some differences between the Finsler and Riemannian worlds: Analysis: Tangent / cotangent spaces can t be identified No natural Dirichlet form Geometry: no Abresch-Gromoll inequality no Splitting theorem

Some observations For a given Finsler manifold the following are equivalent: The manifold is Riemannian The Sobolev space W 1,2 is Hilbert The heat flow is linear The heat flow can be seen as: Gradient flow of the Dirichlet energy w.r.t. L 2 Gradient flow of the relative entropy w.r.t. W2

The idea Restrict to the class of CD(K, ) spaces such that W 1,2 is Hilbert, because if we were able to make computations as in Riemannian manifolds, we should be able to prove the same results.

The idea Restrict to the class of CD(K, ) spaces such that W 1,2 is Hilbert, because if we were able to make computations as in Riemannian manifolds, we should be able to prove the same results. The non-trivial task here is to prove that the resulting notion is stable w.r.t. mgh convergence.this will be achieved by: noticing that, even in the non-smooth case, W 1,2 is Hilbert if and only if the L 2 gradient flow of the Dirichlet energy is linear proving that the L 2 gradient flow of the Dirichlet energy coincides with the W 2 -gradient flow of the relative entropy proving that the W 2 -gradient flow of the relative entropy is stable

Definition of Gradient Flow: the smooth case Let (x t ) R d a smooth curve and f : R d R a smooth functional. Then f (x 0 ) f (x t ) t 1 2 0 t x s f (x s ) ds 0 x s 2 + f 2 (x s ) ds.

Definition of Gradient Flow: the smooth case Let (x t ) R d a smooth curve and f : R d R a smooth functional. Then f (x 0 ) f (x t ) t 1 2 Therefore 0 t x t = f (x t ), t 0 f (x 0 ) = f (x t ) + 1 2 t 0 x s f (x s ) ds 0 x s 2 + f 2 (x s ) ds. x s 2 + f 2 (x s ) ds, t > 0.

Definition of Gradient Flow: the metric setting d(x t+h, x t ) ẋ t := lim for an abs.cont. curve (x t ) h 0 h (F (x) F(y)) + F (x) := lim y x d(x, y) The weak chain rule F (x 0 ) F(x t ) + 1 2 t 0 ẋ s 2 + F 2 (x s ) ds, t > 0. holds for K -convex and l.s.c. F : X R {+ }. Definition (x t ) is a Gradient Flow for the K -conv. and l.s.c. functional F provided (x t ) {F < } is a loc.abs.cont. curve and F(x 0 ) = F (x t ) + 1 2 t 0 ẋ s 2 + F 2 (x s ) ds, t > 0.

General results about GF of K -convex functionals Existence Granted if the space is compact and F(x 0 ) < (Ambrosio, G., Savaré 04 (after De Giorgi)) Uniqueness False in general

Basic facts about the GF of the Entropy Thm. (G. 09) Let (X, d, m) be a compact CD(K, ) space. Then for µ P 2 (X) with Ent m (µ) < the GF of Ent m starting from µ exists and is unique.

Basic facts about the GF of the Entropy Thm. (G. 09) Let (X, d, m) be a compact CD(K, ) space. Then for µ P 2 (X) with Ent m (µ) < the GF of Ent m starting from µ exists and is unique. Furthermore, such flow is stable w.r.t. mg-convergence of the base space.

Proof of uniqueness Key Lemma Let (X, d, m) be a compact CD(K, ) space. Then Ent m 2 ( ) is convex w.r.t. linear interpolation.

Proof of uniqueness Key Lemma Let (X, d, m) be a compact CD(K, ) space. Then Ent m 2 ( ) is convex w.r.t. linear interpolation. Then by contradiction: assume (µ t ), (ν t ) are two GF starting from µ with Ent m ( µ) < and define σ t := µt +νt 2. Then for every t such that µ t ν t we have Ent m ( µ) > Ent m (σ t ) + 1 2 t 0 σ s 2 + Ent m 2 (σ s ) ds

mg convergence of compact spaces (X n, d n, m n ) converges to (X, d, m ) in the mg sense if there is (Y, d Y ) and isometric embeddings ι n, ι of the X s into Y such that (ι n ) m n weakly converges to (ι ) m We say that n µ n P(X n ) weakly converges to µ P(X ) provided (ι n ) µ n weakly converges to (ι ) µ

Γ-convergence of the entropies Thm. (Lott-Sturm-Villani) Let (X n, d n, m n ) be converging to (X, d, m ). Then: Γ lim inequality: for every sequence n µ n P(X n ) weakly converging to µ P(X ) we have Ent m (µ ) lim Ent mn (µ n ). n Γ lim inequality: for every µ P(X ) there is a sequence n µ n P(X n ) weakly converging to µ such that Ent m (µ ) lim n Ent mn (µ n ).

