Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S 2

Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S A. Figalli L. Rifford Abstract Given a compact Riemannian manifold, we study the regularity of the optimal transport map between two probability measures with cost given by the squared Riemannian distance. Our strategy is to define a new form of the so-called Ma-Trudinger-Wang condition and to show that this condition, together with the strict convexity on the nonfocal domains, implies the continuity of the optimal transport map. Moreover our new condition, again combined with the strict convexity of the nonfocal domains, allows to prove that all injectivity domains are strictly convex too. These results apply for instance on any small C 4 -deformation of the two-sphere. Introduction Let µ, ν be two probability measures on a smooth compact connected Riemannian manifold M, g equipped with its geodesic distance d. Given a cost function c : M M R, the Monge-Kantorovich problem consists in finding a transport map T : M M which sends µ onto ν i.e. T # µ = ν and which minimizes the functional cx, Sx dx. min S # µ=ν M In [] McCann generalizing [] from the Euclidean case proved that, if µ gives zero mass to countably n -rectifiable sets, then there is a unique transport map T solving the Monge- Kantorovich problem with initial measure µ, final measure ν, and cost function c = d /. The purpose of this paper is to study the regularity of T. This problem has been extensively investigated in the Euclidean space [3, 4, 5, 9, 5, 6], in the case of the flat torus or nearly flat metrics [8, 0], on the standard sphere and its perturbations [, 7, 9], and on manifolds with nonfocal cut locus [0] see [8, Chapter ] for an introduction to the problem of the regularity of the optimal tranport map for a general cost function. Definition.. Let M, g be a smooth compact connected Riemannian manifold. We say that M, g satisfies the transport continuity property abbreviated T CP if, whenever µ and ν satisfy µb i lim rx r 0 r = 0 for any x M, n νb ii inf x M lim inf rx r 0 r > 0, n Centre de Mathématiques Laurent Schwartz, Ecole Polytechnique, Palaiseau, France figalli@math.polytechnique.fr Université de Nice-Sophia Antipolis, Labo. J.-A. Dieudonné, UMR 66, Parc Valrose, 0608 Nice Cedex 0, France rifford@unice.fr

then the unique optimal transport map T between µ and ν is continuous. Note that the above definition makes sense: by a standard covering argument one can prove that assumption i implies that µ gives zero mass to countably n -rectifiable sets. Thus, by McCann s Theorem, the optimal transport map T from µ to ν exists and is unique. If M, g is a given Riemannian manifold, we call C 4 -deformation of M, g any Riemannian manifold of the form M, g with g close to g in C 4 -topology. Loeper [9] proved that the round sphere S n, g can satisfies T CP. Then, Loeper and Villani [0] showed that any C 4 - deformation of quotients of the sphere like RP n satisfies T CP. Furthermore, Delanoe and Ge [] proved a regularity result under restriction on the measures on C 4 -deformation of the round spheres see also [9]. The main aim of this paper is to prove the following result: Theorem.. Any C 4 -deformation of the round sphere S, g can satisfies T CP. We notice that the above theorem is the first regularity result for optimal transport maps allowing for perturbations of the standard metric on the sphere without any additional assumption on the measures. In particular this shows that, if we sligthly perturbs the sphere into an ellipsoid, then T CP holds true. Furthermore, quite surprisingly the method of our proof allows to easily deduce as a byproduct the strict convexity of all injectivity domains on perturbations of the two sphere. This geometric result is to our knowledge completely new see [0] where the authors deal with nonfocal manifolds: Theorem.3. On a C 4 -deformation of the round sphere S, g can, all injectivity domains are strictly convex. It is known [8, 8] that a necessary condition to prove the continuity of optimal transport maps is the so-called Ma-Trudinger-Wang condition in short MTW condition. This condition is expressed in terms of the fourth derivatives of the cost function, and so makes sense on the domain on smoothness of the distance function, that is outside the cut locus. Another important condition to prove regularity results is the so-called c-convexity of the target domain see [, 8], which in the case of the squared Riemannian distance corresponds to the convexity of all injectivity domains see.4. So, to obtain regularity results on small deformations of the sphere, on the one hand one has to prove the stability of the MTW condition, and on the other hand one needs to show that the convexity of the injectivity domain is stable under small perturbations. Up to now it was not known whether the convexity of the injectivity domains is stable under small perturbations of the metric, except in the nonfocal case see [7, 5, 0]. Indeed the boundaries of the injectivity domains depend on the global geometry of the manifold, and this makes the convexity issue very difficult. Theorem.3 above is the first general result in this direction. Our strategy to deal with these problems is to introduce a variant of the MTW condition, which coincides with the usual one up to the cut locus, but that can be extended up to the first conjugate point see Paragraph.. In this way, since our extended MTW condition is defined up to the first conjugate time, all we really need is the convexity of the nonfocal domains see., which can be shown to be stable under small C 4 -perturbation of the metric see Paragraph 5.. Thus, in Theorems 3. and 3.6 we prove that the strict convexity of nonfocal domains, together with our extended MTW condition, allows to adapt the argument in [0] changing in a careful way the function to which one has to apply the MTW condition to conclude the validity of T CP. Moreover, as shown in Theorem 3.4 and Remark 3.5, the strategy of our proof of Theorem 3. allows to easily deduce the strict convexity of the injectivity domains. Since the assumptions of Theorem 3. are satisfied by C 4 -deformation of S, g can, Theorems. and.3 follow.

The paper is organized as follows: in Section we recall some basic facts in symplectic geometry, and we introduce what we call the extended Ma-Trudinger-Wang condition MT WK, C. In Section 3 we show how MT WK, C, together with the strict convexity of the cotangent nonfocal domains, allows to prove the strict convexity of the injectivity domains and T CP on a general Riemannian manifold. In Section 4 we prove the stability of MT WK, C under C 4 -deformation of S, g can. Then, in Section 5 we collect several remarks showing other cases when our results apply, and explaining why our continuity result cannot be easily improved to higher regularity. Finally, in the appendix we show that the standard sphere S n, g can satisfies MT WK 0, K 0 for some K 0 > 0. Acknowledgements: we warmly thank Cédric Villani for stimulating our interest on the problem. We also acknowledge the anonimous referee for useful comments and for spotting a mistake in a preliminary version of the paper. The extended MTW condition. In the sequel, M, g always denotes a smooth compact connected Riemannian manifold of dimension n, and we denote by d its Riemannian distance. We denote by T M the tangent bundle and by π : T M M the canonical projection. A point in T M is denoted by x, v, with x M and v T x M = π x. For v T x M, the norm v x is g x v, v /. For every x M, exp x : T x M M stands for the exponential mapping from x, and cutx for the cut locus from x i.e. the closure of the set of points y x where the distance function from x dx, is not differentiable. We denote by T M the cotangent bundle and by π : T M M the canonical projection. A point in T M will be denoted by x, p, with x M and p T M a linear form on the vector space T x M. For every p T x M and v T x M, we denote by p, v the action of p on v. The dual metric and norm on T M are respectively denoted by g x, and x. The cotangent bundle is endowed with its standard symplectic structure ω. A local chart ϕ : U M ϕu R n for M induces on T M a natural chart T ϕ : T U T ϕu = ϕu R n. This gives coordinates x,..., x n on U, and so coordinates x,..., x n, p,, p n on T U such that the symplectic form is given by ω = dx dp on T U. Such a set of local coordinates on T M is called symplectic. Fix θ = x, p T M. We recall that a subspace E T θ T M is called Lagrangian if it is a n-dimensional vector subspace where the symplectic bilinear form ω θ : T θ T M T θ T M R vanishes. The tangent space T θ T M splits as a direct sum of two Lagrangian subspaces: the vertical subspace V θ = kerd θ π and the horizontal subspace H θ given by the kernel of the connection map C θ : T θ T M T x M defined as C θ χ := D t Γ0 χ T θ T M, where t, γt, Γt T M is a smooth curve satisfying γ0, Γ0 = x, p and γ0, Γ0 = χ, and where D t Γ denotes the covariant derivative of Γ along the curve γ. Using the isomorphism K θ : T θ T M T x M Tx M χ d θ π χ, C θ χ, we can identify any tangent vector χ T θ T M with its coordinates χ h, χ v := K θ χ in the splitting H θ, V θ. Therefore we have H θ T x M {0} R n {0} 3

