ON THE MA TRUDINGER WANG CURVATURE ON SURFACES

Transcription

1 Author manuscript, published in "Calculus of Variations and Partial Differential Equations 39, " ON THE MA TRUDINGER WANG CURVATURE ON SURFACES A. FIGALLI, L. RIFFORD, AND C. VILLANI Abstract. We investigate the properties of the Ma Trudinger Wang nonlocal curvature tensor in the case of surfaces. In particular, we prove that a strict form of the Ma Trudinger Wang condition is stable under C 4 perturbation if the nonfocal domains are uniformly convex; and we present new examples of positively curved surfaces which do not satisfy the Ma Trudinger Wang condition. As a corollary of our results, optimal transport maps on a sufficiently flat ellipsoid are in general nonsmooth. Contents Introduction. Two-dimensional Ma Trudinger Wang curvature 4. Curvature of tangent focal locus 9 3. MTW conditions at the edge 4. Stability 6 5. New counterexamples 7 Appendix: On the Riemannian cut locus of surfaces 7 References 3 Introduction The Ma Trudinger Wang MTW tensor is a nonlocal generalization of sectional curvature, involving fourth-order derivatives of the squared distance function [4, 7, 9,,, 8,, 5, 6, 7, 8, 9]. Various positivity or nonnegativity conditions on this tensor have been introduced and identified as a crucial tool in the regularity theory for the optimal transport in curved geometry [8,,, 9, 3, 4]; see [6] or [8, Chapter ] for a presentation and survey. Besides, this tensor has led to results of a completely new kind concerning the geometry of the cut locus [,, 4].

2 A. FIGALLI, L. RIFFORD, AND C. VILLANI It is therefore natural to investigate the stability of the Ma Trudinger Wang conditions. But while the condition of, say, strictly positive sectional curvature is obviously stable under C perturbation of the metric, it is not obvious at all that the condition of positive Ma Trudinger Wang curvature tensor is stable under C 4 perturbation, because this tensor is nonlocal. Partial results have already been obtained: roughly speaking, stability of the nonnegative curvature condition under Gromov Hausdorff limits [9]; stability of the positive curvature condition under C 4 perturbation, away from the focal locus [4, 9]; stability of the positive curvature condition under C 4 perturbation of the round spheres [, ]. In the present paper we shall continue these investigations, sticking to the case of surfaces. Without making the problem trivial, this assumption does allow for more explicit calculations. The main results of this paper are: simplified analytic expressions for the MTW curvature tensor on surfaces formulas.8 and 3.3, and the discovery of a strict connection between the MTW tensor and the curvature of the tangent focal locus near the tangent cut locus Proposition 3. and formula 3.4; the stability of the strict MTW condition under an assumption of uniform convexity near the tangent cut locus of nonfocal domains Theorem 4.; 3 new counterexamples of positively curved surfaces which do not satisfy the MTW condition Section 5. We remark that a formula for the MTW curvature tensor on surfaces analogous to.8 has been found independently in [5]. Our stability results should be compared to those in []. In the latter paper we proved the stability around the round sphere M = S n ; in the present paper, we only consider surfaces, but the assumption on M convexity of nonfocal domains near the tangent cut locus is much less restrictive. Moreover, the link we find between the MTW condition and the curvature of the tangent cut locus puts a new light on what the MTW condition geometrically means, and it explains why stability holds on the two-sphere a result first proven in [], see also [5]. It would be interesting to find a similar connection between the MTW condition and some convexity properties of the focal cut locus in higher dimension. Let us further recall that on perturbations of S n, the stability of the MTW condition has strong geometric consequences, namely it implies the uniform convexity of all injectivity domains [, ]. As far as counterexamples are concerned, we shall see in particular that some ellipsoids do not satisfy the MTW condition. A striking consequence is that the

