A few words about the MTW tensor

Transcription

1 A few words about the Ma-Trudinger-Wang tensor Université Nice - Sophia Antipolis & Institut Universitaire de France Salah Baouendi Memorial Conference (Tunis, March 2014)

2 The Ma-Trudinger-Wang tensor Let M be a smooth connected (compact) manifold of dimension n equipped with a smooth Riemannian metric g. For any x, y M, we define the geodesic distance between x and y, denoted by d(x, y), as the minimum of the lengths of the curves joining x to y. Definition (The Ma-Trudinger-Wang tensor) The MTW tensor S is defined as S (x,v) (ξ, η) = 2 3 d 2 d 2 d 2 ( ) ds 2 s=0 dt 2 exp t=0 2 x (tξ), exp x (v+sη) for every x M, v I(x), and ξ, η T x M.

3 Transport maps Let µ 0 and µ 1 be probability measures on M. We call transport map from µ 0 to µ 1 any measurable map T : M M such that T µ 0 = µ 1, that is µ 1 (B) = µ 0 ( T 1 (B) ), B measurable M. T µ 0 µ 1

4 Quadratic Monge s Problem Given two probabilities measures µ 0, µ 1 sur M, find a transport map T : M M from µ 0 to µ 1 which minimizes the quadratic cost (c = d 2 /2) c(x, T (x))dµ 0 (x). M

5 Quadratic Monge s Problem Given two probabilities measures µ 0, µ 1 sur M, find a transport map T : M M from µ 0 to µ 1 which minimizes the quadratic cost (c = d 2 /2) c(x, T (x))dµ 0 (x). M Theorem (McCann 01) If µ 0 is absolutely continuous w.r.t. Lebesgue, then there exists a unique optimal transport map T from µ 0 to µ 1. In fact, there is a c-convex function ϕ : M R satisfying T (x) = exp x ( ϕ(x)) µ 0 a.e. x M. (Moreover, for a.e. x M, ϕ(x) belongs to the injectivity domain at x.)

6 The case of R n Quadratic Monge s Problem in R n : Given two probability measures µ 0, µ 1 with compact supports in R n, we are concerned with transport maps T : R n R n pushing forward µ 0 to µ 1 which minimize the transport cost R n T (x) x 2 dµ 0 (x). Theorem (Brenier 91) If µ 0 is absolutely continuous with respect to the Lebesgue measure, there exists a unique optimal transport map with respect to the quadratic cost. In fact, there is a convex function ψ : M R such that T (x) = ψ(x) µ 0 a.e. x R n.

7 The case of R n Quadratic Monge s Problem in R n : Given two probability measures µ 0, µ 1 with compact supports in R n, we are concerned with transport maps T : R n R n pushing forward µ 0 to µ 1 which minimize the transport cost R n T (x) x 2 dµ 0 (x). Theorem (Brenier 91) If µ 0 is absolutely continuous with respect to the Lebesgue measure, there exists a unique optimal transport map with respect to the quadratic cost. In fact, there is a convex function ψ : M R such that T (x) = ψ(x) µ 0 a.e. x R n. Necessary and sufficient conditions for regularity?

8 An obvious counterexample Supp(µ 0 ) Supp(µ 1 ) T

9 The convexity of the targer is necessary Supp(µ 0 ) Supp(µ 1 ) T

10 The convexity of the targer is necessary Supp(µ 0 ) Supp(µ 1 ) 49% 70% 29% 49%

11 The convexity of the targer is necessary Supp(µ 0 ) Supp(µ 1 ) 49% 70% 29% T 49%

12 The convexity of the targer is necessary Supp(µ 0 ) Supp(µ 1 )

13 The convexity of the targer is necessary Supp(µ 0 ) Supp(µ 1 ) T gradient of a convex function

14 The convexity of the targer is necessary Supp(µ 0 ) Supp(µ 1 ) T gradient of a convex function = y x, T (y) T (x) 0!!!

15 Caffarelli s Regularity Theory If µ 0 and µ 1 are associated with densities f 0, f 1 w.r.t. Lebesgue, then T µ 0 = µ 1 ζ ( T (x) ) f 0 (x)dx = ζ(y)f 1 (y)dy R n R n ψ weak solution of the Monge-Ampère equation : Theorem (Caffarelli 90s) det ( 2 ψ(x) ) = f 0(x) f 1 ( ψ(x)). ζ. Let Ω 0, Ω 1 be connected and bounded open sets in R n and f 0, f 1 be probability densities resp. on Ω 0 and Ω 1 such that f 0, f 1, 1/f 0, 1/f 1 are essentially bounded. Assume that µ 0 and µ 1 have respectively densities f 0 and f 1 w.r.t. Lebesgue and that Ω 1 is convex. Then the quadratic optimal transport map from µ 0 to µ 1 is continuous.