and so that V θ {0} T x M {0} R n, T θ T M H θ V θ R n R n. If a given n-dimensional vector subspace E T θ T M is transversal to V θ i.e. E V θ = {0}, then E is the graph of some linear map S : H θ V θ. It can be checked that E is Lagrangian if and only if S is symmetric in a symplectic set of local coordinates. The Hamiltonian vector field X H of a smooth function H : T M R is the vector field on T M uniquely defined by ω θ XH θ, = d θ H for any θ T M. In a symplectic set of local coordinates, the Hamiltonian equations i.e. the equations satisfied by any solution of ẋ, ṗ = X H x, p are given by ẋ = H p, ṗ = H x. Finally, we recall that the Hamiltonian flow φh t of X H preserves the symplectic form ω. We refer the reader to [, 6] for more details about the notions of symplectic geometry introduced above.. Let H : T M R be the Hamiltonian canonically associated with the metric g, i.e. Hx, p = p x x, p T M. We denote by φ H t the Hamiltonian flow on T M, that is the flow of the vector field written in a symplectic set of local coordinates as { ẋ = H p x, p, ṗ = H x x, p. For every x, p T M, we define the Lagrangian subspace J x,p T x,p T M T x M T x M as the pullback of the vertical distribution at φ H x, p by φ H, that is J x,p := φ H Vφ H = φ x,p H Vφ H x,p x, p T M. Let x M be fixed. We call cotangent nonfocal domain of x the open subset of T x M defined as N F x := { p T x M J x,tp V x,tp = {0} t 0, ] }.. It is the set of covectors p T x M \ {0} such that the corresponding geodesic γ : [0, ] M defined as γt := π φ H t x, p has no conjugate points on the interval 0, ]. By construction, for every p N F x, the Lagrangian subspace J x,p is transversal to the vertical subspace V x,p in T x,p T M. Hence, there is a linear operator Kx, p : T x M T x M such that J x,p = { h, Kx, ph T x M T x M h T x M }. We are now ready to define our extended Ma-Trudinger-Wang tensor. Definition.. We call extended Ma-Trudinger-Wang tensor abbreviated MT W tensor, the mixed tensor field given by d Ŝx, p ξ, η := 3 ds Kx, p + sηξ, ξ s=0 ξ T x M, η Tx M, for every x, p N F x. 4

The above definition extends the definition of the Ma-Trudinger-Wang tensor, which was first introduced in [] and extensively studied in [,, 3, 7, 8, 9, 0, 3, 4, 9]. Indeed, let x y M be such that y / cutx, and take ξ T x M, η T x M. There is a unique p T x M \ {0} such that the curve γ : [0, ] M defined by γt := π φ H t x, p t [0, ] is a minimizing geodesic between x and y. Since such a curve contains no conjugate points, the covector p necessarily belongs to N F x. Let v T x M resp. η T x M be the unique vector such that p, w = g x v, w resp. η, w = g x η, w for any w T x M. By the definition of K,, if we define cx, y = d x, y/, then for s small one has Kx, p + sηξ, ξ = d dt c exp x tξ, exp x v + s η t=0.. Thus, differentiating both sides yields Ŝx, p ξ, η = 3 d d ds dt c exp x tξ, exp x v + s η = Sx, y ξ, η,.3 s=t=0 where S denotes the classical Ma-Trudinger-Wang tensor see for instance [8, Chapter ]. Observe that, although the MT W tensor is not defined at x, 0, the above formula shows that Ŝx, 0 is well-defined by continuity, and it is a smooth function near x, 0. Denote by I x the cotangent injectivity domain of x defined as I x := { p T x M π φ H t x, p / cutx t [0, ] },.4 and observe that I x N F x {0}. The discussion above shows that, up to identify p, η Tx M with v, η T x M using the Riemannian metric g x, the MT W tensor Ŝ and the classical MTW tensor S coincide on the injectivity domains. For this reason, our tensor can be seen as an extension of the MTW tensor beyond the injectivity domain until the boundary of the nonfocal domain. It is worth mentioning that in Definition. it is not necessary to work with the horizontal spaces which are given by the Riemannian connection associated with the metric g. Let ϕ : U M ϕu R n be a local chart in M and x,..., x n, p,, p n be a symplectic set of local coordinates on T U. As we already said before, for every θ = x, p T U = R n R n, the horizontal space H θ canonically associated with g is defined as the set of pairs χ = h, v R n R n such that v = Γ0, where t ɛ, ɛ γt, Γt is the smooth curve satisfying γ0, Γ0 = x, p, γ0 = h, and Γt is obtained by parallel transport of the covector Γ0 = p along the curve γ. Writing the ordinary differential equations of parallel transport in local coordinates yields that there exists a bilinear mapping L x : R n R n R n The equality is a simple consequence of the following fact: for each x t := exp x tξ, denote by p t, q t the covectors in Tx t M and Ty M satisfying φ H y, qt = xt, pt with y := exp x v + s η, p 0 = p + sη, and p t x t = q t y = dx t, y. Then so that differentiating again at t = 0 we obtain d dτ cx t+τ, y τ=0 = d xcx t, y, ẋ t = p t, ẋ t, d dt cxt, y t=0 = ṗ 0, ξ, where ṗ 0 denotes the covariant derivate of p t along the curve x t, and we used that the covariant derivative of ẋ t along x t is zero. Hence, since φ H xt, pt = y, qt, by the definition of Kx, p + sη we easily get ṗ 0 = Kx, p + sηξ. 5