Γ lim for the slopes Thm. (G. 09) Let X n be CD(K, ) spaces mg-converging to X and µ n weakly converging to µ. Then Ent m (µ ) lim Ent mn (µ n ). n

Γ lim for the slopes Thm. (G. 09) Let X n be CD(K, ) spaces mg-converging to X and µ n weakly converging to µ. Then Ent m (µ ) lim Ent mn (µ n ). n Cor. 1 Let X n be mg-converging to X and µ n be weakly converging to µ be such that lim Ent m n (µ n ) = Ent m (µ ) <. n Then the GF of Ent mn starting from µ. starting from µ n converge to the GF of Ent m

Variational definition of Df on R d Let f : R d R be smooth. Then Df is the minimum continuous function G for which f (γ 1 ) f (γ 0 ) holds for any smooth curve γ 1 0 G(γ t ) γ t dt

Test plans Let π P(C([0, 1], X)). We say that π is a test plan provided: for some C > 0 it holds e t π Cm, t [0, 1]. it holds 1 0 γ t 2 dt dπ <

The Sobolev class S 2 (X, d, m) We say that f : X R belongs to S 2 (X, d, m) provided there exists G L 2 (X, m), G 0 such that for any test plan π. f (γ1 ) f (γ 0 ) 1 dπ(γ) G(γ t ) γ t dt dπ(γ) 0 Any such G is called weak upper gradient of f. The minimal G in the m-a.e. sense will be denoted by Df

Basic properties Lower semicontinuity From f n f m-a.e. with f n S 2 and Df n G weakly in L 2 we deduce f S 2, Df G Locality Df = Dg m-a.e. on {f = g} Chain rule for ϕ Lipschitz D(ϕ f ) = ϕ f Df Leibniz rule for f, g S 2 L D(fg) f Dg + g Df

The Energy E and the Sobolev space W 1,2 We define E : L 2 (X, m) [0, + ] as E(f ) := 1 Df 2 dm if f S 2 (X), + otherwise. 2 Then E is convex and lower semicontinuous.

The Energy E and the Sobolev space W 1,2 We define E : L 2 (X, m) [0, + ] as E(f ) := 1 Df 2 dm if f S 2 (X), + otherwise. 2 Then E is convex and lower semicontinuous. The Sobolev space W 1,2 (X) is W 1,2 (X) := L 2 (X) S 2 (X) endowed with the norm f 2 W 1,2 := f 2 L 2 + Df 2 L 2 W 1,2 (X) is a Banach space.

Laplacian (first definition) We say that f D( ) W 1,2 (X) if E(f ) 0. In this case we define f := v, where v is the element of minimal norm in E(f ).

Integration by parts For f D( ) and g W 1,2 (X) we have g f dm Dg Df dm. For a C 1 map u : R R we have u(f ) f dm = u (f ) Df 2 dm.

Gradient flow of E w.r.t. L 2 For any f 0 L 2 (X, m) there exists a unique map t f t L 2 (X, m) such that d + dt f t = f t, t > 0.

The result Thm. (G., Kuwada, Ohta 10. Ambrosio, G. Savaré 11) Let (X, d, m) be a CD(K, ) space and µ = f m P 2 (X) with f L 2 (X, m). Let t f t be the GF of E w.r.t. L 2 starting from f t µ t be the GF of Ent m w.r.t. W 2 starting from µ Then µ t = f t m t 0.

Main steps of the proof Let t f t be a Gradient Flow of E starting from a probability density f 0 and define µ t = f t m.

Main steps of the proof Let t f t be a Gradient Flow of E starting from a probability density f 0 and define µ t = f t m. We will prove: Mass preservation and maximum/minimum principle to show that c f 0 C implies c f t C for every t 0 That t Ent m (µ t ) is a.c. and The slope estimate: That (µ t ) is a.c. w.r.t. W 2 and Dft 2 t Ent m (µ t ) = dm Ent m 2 Dft 2 (µ t ) dm f t f t µ t 2 Dft 2 dm f t

Maximum/minimum principle ad mass preservation We claim that for any f L 2 such that f c and any τ > 0 the minimum of g 1 Dg 2 dm + f g 2 L 2, 2 2τ satisfies g c.