3 MA TRUDINGER WANG CONDITIONS 3 smoothness of optimal transport on Riemannian manifolds may fail even on ellipsoids. Finally, let us observe that to show the stability of the strict MTW condition near the tangent cut locus, we will need to prove a result on the focalization time at the tangent focal cut locus, which we believe being of independent interest: if v T x M is a focal velocity belonging to the tangent cut locus, then the tangent focal locus and the segment joining to v are orthogonal Proposition A.6. As a corollary of this fact, we obtain that the focal cut locus of any point x M has zero Hausdorff dimension Corollary A.8. Notation: Throughout all this paper M, g is a given C compact Riemannian manifold of dimension, equipped with its geodesic distance d, its exponential map exp : x, v exp x v, and its Riemann curvature tensor Riem. We write gx = g x, g x v, w = v, w x. We further define t C x, v: the cut time of x, v: { } t C x, v = max t ; exp x sv s t is a minimizing geodesic. t F x, v: the focalization time of x, v: { } t F x, v = inf t ; detd tv exp x =. TCLx: the tangent cut locus of x: cutx: the cut locus of x: TCLx = { t C x, vv; v T x M \ {} }. cutx = exp x TCLx. TFLx: the tangent focal locus of x: TFLx = { t F x, vv; v T x M \ {} }. TFCLx: the tangent focal cut locus of x: TFCLx = TFLx TCLx. fcutx: the focal cut locus of x: fcutx = exp x TFCLx.

4 4 A. FIGALLI, L. RIFFORD, AND C. VILLANI Ix: the injectivity domain of the exponential map at x; so Ix = { tv; t < t C x, v, v T x M }. NFx: the nonfocal domain of the exponential map at x: NFx = { tv; t < t F x, v, v T x M }. exp : the inverse of the exponential map; by convention exp x minimizing velocities v such that exp x v = y. In particular TCLx = exp x and Ix = exp x M \ cutx. y is the set of cutx, Recall that t F t C, or equivalently IM NFM [5, Corollary 3.77]. The injectivity domain is included in the nonfocal domain.. Two-dimensional Ma Trudinger Wang curvature In this section we particularize to dimension two the general recipe for the computation of the Ma Trudinger Wang curvature, as given in [, Section and Paragraph 5.]. W.. Jacobi fields and Hessian operator. Let us fix a geodesic γt t T with γ = x, γ = y, γ = σ, σ =, and T = t F σ = t F x, σ. We choose a unit vector σ orthogonal to σ, and identify tangent vectors at x to their coordinates in the g-orthonormal basis σ, σ. Thus, modulo identification, T x M = R, σ =,, g x = Id R. Next, we let γ α t t be the geodesic starting at x with initial velocity σ α = cos α, sin α. We further define σα = sin α, cos α. For any α [, π] and τ we let kα, τ be the Gauss curvature of M at γ α τ. This function determines two fundamental solutions, f given by f i α, τ + kα, τ f i α, τ = i =,,. α, =, f α, =, f α, =, f α, =. Here as in the sequel, dots stand for τ-derivatives, while we shall use primes for α-derivatives. We further let Fα, τ = f α, τ α, τ.

5 MA TRUDINGER WANG CONDITIONS 5 Our goal is to express the MTW curvature in terms of, F, and their derivatives with respect to α. In the case of the unit sphere, f τ = cos τ, τ = sin τ, Fτ = cot τ, and there is no α-dependence. For this we only need to work with α close to. For any α, we define an orthonormal basis by setting e α, = σ α, e α, = σα, and from this we deduce e α, τ, e α, τ by parallel transport along γ. Then we define fields J and J by their matrix in this local basis: J α, τ = [ ] τ α, τ [ ] J α, τ =. f α, τ Each J i should be thought of as an array of two Jacobi fields. Let w = τσ α with τ < t F σ α, and let S x,w be the symmetric operator whose matrix, in the basis σ α, σα, is [ ] Sα, τ = τj α, τ J α, τ =. τfα, τ Then S x,w is the extended Hessian operator, as defined in [, Equation.6]. If w Ix then S x,w coincides with xd, exp x w /, see [8, Chapter 4, Third Appendix]... MTW tensor. The Ma Trudinger Wang tensor is obtained basically by differentiating the Hessian operator twice. Let [ ] cos α sin α Qα =, sin α cos α so that for v = τ σ α, the matrix of S x,v in the standard basis of R is Q αsα, τqα. Equivalently,. S x,τ σαξ, ξ = Sα, τ Qαξ, Qαξ. Let now v = t,, η = η, η R intrinsically, this means η = η σ + η σ, and s R small enough. Then v + sη = τσ α, where.3 τ = v + sη = t + sη + sη, α = tan sη. t + sη