such that the horizontal space H θ in local coordinates is given by H θ = { h, L x h, p h R n}. Denote by H θ the horizontal space given by the base space in the symplectic set of local coordinates, that is H θ := R n {0}. Since H θ is a Lagrangian subspace of T θ T M = R n R n, there is a linear operator Kx, p : R n R n such that Then, for every h R n we have J x,p = { h, Kx, ph R n R n h R n}. Kx, ph = L x h, p + Kx, ph. Since L x is linear in the p variable, this shows that for every x, p N F x d Ŝx, p ξ, η = 3 ds Kx, p + sηξ, ξ s=0 ξ R n, η R n this argument is the symplectic analogous of [8, Remark.3]. It has also to be noticed that the MT W tensor may be locally associated with an extended cost function through formulas like.-.3. More precisely, fix θ = x, p T M with p N F x. Since the point y := π φ H x, p is not conjugated with x, thanks to the Inverse Function Theorem there exist an open neighborhood V of x, p in T M, and an open neighborhood W of x, y in M M, such that the function Ψ θ : V T M W M M x, p x, π φ H x, p, is a smooth diffeomorphism from V to W. The extended cost function ĉ θ : W R which can be locally associated with the MT W tensor at θ = x, p is uniquely defined by ĉ θ x, y := Ψ θ x, y x, y W..5 x For the same reasons as before, we have for any ξ T x M and η T x M, which yields Kx, p + sηξ, ξ = d dt ĉθ expx tξ, π φ H x, p + sη t=0 Ŝx, p ξ, η = 3 d d ds dt ĉθ expx tξ, π φ H x, p + sη..7 s=t=0 Moreover, if instead we work in a symplectic set of local coordinates x,..., x n, p,, p n on T U, then for any θ = x, p T U with p N F x, and any ξ R n, η R n, there holds.6 Kx, p + sηξ, ξ = x ĉ θ x, π φ H x, p + sη.8 ξ Set, for t small, x t := x + tξ, and denote by p t, q t the covectors in T x t M = R n and T y M satisfying φ H y, qt = xt, pt with y := φh x, p + sη, p 0 = p + sη, and p t x t = q t y = ĉ θ x t, y. Then d dτ ĉθx t+τ, y τ=0 = d xĉ θ x t, y, ẋ t = p t, ẋ t = p t, ξ, so that differentiating again at t = 0 we obtain x ξ ĉ θ`x, φ H x, p + sη = d dt ĉθx t, y t=0 = ṗ 0, ξ, where ṗ 0 denotes the classical derivative of p t in the direction η in local coordinates. Since φ H xt, pt = y, qt, by the definition of Kx, p + sη we obtain ṗ 0 = Kx, p + sηξ. 6

and Ŝx, p ξ, η = 3 p η x ξ ĉ θ x, π φ H x, p,.9 where resp. x p denotes the classical second derivative in coordinates in the x variable ξ η in the direction ξ resp. in the p variable in the direction η. The following definition extends the definition of MTWK, C introduced in [0]: Definition.. Let K, C 0. We say that M, g satisfies MT WK, C if, for any x, p T M with p N F x, Ŝx, p ξ, η K ξ x η x C η, ξ ξ x η x ξ T x M, η T x M. In [8] it was observed that, if ξ and η are orthogonal unit vectors in T x M and η := g x η, Tx M, then Ŝx, 0 ξ, η = Sx, x ξ, η coincides with the sectional curvature at x along the plane generated by ξ and η. More precisely one can prove that, for all ξ T x M, η Tx M, Ŝx, 0 ξ, η = σ x P ξ x η x η, ξ, where σ x P denotes the sectional curvature at x along the plane generated by ξ and η. In particular, if M, g satisfies MT WK, C, then its sectional curvature is bounded from below by K. Therefore, if M, g satisfies MT WK, C with K > 0, by Bonnet-Myers Theorem its diameter is bounded, so that M is compact, and in addition the set x M x, N F x T M is compact. Furthermore, by the above formula we also see that Ŝx, 0 ξ, η = 0 whenever g x ξ, is parallel to η since in this case η, ξ = ξ x η x. Therefore, if M, g satisfies MT WK, C, then C K. The round sphere S n, g can and its quotients satisfy MT WK 0, K 0 for some K 0 > 0 see Appendix. Moreover, since the MT W tensor depends only on the Hamiltonian geodesic flow, if a given Riemannian manifold M, g satisfies MT WK, C, then its quotients as well as its coverings satisfy MT WK, C. The aim of this paper is to show that the MT WK, C condition, together with the strict convexity of the cotangent nonfocal domains, allows to prove the strict convexity of all cotangent injectivity domains and T CP. As a corollary, we will obtain Theorems. and.3. 3 Extended regularity, convexity of injectivity domains, and T CP 3. Our strategy is to show that an extended version of the uniform regularity property introduced by Loeper and Villani in [0] is sufficient to obtain T CP. Our definition of extended regularity is in some sense stronger than the one given by Loeper and Villani, as it takes into account what happens until the boundary of the cotangent nonfocal domain, and besides requires its convexity. On the other hand we do not require the uniform convexity of the injectivity domains, which is an assumption much more complicated to check than the convexity of the nonfocal domains see [7]. Moreover our definition has the advantage that it allows to deduce the convexity of all injectivity domains as an immediate corollary see Theorem 3.4. We notice that, since our definition of extended regularity involves the geodesic Hamiltonian flow as we want to be able to cross the cut locus, it cannot be expressed only in term of the cost function c = d /. For 7

this reason we will say what means for a Riemannian manifold M, g to be stricly regular, while in [0] the authors defined what means d / being uniformly regular. Definition 3. Extended regularity. We say that M, g is regular resp. strictly regular if there are ρ, κ > 0, and a function f Cc [0, ] with f 0 and {f > 0} = /4, 3/4, such that a for every x M, N F x is convex resp. strictly convex, b for every x M, let p t 0 t be a C curve drawn in N F x {0} with p 0 p I x, and let y t := π φ H x, p t. Then there exists λ > 0 such that the following holds: let x M. If p t x ρ d x, x ẏ t y t t 0,, 3. then, for any t [0, ], dx, y t p t x min dx, y 0 d x, y 0, dx, y d x, y +λft d x, x. 3. Moreover, given a family of curves p t 0 t as above such that p t 0 t vary inside a compact subset of N F x {0} and p p 0 x is uniformly bounded away from 0, the constant λ > 0 can be chosen to be the same for all curves. One of the motivations of the above definition is that, roughly speaking, the extended regularity is an integral manifestation of the extended MT W condition: Theorem 3.. Assume that there exist K, C > 0 such that i for every x M, N F x is convex, ii M, g satisfies MT WK, C. Then M, g is regular. The above theorem is indeed a simple consequence of the following lemma, combined with an approximation argument: Lemma 3.3. Let M, g be a Riemannian manifold satisfying MT WK, C for some K, C > 0, and assume that N F x is convex for all x M. Let x M, and let p t 0 t be a C curve drawn in N F x {0} Tx M, with p 0 p I x. For any t 0,, set y t := π φ H x, p t, and suppose that p t x K 6 d x, x ẏ t y t t 0,. 3.3 Assume further that x cuty t only for a finite set of times 0 < t <... < t N <. Finally, let f Cc [0, ] be as in Definition 3.. Then, for any t [0, ], dx, y t p t x min dx, y 0 d x, y 0, dx, y d x, y + λftd x, x, 3.4 where { λ := min K C diamm f K inf ẏt yt, /4 t 3/4 f inf ẏt } y /4 t 3/4 t. Note that, since M has sectional curvature bounded from below by K > 0 thanks to MT WK, C, then diamm is finite. 8