16 Back to Riemannian manifolds Given two probabilities measures µ 0, µ 1 sur M, find a transport map T : M M from µ 0 to µ 1 which minimizes the quadratic cost (c = d 2 /2) c(x, T (x))dµ 0 (x). M

17 Back to Riemannian manifolds Given two probabilities measures µ 0, µ 1 sur M, find a transport map T : M M from µ 0 to µ 1 which minimizes the quadratic cost (c = d 2 /2) c(x, T (x))dµ 0 (x). M Definition We say that the Riemannian manifold (M, g) satisfies the Transport Continuity Property (TCP) if for any pair of probability measures µ 0, µ 1 associated locally with continuous positive densities ρ 0, ρ 1, that is µ 0 = ρ 0 vol g, µ 1 = ρ 1 vol g, the optimal transport map from µ 0 to µ 1 is continuous.

18 Exponential mapping and injectivity domains Let x M be fixed. For every v T x M, we define the exponential of v by exp x (v) = γ x,v (1), where γ x,v : [0, 1] M is the unique geodesic starting at x with speed v.

19 Exponential mapping and injectivity domains Let x M be fixed. For every v T x M, we define the exponential of v by exp x (v) = γ x,v (1), where γ x,v : [0, 1] M is the unique geodesic starting at x with speed v. We call injectivity domain at x, the subset of T x M defined by { t > 1 s.t. γ I(x) := v T x M tv is the unique minim. geod. between x and exp x (tv) It is a star-shaped (w.r.t. 0 T x M) bounded open set with Lipschitz boundary. }

20 The distance to the cut locus Using the exponential mapping, we can associate to each unit tangent vector its distance to the cut locus. In that way, we define the so-called injectivity domain I(x) whose boundary is the tangent cut locus TCL(x).

21 A necessary condition for TCP Theorem (Figalli-R-Villani 10) Let (M, g) be a smooth compact Riemannian manifold satisfying TCP. Then the following properties hold: all the injectivity domains are convex, for any x, x M, the function F x,x : v I(x) c ( x, exp x (v) ) c ( x, exp x (v) ) is quasiconvex (its sublevel sets are always convex).

22 Almost characterization of quasiconvex functions Lemma (Sufficient condition) Let U R n be an open convex set and F : U R be a function of class C 2. Assume that for every v U and every w R n \ {0} the following property holds : Then F is quasiconvex. v F, w = 0 = 2 vf w, w > 0.

23 Almost characterization of quasiconvex functions Lemma (Sufficient condition) Let U R n be an open convex set and F : U R be a function of class C 2. Assume that for every v U and every w R n \ {0} the following property holds : Then F is quasiconvex. v F, w = 0 = 2 vf w, w > 0. Lemma (Necessary condition) Let U R n be an open convex set and F : U R be a function of class C 2. Assume that F is quasiconvex. Then for every v U and every w R n \ {0} the following property holds : v F, w = 0 = 2 vf w, w 0.

24 Proof of the easy lemma Proof. Let v 0, v 1 U be fixed. Set v t := (1 t)v 0 + tv 1, for every t [0, 1]. Define h : [0, 1] R by h(t) := F (v t ) t [0, 1]. If h max{h(0), h(1)}, there is τ (0, 1) such that h(τ) = max h(t) > max{h(0), h(1)}. t [0,1]

25 Proof of the easy lemma Proof. Let v 0, v 1 U be fixed. Set v t := (1 t)v 0 + tv 1, for every t [0, 1]. Define h : [0, 1] R by h(t) := F (v t ) t [0, 1]. If h max{h(0), h(1)}, there is τ (0, 1) such that There holds h(τ) = max h(t) > max{h(0), h(1)}. t [0,1] ḣ(τ) = vτ F, v τ et ḧ(τ) = 2 v τ F v τ, v τ. Since τ is a local maximum, one has ḣ(τ) = 0. Contradiction!!

26 The Ma-Trudinger-Wang tensor The MTW tensor S is defined as S (x,v) (ξ, η) = 3 d 2 d 2 d 2 ( ) 2 ds 2 s=0 dt 2 exp t=0 2 x (tξ), exp x (v+sη), for every x M, v I(x), and ξ, η T x M.

27 The Ma-Trudinger-Wang tensor The MTW tensor S is defined as S (x,v) (ξ, η) = 3 d 2 d 2 d 2 ( ) 2 ds 2 s=0 dt 2 exp t=0 2 x (tξ), exp x (v+sη), for every x M, v I(x), and ξ, η T x M. Proposition (Villani 09, Figalli-R-Villani 10) Let (M, g) be such that all injectivity domains are convex. Then the following properties are equivalent: All the functions F x,x are quasiconvex. The MTW tensor S is 0, that is for any x M, v I(x), and ξ, η T x M, ξ, η x = 0 = S (x,v) (ξ, η) 0.