Maximum/minimum principle ad mass preservation We claim that for any f L 2 such that f c and any τ > 0 the minimum of g 1 Dg 2 dm + f g 2 L 2, 2 2τ satisfies g c. If not, the function max{g, c} provides a better competitor. The minimum principle follows from the convergence of the implicit Euler scheme to the Gradient Flow

Entropy dissipation For 0 < c f 0 C < the curve t f t L 2 (X, m) is AC.

Entropy dissipation For 0 < c f 0 C < the curve t f t L 2 (X, m) is AC. Then so is the curve t u(f t ) L 2 (X, m) L 1 (X, m) for u(z) = z log z.

Entropy dissipation For 0 < c f 0 C < the curve t f t L 2 (X, m) is AC. Then so is the curve t u(f t ) L 2 (X, m) L 1 (X, m) for u(z) = z log z. Computing we have t u(f t ) dm = u (f t ) f t dm = u (f t ) Df t 2 dm

Slope of the entropy and Fisher information Lemma (Lott-Villani) For µ = f m with f Lipschitz we have lip Ent m 2 2 f (µ) f dm where lip f (x) := lim y x f (y) f (x) d(x, y)

Slope of the entropy and Fisher information Lemma (Lott-Villani) For µ = f m with f Lipschitz we have lip Ent m 2 2 f (µ) f dm where lip f (x) := lim y x f (y) f (x) d(x, y) By relaxation, using the weak lower semicontinuity of the slope we deduce Df Ent m 2 2 (µ) dm = 4 D f 2 dm f for every µ = f m with f W 1,2.

The non-trivial property: Kuwada s lemma Suppose that µ 0 := f m is in P 2 (X).

The non-trivial property: Kuwada s lemma Suppose that µ 0 := f m is in P 2 (X). Then µ t := f t m is in P 2 (X) and the curve t µ t is absolutely continuous w.r.t. W 2 and µ t 2 Dft 2 dm f t

Hamilton-Jacobi semigroup alias Hopf-Lax formula alias Moreau-Yosida approximation alias inf-convolution Let (X, d) be a metric space. For ψ : X R Lipschitz and bounded t > 0 we define Q t ψ : X R by Q t ψ(x) := inf ψ(y) + d 2 (x, y) y X 2t Then: Q t ψ is Lipschitz for every t > 0, t Q t ψ is a Lipschitz curve w.r.t. the sup distance (Q 0 ψ := ψ),

Solutions of Hamilton-Jacobi equation For any x X the map t Q t ϕ(x) is locally Lipschitz and it holds ( d dt Q lipqt ϕ(x) ) 2 tϕ(x) + 0, 2 for every t 0 with the possible exception of a countable set.

Estimating W 2 (µ t, µ t+s ) by duality 1 2 W 2 2 (µ t, µ t+s ) = sup ϕ LIP = sup ψ LIP ϕf t dm + Q 1 ψf t+s dm ϕ c f t+s dm ψf t dm

Key computation r Q r ψ and r f t+rs are Lipschitz curves with values in L 2 (X, m).

Key computation r Q r ψ and r f t+rs are Lipschitz curves with values in L 2 (X, m). Thus Q 1 ψf t+s ψf t dm 1 d ) = (Q r ψf t+rs dr dm dr 0 1 = 0 1 0 (lipq r ψ) 2 f t+rs + sq r ψ f t+rs dr dm 2 (lipq r ψ) 2 2 1 (lipq r ψ) 2 0 2 1 s2 Df t+rs 2 2 f t+rs 0 f t+rs + DQ r ψ s Df t+rs f t+rs dr dm f t+rs f t+rs + DQ r ψ 2 f t+rs + s2 Df t+rs 2 dr dm 2 2 f t+rs dr dm.

Conclusion of the argument W 2 2 (µ t, µ t+s ) s 2 1 s2 c s2 c 0 1 0 Df 2 t+rs f t+rs Df t 2 dm dm dr Df t+rs 2 dm dr

Spaces with Riemannian Ricci curvature bounded from below RCD(K, N) := CD(K, N) + linearity of the heat flow = CD(K, N) + W 1,2 is Hilbert

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