Proof of Theorem 3.. Let x, x, p t 0 t and y t 0 t be as in Definition 3.. Up to slightly reduce ρ, by density and the approximation lemma proved in [4, Section ] we may assume that y 0, y / cutx and that y t meets cutx only at finitely many times t,..., t N, all the other conditions in Definition 3. being unchanged. Since p t N F x {0} for all t [0, ], we have dist p t, N F x {0} > 0 t [0, ]. 3.5 Moreover, as ẏ t yt C 0 ṗ t x for some constant C 0 > 0 depending only on M 3, thanks to 3.3 we deduce that ṗ t 0 for all t 0,. Indeed, if not, by 3. we would get d dt ṗ t x ρ d x, xc 0 ṗ t x, and Gronwall Lemma would imply ṗ t 0, which contradicts p 0 p. Hence, combining 3.5 with the fact that ṗ t 0 for t 0,, we obtain ẏ t yt > 0 t 0,, 3.6 which by continuity implies inf /4 t 3/4 ẏt yt > 0. Moreover, if we take a family of curves p t 0 t inside a compact subset of N F x {0}, with p p 0 x is uniformly bounded away from 0, it is easy to see by compactness that there exists a constant δ 0 > 0 such that inf /4 t 3/4 ẏt yt δ0 for all curves p t 0 t. Then the theorem follows easily from Lemma 3.3. The proof of Lemma 3.3 is strongly inspired by the proof of [0, Theorem 3.], which uses a variant of the tecniques introduced in [7, Section 4]. However the main difference with respect to the preceding proofs is in the fact that, since our curve t p t can exit from I x, we have to change carefully the function to which one applies the MT WK, C condition. The advantage of our choice of such a function is that it allows to deduce a stronger result, where we bound from below dx, y t p t x instead of dx, y t d x, y t. This fact is crucial to deduce the strict convexity of all cotangent injectivity domains. Proof of Lemma 3.3. First of all, without loss of generality we can assume that x x. Indeed, if x = x we simply write 3.4 for a sequence x k k N, with x k x, and then let x k x. Thus, we suppose d x, x > 0. Since p t N F x {0} for all t [0, ], as in the proof of Theorem 3. we have dist p t, N F x {0} > 0 and ẏ t yt > 0 t [0, ]. Set t 0 = 0, t N =, and define h : [0, ] R by ht := cx, y t + p t x + δft t [0, ], where cx, y = d x, y/ and δ := λd x, x. Let us first show that h cannot have a maximum point on an interval of the form t j, t j+. For every t t j, t j+, since y t / cutx, h is a smooth function of t. We fix t t j, t j+, and we compute ḣt and ḧt. As in Paragraph., define the extended cost function ĉ := ĉ x,pt in an open set W of M M containing x, y t as ĉz, y := exp z y z = exp y z y z, y W, 3 Actually, since MT WK, C implies that the sectional curvature of M is bounded from below by K > 0 see the discussion after Definition., C 0 = would work. 9

where exp z resp. exp y denotes a local inverse for exp z resp. exp y near x resp. y t. Hence, for s close to t, we can write hs = cx, y s + ĉ x, y s + δfs. Moreover the identity p s = d x ĉ x, y s holds. Take a local chart in an open set U M contiaining y t and consider the associated symplectic set of local coordinates y,, y n, q,, q n in T U. Then, as in [0, Proof of Theorem 3.] we can easily compute ẏ t and ÿ t at time t: using Einstein convention of summation over repeated indices, we get ẏ i = ĉ xiyj ṗ j, ÿ i = ĉ xiykĉ xk y l y j ẏ l ẏ j ĉ xiyj p j, everything being evaluated at x, y t and at time t. Here we used the notation ĉ xiyj for the inverse of ĉ xiy j = d xy ĉ ij, which denotes the second partial derivatives of ĉ in the x i and y j variables. Let us define q t := d y cx, y t, q t := d y ĉ x, y t. Then we easily get ḣt = ˆq t, ẏ t + δ ft, [cyiy ḧt = j x, y t ĉ yiy j x, y t ] + ĉ xk y iy j x, y t ĉ x ly k x, y t ˆq t ẏ i ẏ j l ĉ xiyj x, y t ˆq t where ˆq t := q t q t. Now, using.8, we obtain i pt j + δ ft, ḧt = Ky t, q t ẏ t, ẏ t Ky t, q t ẏ t, ẏ t d ds Ky t, q t + sˆq t ẏ t, ẏ t s=0 where v t := d xy ĉ x, y t ˆqt. Recalling.9, that is 3.7 can be written as ḧt = + v t, p t + δ ft, 3.7 d ds Ky t, q t + sˆq t ẏ t, ẏ t = d ds Ky t, q t + sˆq t ẏ t, ẏ t, = 3 0 d s 0 ds Ky t, q t + sˆq t ẏ t, ẏ t ds + v t, p t + δ ft s Ŝy t, q t + sˆq t ẏ t, ˆq t ds + vt, p t + δ ft. By MT WK, C and sds = /, we get 0 ḧt K ˆq t yt ẏ t yt C ˆq t, ẏ t ˆq t yt ẏ t yt + v t, p t + δ 3 ft. We now claim that the function h cannot have any maximum on t j, t j+. Indeed, if ḣt = 0 for some t t j, t j+, we have which implies ˆq t, ẏ t δ ft. Thus ḧt 3 0 = ḣt = ˆq t, ẏ t + δ ft, K ˆq t yt ẏ t yt Cδ ft ˆq t yt ẏ t yt v t, p t δ ft, 0

and so by 3.3 ḧt 3 K ˆq t yt ẏ t yt Cδ ft ˆq t yt ẏ t yt K 6 v t x d x, x ẏ t y t δ ft. Since MT WK, C implies that the sectional curvature of M is bounded below by K > 0 see the discussion after Definition., the exponential mapping exp yt is -Lipschitz, which implies that the norm of the operator d xy ĉ x, y t : T yt M T x M is bounded by. Hence, we have v t x ˆq t yt, d x, x ˆq t yt, 3.8 which give ḧt K ˆq t yt ẏ t yt Cδ ft 3 ˆq t yt ẏ t yt K 6 ˆq t y t ẏ t y t δ ft K 6 ˆq t yt ẏ t yt C 3 δ ft ˆq t yt ẏ t yt δ ft. If t / [/4, 3/4] then ft = ft = 0, which combined with 3.8 implies ḧt K 6 ˆq t y t ẏ t y t K 6 d x, x ẏ t y t. On the other hand, if t [/4, 3/4], recalling that δ = λd x, x and the definition of λ we obtain C 3 δ ft = C 3 λd x, x ft K d x, x ẏ t yt K ˆq t yt ẏ t yt, which using again 3.8 and the definition of λ yields ḧt K ˆq t y t ẏ t y t δ ft K d x, x ẏ t y t δ ft K 4 d x, x ẏ t y t. 3.9 In any case, thanks to 3.6, we have ḧt > 0, which shows that h cannot have a maximum on any interval t j, t j+. Thus, as h is continuous on [0, ], it has to achieve its maximum at one of the times t j 0 j N. The goal is to show that necessarily j = 0 or j = N. Indeed, let j {,..., N }. We first note that, since t cx, y t = d x, y t / is semiconcave and t p t x is smooth, ht is semiconvex. If ḣ is continuous at t j and ḣt j 0, clearly t j cannot be a maximum of h. The same is true if ḣ is discontinuous at t j, because by semiconvexity necessarily ḣt+ j > ḣt j. Finally, if ḣ is continuous at t j and ḣt j = 0, the same computations as before show that ḧt is strictly positive when t is close to but different from t j, which implies that h cannot have a maximum at t j. The only possibility left out for h is to achieve its maximum at t 0 = 0 or t N =, and we obtain 3.. 3. One main feature of our definition of regularity is that it immediately implies the convexity of all injectivity domains: Theorem 3.4. Let M, g be a regular Riemannian manifold. Then I x is convex for all x M. Proof. It is sufficient to show that I x is convex for all x M. We fix x M, and choose x = x in the definition of regularity. Then, considering p t := tp + tp 0 with p 0 p I x N F x, we get d x, y t p t x t [0, ]. This gives p t I x, that is I x is convex.