28 When geometry enters the problem Theorem (Loeper 06) For every x M and for any pair of unit orthogonal tangent vectors ξ, η T x M, there holds S (x,0) (ξ, η) = κ x (ξ, η), where the latter denotes the sectional curvature of M at x along the plane spanned by {ξ, η}. Corollary (Loeper 06) TCP = S 0 = κ 0.

29 When geometry enters the problem Theorem (Loeper 06) For every x M and for any pair of unit orthogonal tangent vectors ξ, η T x M, there holds S (x,0) (ξ, η) = κ x (ξ, η), where the latter denotes the sectional curvature of M at x along the plane spanned by {ξ, η}. Corollary (Loeper 06) Caution!!! κ 0 S 0. TCP = S 0 = κ 0.

30 Sufficient conditions for TCP Theorem (Figalli-R-Villani 10) Let (M, g) be a compact smooth Riemannian surface. It satisfies TCP if and only if the two following properties holds: all the injectivity domains are convex, S 0.

31 Sufficient conditions for TCP Theorem (Figalli-R-Villani 10) Let (M, g) be a compact smooth Riemannian surface. It satisfies TCP if and only if the two following properties holds: all the injectivity domains are convex, S 0. Theorem (Figalli-R-Villani 10) Assume that (M, g) satisfies the two following properties: all its injectivity domains are strictly convex, the MTW tensor S is 0, that is for any x M, v I(x), and ξ, η T x M, Then, it satisfies TCP. ξ, η x = 0 = S (x,v) (ξ, η) > 0.

32 Examples and counterexamples Examples: Flat tori (Cordero-Erausquin 99) Round spheres (Loeper 06) Product of spheres (Figalli-Kim-McCann 13) Quotients of the above objects Perturbations of round spheres (Figalli-R-Villani 12)

33 Examples and counterexamples Examples: Flat tori (Cordero-Erausquin 99) Round spheres (Loeper 06) Product of spheres (Figalli-Kim-McCann 13) Quotients of the above objects Perturbations of round spheres (Figalli-R-Villani 12) Counterexamples:

34 Examples and counterexamples Examples: Flat tori (Cordero-Erausquin 99) Round spheres (Loeper 06) Product of spheres (Figalli-Kim-McCann 13) Quotients of the above objects Perturbations of round spheres (Figalli-R-Villani 12) Counterexamples:

35 Questions How is the set of metrics satisfying S 0? Does S 0 imply convexity of injectivity domains? Does S 0 imply more topological obstructions than κ 0?

36 Questions How is the set of metrics satisfying S 0? Does S 0 imply convexity of injectivity domains? Does S 0 imply more topological obstructions than κ 0? Focalization is the major obstacle

37 Focalization Definition Let x M and v be a unit tangent vector in T x M. The vector v is not conjugate at time t 0 if for any t [0, t + δ] (δ > 0 small) the geodesic from x to γ(t ) is locally minimizing. x x

38 The distance to the conjugate locus Again, using the exponential mapping, we can associate to each unit tangent vector its distance to the conjugate locus. In that way, we define the so-called nonfocal domain N F(x) whose boundary is the tangent conjugate locus TFL(x).

39 Fundamental inclusion The following inclusion holds injectivity domain nonfocal domain.

40 Spheres Loeper checked that the MTW tensor of the round sphere S 2 satisfies for any x S 2, v I(x) and ξ, η T x S 2, Theorem (Loeper 06) ξ, η x = 0 = S (x,v) (ξ, η) ξ 2 η 2. The round sphere S 2 satisifes TCP.

41 Small deformations of S 2 On S 2, the MTW tensor is given by ( S ) (x,v) ξ, ξ [ 1 = 3 r cos(r) ] 2 r sin(r) [ ξ sin 2 (r) r cos(r) sin 3 (r) [ 6 r 2 + cos(r) r sin(r) + 5 sin 2 (r) ] ξ 4 2 ] ξ 2 1ξ 2 2, with x S 2, v I(x), r := v, ξ = (ξ 1, ξ 2 ), ξ = ( ξ 2, ξ 1 ). Theorem (Figalli-R 09) Any small deformation of S 2 in C 4 topology satisfies TCP.

42 The Villani Conjecture Conjecture (Villani s Conjecture) Any Riemannian manifold satisfying S 0 has convex injectivity domains.

43 The Villani Conjecture Conjecture (Villani s Conjecture) Any Riemannian manifold satisfying S 0 has convex injectivity domains. Theorem (Figalli-Gallouët-R 14) Let (M, g) be a smooth compact Riemannian manifold, assume that it is nonfocal. Then all its injectivity domains are convex.

44 Thank you for your attention!!