Remark 3.5. One can actually prove that, if M, g is strictly regular, then strict convexity of all injectivity domains holds. Indeed, let us assume by contradiction that there are p 0 p N F x I x such that that tp + tp 0 I x for all t 0,. Consider x = x in the proof of Lemma 3.3, and using the same notation we perturb the segment into a curve p t 0 t such that p t N F x \ I x for all t /4, 3/4, and p t yt K 6 q t q t yt ẏ t y t, where q t := d y c x, y t q t for t /4, 3/4 this can always be done as the segment tp + tp 0 lies at positive distance from N F x for t /4, 3/4. The function ht = c x, y t + p t x/ is identically zero on [0, /4] [3/4, ], and it is smooth on /4, 3/4. Since now δ = 0, by the first inequality in 3.9 we get ḧt K q t q t y t ẏ t y t > 0 t /4, 3/4 whenever ḣt = 0. This fact implies that h cannot attain a maximum on /4, 3/4. Hence, for any t /4, 3/4, a contradiction. 3.3 0 = d x, y t d x, y t < p t x d x, y t max hs = 0, s [0,] Here is another main motivation for our definition of extended regularity: Theorem 3.6. Any Riemannian manifold M, g which is stricly regular satisfies T CP. The proof of this theorem closely follows the proof of [0, Theorem 5.]. Proof. Condition i in the definition of T CP insures that µ gives no mass to set with σ-finite n -dimensional Hausdorff measure. Thanks to McCann s Theorem read in the Hamiltonian formalism, there exists a unique optimal transport map between µ and ν, which is given by T x = π φ H x, d x ψ, where ψ is a semiconvex function. Moreover d x ψ I x N F x at all point of differentiability of ψ. Since N F x is convex for all x, the subdifferential of ψ satisfies ψx N F x for all x M. To prove that ψ is C, we need to show that ψx is everywhere a singleton. The proof is by contradiction. Assume that there is x M such that p 0 p ψ x. Let y 0 = exp x p 0, y = exp x p. Thus y i c ψ x, i.e. In particular ψ x + d x, y i = min x M { ψx + } d x, y i, i = 0,. d x, y i d x, y i ψ x ψx, x M, i = 0,. 3.0 Fix η 0 > 0 small the smallness to be chosen later. For 0,, we define D N F x as follows: D consists of the set of points p T x M such that there exists a path p t 0 t N F x from p 0 to p such that, if we set y t := π φ H x, p t, we have p t = 0 for t [/4, 3/4], p t yt η 0 ẏ t y t for t [/4, 3/4], and p = p t for some t [/4, 3/4]. By the strict convexity of N F x, if η 0 is sufficiently small then D lies a positive distance σ away from N F x for all 0,. Thus all paths p t 0 t used in the definition of D satisfy ẏ t yt c p 0 p x t [/4, 3/4],

with c independent of 0,. Moreover condition 3. is satisfied if η 0 ρ and d x, x. By simple geometric consideration, we see that D contains a parallelepiped E centered at p 0 + p / with one side of length p 0 p x, and the other sides of length p 0 p x, such that all points y in such parallelepiped can be written as y t for some t [/3, /3], with y t as in the definition of D. Therefore L n E c n, with L n denoting the Lebesgue measure on T x M. Since E lies a positive distance from N F x, we obtain vol Y L n E c n, Y := π φ H x, E. We then apply Theorem 3. to the paths p t 0 t used in the definition of D to obtain that, for any y Y and x M \ B x, dx, y d x, y min dx, y 0 d x, y 0, dx, y d x, y +λ inf ft d x, x, t [/3,/3] with inf t [/3,/3] ft > 0. Combining this inequality with 3.0, we conclude that for any y Y, y c ψx x M \ B x. This implies that all the mass brought into Y by the optimal map comes from B x, and so µb x νy. We now remark that by condition ii in the definition of T CP and a standard covering argument, there exists a constant c > 0 such that νa c vol A for all Borel sets A M. Thus, as µb x o n and νy c vol Y c n, we obtain a contradiction as 0. 4 Stability of MT W near the sphere In this section, we show that any C 4 -deformation of the standard -sphere satisfies MT WK, C for some K, C > 0. Let M, g be a smooth, compact and positively curved surface. It is easy to show that, for every x M, the set N F x T x M is a compact set with smooth boundary see [7]. In fact it can even be shown that, if M = S and g is C 4 -close to the round metric g can, then all the N F x are uniformly convex. Thus Theorems. and.3 are a consequence of Theorems 3., 3.6, 3.4 and Remark 3.5, together with the following result: Theorem 4.. There exist K, C > 0 such that any C 4 -deformation of S, g can satisfies MT WK, C. Proof. For 0, let g be a smooth metric on S such that g g can C 4 so that g 0 = g can. We see S as the sphere centered at the origin with radius one in R 3, so that we can identify covectors with vectors. Let x S ; we observe that for g 0 the set N F x {0} corresponds to the open ball centered at x with radius π intersected with the hyperplan tangent to S at x, while for g the nonfocal domain N F x {0} is a C -perturbation of the ball. Our aim is to show that there exist K, C > 0 such that, if > 0 is sufficiently small, then for every x S and every p N F x one has Ŝx, p ξ, η Kg xξ, ξg xη, η C ξ, η g x ξ, ξ / g xη, η / for all ξ T x S n, η T x S n. Since the property holds true on S, g can with K = C = K 0 for some K 0 > 0 see Appendix, the above inequality holds by continuity on S, g with 3

K = K 0 / and C = K 0 when p is uniformly away from the boundary of N F x {0}, and is sufficiently small. Thus all we have to prove is that the above inequality remains true when p is close to N F x {0}. Moreover, by the homogeneity of S, g can, it will suffice to prove the estimate only for a fixed point x S and along a fixed geodesic t π φ H t x, p. Consider the stereographic projection of the sphere S R 3 from the north pole N = 0, 0, onto the space R R {0} R 3. The projection of some point x = x, x, x 3 S is given by σx = x x 3, x x 3 The function σ is a smooth diffeomorphism from S \ {N} onto R, whose inverse is σ y y = + y, y + y, y + y y = y, y R, where denotes the Euclidean norm on R. The pushforward of the metric g under σ induces a metric on R, that we still denote by g, and which for = 0 is given by g 0 yv, v =. 4 + y v y, v R. Note that since we work in R, we can identify covectors with vectors. We denote by H y, p the Hamiltonian canonically associated to g, which for = 0 is given by We observe that H H 0 C 4 H 0 y, p = + y p y, p R. 8. The Hamiltonian system associated to H is { ẏ = H p y, p ṗ = H y y, p, and the linearized Hamiltonian system along a given solution y t, p t is { ḣ = H p y, p q + H y p y, p h q = H y p y, p q H y y, p h We note that h is a Jacobi vector field along the geodesic t y t. Set Y =, 0 R, and consider the geodesic θα starting from Y with velocity of norm and making angle α computed with respect to g with the line {x = 0}. For = 0 this geodesic is given by θαt 0 = cost π cosα sint π, sinα sint π cosα sint π, 0 and it is a minimizing geodesic between Y and, 0. Since the first conjugate time for θ α is t = π for all α, and we are perturbing the metric in the C 4 topology, there exists a smooth function α t cα such that t cα is the first conjugate time of θ α, and t cα π C. Fix t α 0, t cα, and set V α := t α θ α0. As we notice in Paragraph., in order to compute the MT W tensor at Y, V α, we can use the horizontal space given by any choice of a symplectic set of local coordinates. Therefore, we can work with the standard splitting R 4 = R R and take as horizontal vertical spaces, H Y,V α = R {0} R 4 and V Y,V α = {0} R R 4. 4

In the sequel, we shall denote by K Y, V α which corresponds to the operator K Y, V α of Paragraph. the matrix such that J Y,V α = { h, K Y, V α h R R h R }. By an easy rescaling, it is not difficult to see that K Y, V α is given by K Y, V α = t αs α t α where S α t α is the symmetric matrix such that any solution of the linearized Hamiltonian system along θ α starting from h, S α th satisfies h t α = 0. Let us compute S α t α. Let Et, α := θ αt and Et, α be a basis of parallel vector fields along θα such that g E0, α, E0, α = 0. For = 0 they are given by Et, 0 α := θ cosα sint π α t = cosα sint π, sinα cost π cosα sint π, 0, sinα cost π Et, 0 α := cosα sint π, cosα sint π cosα sint π, 0, Let h α, q α be a solution of the linearized Hamiltonian system along θ α such that h α t α = 0 for some t α 0, t cα. Since E t, α, E t, α form a basis of parallel vector fields along θ α, there are two smooth functions u α,t, u α,t such that h αt = u α,te t, α + u α,te t, α. If we denote by u αt R the vector u α,t, u α,t, and by A αt the matrix having E t, α and E t, α as column vectors, we can write h αt = A αtu αt. As h αt is a Jacobi vector field along θ α we have ḧ α + R h α, θ α θ α = 0, where R denotes the Riemann tensor, and using the symmetries of R we get R h α, θ α θ α = R h α, E E = R u α,e + u α,e, E E = R u α,e, E E = u α, R E, E E, E E + u α, R E, E E, E E = u α, R E, E E, E E. This gives { ü α, t, α = 0 ü α,t, α = g R E, E E, E u α, t, α, so that { u α, t, α = λ + λ t ü α,t, α = g R E, E E, E u α, t, α = a t, αu α,t, α, where t, α a t, α is close to in C -topology. Hence we can write u αt = U t, αu α0 + U t, α u α0, with U t, α = 0 0 f t, α, U t 0 t, α = 0 f t, α, 5

with f t, α and f t, α are close to cost and sint in the C -norm, respectively. Recalling that h α t α = 0, we have 0 = U t α, αu α0 + U t α, α u α0 = u α0 = [ U t α, α ] U t α, αu α0. and as u α0 = A α0 h α 0 we get ḣ α0 = A α0u α0 + A α0 u α0 = [ A α0 A α0 [ U t α, α ] ] U t α, α u α0 = [ A α0 A α0 [ U t α, α ] U t α, α] A α 0 h α 0. Hence from the linearized Hamiltonian system we finally obtain with where Defining qα0 = [ H p Y, θ ] ḣ [ α0 H α 0 p Y, θ ] α0 H y p Y, θ α0h α0 = Sαth α0, we can write [ Sαt = Cα H p Y, θ ] A α0 α 0 [ U t α, α ] U t α, α A α0, [ Cα H = p Y, θ ] [ α0 A α0 A α0 H y p Y, θ ] α0. N αt := tf t, α [ U t, α ] U t, α = f t, α 0 0 tf t, α Sαt = Cα [ H tf t, α p Y, θ ] A α0 α 0Nαt A α0. We observe that Nαt is smooth up to t = t cα. As a matter of fact we remark that, for = 0, [ ] Cα 0 cosα 0 cosα sinα = I R α R α =, sinα 0 sinα cosα and A 0 α0 = R α, where we used the notation cosα sinα R α = sinα cosα Let us now focus on the matrix [ H p Y, θ ] A α0 α 0Nαt A α0. Denoting by G the matrix associated to the metric g at the point Y, we have [ H p Y, θ α0 ] = G. Moreover G A α0n αt A α0. = G /[ G / A α0 ] N αt A α0 = G /[ G / A α0 ] Nαt [ G / A α0 ] G /. Recalling that A α0 = E0, α, E0, α, we have G / A α0 = G / E0, α, G / E0, α, 6

and since = g E 0, α, E 0, α = g E 0, α, E 0, α, 0 = g E 0, α, E 0, α, we immediately get that G / A α0 is an orthogonal matrix for all α. Thus, there exists ᾱ and α such that G / A α0 = G / A cosα 00 sinα sinα cosα =: RᾱR. α that is, ᾱ is the angle between, 0 = θ 0 and G /, 0 = G / θ 0, and we obtain tf t, α G A α0nαt A α0 [ = G / tf t, α G / A α0nαt G / A α0 ] G / [ = G / Rᾱ tf t, α R α NαtR α ] R ᾱ G /. A simple computations gives that R α N t, α R α is equal to the matrix cos α f t, α + t sin α f t, α cosα sinα f t, α tf t, α cosα sinα f t, α tf t, α sin α f t, α + t cos α f t, α so that tf t, Rα N t, α R α = α t + f t, α cos α cosα sinα cosα sinα sin α sin α f t, α cosα sinα f t, α cosα sinα f t, α cos α f t, α,. We now define T s as the g -norm at Y of the vector v, 0 + sη, and α s is the angle computed with respect to g between v, 0 and v, 0 + sη. In the sequel we denote by ˆf resp. ˆf the function s f T s, α s resp. s f T s, α s, and by ˆf, ˆf resp. ˆf, ˆf its first and second derivative. We want to compute the second derivative of K s := K ɛ Y, v, 0 + sη for T 0 close to t cα π, so that / ˆf 0 / sint cα will be dominant with respect to all other terms. Thanks to the computations made above, we have d { } ds {K s} s=0 = d ds T scα s + G / Rᾱ s=0 [ M 0 + ˆf 0M + ˆf 0 M + ˆf 03 M 3 ] R ᾱ G / with and and M i = M i Mi Mi M i 3 i = 0,,, 3, M 0 = α 0, M 0 = α 0, M 0 3 = α 0, M = T 0 ˆf 0 α 0, M = ˆf 0 T 0 α 0 T 0 ˆf 0 α 0 T 0 ˆf 0 α 0, 7

and and M 3 = ˆf 0 T 0 ˆf 0 T 0 T 0 ˆf 0 + T 0 ˆf 0 α 0, M = 0, M = T 0 ˆf 0 ˆf 0 α 0, M 3 = T 0 ˆf 0 ˆf 0 + ˆf 0 ˆf 0 T 0 + T 0 ˆf 0 ˆf 0, M 3 = M 3 = 0, M 3 3 = T 0 ˆf 0 ˆf 0. We now observe that α s is given by the angle between the two vectors [ ] G / v and G / v + sη. 0 0 v v Therefore, if we define = R 0 ᾱ G / and η 0 = R ᾱ G / η, we get sη α s = arctan v + sη which implies α 0 = 0, α 0 = η v, α 0 = η η v. Regarding T s, we have T s = G / v + sη = sη R ᾱ G / v + sη sη = v + sη = v + sη + s η, sη hence T 0 = v, T 0 = η, T 0 = η v. Moreover v v v, η η η. Thus we finally obtain M0 = η ηη v ηη η, M = M = ˆf 0 v η ˆf 0η ˆf 0η ˆf 0 v η ˆf 0η v ˆf 0 0 ˆf 0 ˆf 0η ˆf 0 ˆf 0η v ˆf 0 ˆf 0 + ˆf 0 ˆf 0η + v ˆf 0 ˆf 0 0 0 M3 = 0 v ˆf. 0 ˆf 0,, We note that for = 0 d { } ds T 0 scα 0 0 s s=0 = T 0 0 T 0 0 α 0 0 T 0 0 α 0 0 T 0 0 α 0 0 T 0 0 α 0 0 T 0 0 α 0 0 T 0 0 + T 0 0 α 0 0 = 0, 8

which implies d { } ds T scα s s=0 Therefore, defining ξ = R ᾱ G / ξ, we end up with ξ, d ds {K s} s=0 ξ = ξ, M 0 ξ + ˆf 0 ξ, M ξ η. + ˆf 0 ξ, M ξ + ˆf 03 ξ, M 3 ξ + O η ξ. We now observe that, as ˆf 0 cosv, we have ˆf 0 0 for v π/3, and so in this case ˆf M 03 3 3ξ + ˆf M 0 ξξ + ˆf 0M ξ Therefore for v π/3 = ˆf 0 [ v ˆf 0 ˆf 0 ˆf 0ξ + v η ξ] 0 ξ, d ds {K s} s=0 ξ ξ, M 0 ξ Now, easy computations give for i =, + ˆf 0 [ M ξ ξ + M 3ξ ] + ˆf 0 M 3ξ + O η ξ. ˆf i 0 = f i t v, 0η v f i α v, 0η, [ ˆf i f ] [ i 0 = t v, 0 η fi + v α v, 0 + f ] i v t v, 0 η [ fi + v α v, 0 f ] i v α t v, 0 ηη. Let us now observe the following: since as functions of v ˆf 0 and ˆf 0 are close to cosv and sinv in the C -norm respectively, if we define l := t c0 v we easily get ˆf 0 + + l, 0 ˆf 0 + l, ˆf 0 + l η, ˆf 0 + η + l η, ˆf 0 η + l η, ˆf 0 + v η + l η. From these estimates it is easy to see that M 3 η η + l η = M 3 C + l [ η + η ], and M + l η η. 9

Since ˆf + as 0 l 0 i.e. v t cα, we obtain M0 3 + ˆf 0M 3 + C + l ˆf M 0 3 η ˆf, 0 M 0 + ˆf 0M C + l ˆf 0 η η, from now on, C is a positive constant, independent of for > 0 sufficiently small, which may change from line to line. Hence, combining all together, ξ, d ds {K s} s=0 ξ + + v η ξ C + l ˆf 0 η η ξ ξ C + l ˆf 0 [ η + η ] ξ + O η ξ v η ξ C + l ˆf 0 η η ξ ξ C + l η ˆf ξ + O η ξ 0 C + l C + l π η ξ + C + l ˆf 0 η ξ + O η ξ π [ η ξ + η ξ ] + O η ξ. From this formula, since η η C η and ξ ξ C ξ, we finally get 3 d ξ, ds {K s} s=0 ξ 3 C + l [ η π ξ + η ξ ] + O η ξ 3 C + l [ η π ξ + ηξ ] + O η ξ 3 C + l [ η π ξ η, ξ ] + O η ξ 3 C + l π η ξ 3 + C + l π η, ξ. Fix δ > 0. From the above estimate we deduce that, if δπ /3C and l δπ /3C, then MT W3/π δ, 3/π + δ holds for all v N F x {0} such that dist v, N F x {0} δ/c. Since as we already said MT WK, C trivially holds if v is uniformly away from N F x {0} for > 0 small enough, the result follows. 5 Final comments 5. Our approach applies to more general situations than the one we chose to present. In particular, we do not necessarily need the strict convexity of the cotangent nonfocal domains: let M, g be a compact Riemannian manifold, and define M > 0 and m 0, + ] by { } { } M := max p x p I x, x M and m := min p x p / N F x, x M. Assume that the two following conditions are satisfied: 0

i M < m, ii there is K > 0 such that for every x M, ξ T x M, η T x M, η, ξ = 0 = Ŝx, p ξ, η K ξ x η x for any p T x M satisfying p x 0, M ]. Following the proof of [0, Lemma.3], it is not difficult to show that under these assumptions there exists C > 0 such that, for every x M and every p T x M satisfying p x 0, M ], Ŝx, p ξ, η K ξ x η x C η, ξ ξ x η x ξ T x M, η T x M. Then, one can easily check that both the proof of Lemma 3.3 and the proof of Theorem 3.6 still work, and so M, g satisfies T CP, and all its injectivity domains are strictly convex. In particular, this allows to recover in a simple way the result proved independently in [, 0] that any C 4 -deformation of a quotient of the standard sphere S n say for instance RP n satisfies T CP. Indeed ii follows from the fact that our extended MTW condition is stable far from the boundary of N F x {0} while the classical MTW condition is a priori stable only far from the boundary of I x. 5. It can be shown [7] that the cotangent injectivity domains of any smooth complete Riemannian manifold have locally semiconcave boundaries. In fact, if g is a smooth Riemannian metric which is C 4 -close to the round metric on the sphere S n, then for all x S n the sets N F x are uniformly convex [7] while it is not known whether the sets I x are convex or not. As a consequence, if S n, g is a C 4 -deformation of the standard sphere which satisfies MT WK, C for some K, C > 0, then it satisfies T CP, and all its injectivity domains are strictly convex. 5.3 In [, 0] the authors can improve T CP to higher regularity thanks to the stay-away property of the optimal transport map T. More precisely, in [0] the authors assume that the cut locus is nonfocal which is for example that case if one considers C 4 -deformation of a quotient of the standard sphere S n, and combining this hypothesis with T CP one gets the existence of a constant σ > 0 such that dt x, cutx σ for all x M. On the other hand, in [] the authors show that if M, g is C 4 -deformation of S n, g can, and one imposes some boundedness constraint on the measures µ and ν the constraint depending on the size of the perturbation, then the stay-away property of the optimal map holds. Once the stay-away property is established, T CP allows to localize the problem and to apply the a priori estimates of Ma, Trudinger and Wang [], obtaining C regularity on T under C assumptions on the measures. In our case it is not clear whether the stay-away property is true or not, and this is why our result cannot be easily improved to higher regularity. 5.4 As we already said, if a Riemannian manifold M, g satisfies MT WK, C for some K, C > 0, then its sectional curvature is bounded below by K. As it was shown by Kim [6], the converse result is false. Describing the positively curved and simply connected Riemannian manifolds which satisfy MT WK, C for some K, C > 0 is a formidable challenge.

A Appendix : The round sphere The purpose of the appendix is to provide a proof of the following result. Theorem A.. There exists K 0 > 0 such that, for every n, the round sphere S, g can satisfies MT WK 0, K 0. Proof. Let us see the round sphere S n as a submanifold of R n+ equipped with the Riemannian metric induced by the Euclidean metric. More precisely, we see S n as the sphere centered at the origin with radius one in R n+. Since we work in R n+, we can identify covectors with vectors. For every x S n, the set N F x {0} corresponds to the open ball centered at x with radius π intersected with the hyperplan tangent to S n at x. Our aim is to show that there exists a constant K 0 > 0 such that, for every x S n R n+ and p N F x, one has Ŝx, p ξ, η = 3 d Kx, p + sηξ, ξ ds K s=0 0 ξ x η x K 0 ξ, η ξ η, for all ξ T x S n, η Tx S n. v T x S n, This is equivalent to show that, for every x S n and every 3 d d ds dt c exp x tξ, exp x v + sη K t=s=0 0 ξ x η x K 0 ξ, η ξ η ξ, η Tx S n, where c := d /. Since the function t, s c exp x tξ, exp x v + sη depends only on the behavior of the Riemannian distance in the affine space containing x and spanned by the three vectors v, ξ, η, we just have to prove Theorem A. for n = 3. Moreover, by the homogeneity of S, g can, it suffices to prove the estimate only for a fixed point x S n and along a fixed geodesic t exp x tv. Consider the stereographic projection of the sphere S 3 R 4 centered at the origin and of radius from the north pole N = 0, 0, 0, onto the space R 3 R 3 {0} R 4. The projection of some point x = x, x, x 3, x 4 S 3 is given by σx = x x,, x 4 x 4 x 3 x 4 The function σ is a smooth diffeomorphism from S 3 \ {N} onto R 3, whose inverse is σ y y = + y, y + y, y 3 + y, y + y y = y, y, y 3 R 3, where denotes the Euclidean norm on R 3. The pushforward of the round metric on S 3 is given by 4 g y v, v = + y v y, v R 3, and the Hamiltonian canonically associated to g is. Hy, p = + y p y, p R 3. 8 The Hamiltonian system associated to H is { ẏ = H p y, p = + y 4 p ṗ = H y y, p = + y p y,

and the linearized Hamiltonian system along a given solution yt, pt is { ḣ = + y y, h p + + y 4 q q = + y p h p y, h y + y p qy We note that h is a Jacobi vector field along the geodesic t yt. Set x =, 0, 0, 0 and x =, 0, 0, 0, and for α small, let γ α be the minimizing geodesic on S 3 joining x to x defined by γ α t := cost π, sinα sint π, 0, cosα sint π t [0, π]. Its image by the stereographic projection is given by cost π θ α t := σγ α t = cosα sint π, sinα sint π cosα sint π, 0. It is a minimizing geodesic between Y := σx =, 0, 0 and σx =, 0, 0. Fix t 0, π, and set V := t θ α 0. We need to compute the matrix KY, V. By an easy rescaling argument, we have KY, V = ts α t, where S α t is the 3 3 symmetric matrix such that any solution of the linearized Hamiltonian system along θ α starting from h, S α th, satisfies h t = 0. Let us compute S α t. Define three vector fields E, E, E 3 along θ α by E t := θ cosα sint π α t = cosα sint π, sinα cost π cosα sint π, 0, sinα cost π E t := cosα sint π, cosα sint π cosα sint π, 0, E 3 t := 0, 0,. The vectors E t, E t, E 3 t form a basis of parallel vector fields along θ α. Let h, q be a solution of the linearized Hamiltonian system along θ α such that ht = 0 for some t > 0. Since E t, E t, E 3 t form a basis of parallel vector fields along θ α, there are three smooth functions u, u, u 3 such that ht = u te t + u te t + u 3 te 3 t t. Hence, as h is a Jacobi vector field along θ α, its second covariant derivative along θ α is given by D t ht = ü te t + ü te t + ü 3 te 3 t. Therefore, since R 3, g has constant curvature, we have 0 = D t h + Rh, θ α θ α = D t h + g θα, θ α h g h, θ α θα = ü te t + ü te t + ü 3 te 3 t + u te t + u te t + u 3 te 3 t u t θ α t = ü te t + [ü t + u t]e t + [ü 3 t + u 3 t]e 3 t. We deduce that there are six constants λ, λ, λ 3, λ 4, λ 5, λ 6 such that u t = λ + λ t u t = λ 3 cost + λ 4 sint u 3 t = λ 5 cost + λ 6 sint. 3

Since E 0 = cosα, sinα, 0, E 0 = sinα, cosα, 0, E 3 0 = 0, 0,. the equality h0 = u 0E 0 + u 0E 0 + u 3 0E 3 0 yields h 0 = u 0 cosα + u 0 sinα h 0 = u 0 sinα + u 0 cosα h 3 0 = u 3 0, which gives λ = cosαh 0 sinαh 0, λ 3 = sinαh 0 + cosαh 0, λ 5 = h 3 0. Furthermore, Ė 0 = cosα, sinα, Ė 0 = sinα, cosα, Ė 3 0 = 0. Differentiating ht = u te t + u te t + u 3 te 3 t at t = 0, we obtain ḣ 0 = λ cosα + λ 4 sinα λ cosα λ 3 sinα ḣ 0 = λ sinα + λ 4 cosα + λ sinα λ 3 cosα ḣ 3 0 = λ 6. From the linearized Hamiltonian system, since Y 0 =, 0, 0 and P 0 = V 0 = E 0, we have q0 = ḣ0 + h 0E 0. Recalling that h t = 0 ut = 0, we get λ = λ t, λ 4 = λ 3 cos t sin t, λ 6 = λ 5 cos t sin t. Thus we finally obtain q 0 q 0 = q0 = S α th0 = q 3 0 a α t b α t 0 b α t c α t 0 0 0 d t h 0 h 0 h 3 0, where Hence with b α t := c α t := sin α t a α t := cos α t cosα sinα t cos α cos t sin t KY, V = ts α t = cos t sin α sin t cosα sinα cos t sin t cosα, + cosα, sinα, k Y, V k Y, V 0 k Y, V k 3 Y, V 0 0 0 k 4 Y, V d t = cos t sin t., k Y, V = ta α t, k Y, V = tb α t, k 3 Y, V = tc α t, k 4 Y, V = td t. 4