6 6 A. FIGALLI, L. RIFFORD, AND C. VILLANI We differentiate. twice with respect to s: Qξ, Qξ + SQξ, d S x,τσαξ, ξ = ds [ S α Q ξ α + ] α s S τ Qξ, Qξ τ. s.4 d ds S x,τσαξ, ξ [ S = Qξ, Qξ + 4 S Q ξ, Qξ α α α Q Q Q ] α + S ξ, ξ + SQξ, ξ α α α s [ S S Q ] τ + Qξ, Qξ + 4 Qξ, ξ α τ τ α s S τ + Qξ, Qξ τ s [ S Q ] α + Qξ, Qξ + SQξ, ξ α α s S τ + Qξ, Qξ. τ s α s For a function f = fα, τ, we will use a dot to designate a derivative with respect to τ time, and a prime to designate a derivative with respect to α: f = f/ α, f = f/ t, etc. By direct computation, at s = we have [ ] [ ] [ ] S S = tf α = S tf τ = F + tf [ ] [ ] [ ] S α = S tf α τ = S F + tf τ = F + t F, [ ] [ ] [ ] Q Q = α = Q α =, τ = t, τ s = η, τ s = η t,

7 α =, MA TRUDINGER WANG CONDITIONS 7 α s = η t, α s = η η t. Plugging this back in.4, we obtain the following expression for the extended Ma Trudinger Wang tensor, as defined in [] see also [, Definition.]: 3 S x,vξ, η = d.5 ds S x,τsσαs ξ, ξ s= = [ ] tf ξ + 4tF ξ ξ + tfξ + ξ ξ tfξ η [ t + F + tf ξ + 4F + tfξ ] η η ξ t F + t F ξη + [ ] tf ξ + tfξ ξ ξ ξ η η F + tf ξ η. t Whenever v Ix, S x,v coincides modulo identification with the usual MTW tensor, see [] for more details. At this stage we note that the identity implies d f f f dτ = t.6 f f = as the above identity holds at τ =, or equivalently.7 F =. f It follows F = f 3, F =. f 3

8 8 A. FIGALLI, L. RIFFORD, AND C. VILLANI Plugging this in.5 yields.8 3 S x,vξ, η = t F t + t f f 3 ξη + 4 t 4 ξ f ξ η η + 4F t ξη η + ξ η 4f f 3 ξ ξ η t + Ft F t + ξ f η. Now, if we take η = ξ = ξ, ξ we get.9 where. 3 S x,vξ, ξ = At ξ 4 + Bt ξ 3 ξ + Ct ξ ξ + Dt ξ ξ 3 + Et ξ 4, At = t F t Bt = 4F t Ct = 5 6 f t + F t F t Dt = 4f f 3 Et = f t f MTW conditions. We say that the MTW condition in short MTW holds if.9 is non-negative for all ξ, for any x, v in the injectivity domain IM. The strict form of the Ma Trudinger Wang condition amounts to saying that.9 is positive whenever ξ, ξ,, a either for any choice of x, v in the injectivity domain IM; b or for any choice of x, v in the nonfocal domain NFM.

9 MA TRUDINGER WANG CONDITIONS 9 In case a, we say that the strict MTW condition denoted MTW + holds. In case b, we say that the extended strict MTW condition denoted MTW + holds. A useful quantitative form of this inequality is ξ, η T x M, S x,v ξ, η K ξ η C ξ, η, where K, C are positive constants. If this holds true for all x, v IM resp. for all x, v NFM, we say that M satisfies the condition MTWK, C resp. MTWK, C. All these conditions may or may not be satisfied by M. They are anyway strictly stronger than the condition of positive Gauss curvature. Examples and counterexamples are discussed at the end of this paper.. Curvature of tangent focal locus It is near the tangent focal cut locus that the study of the MTW condition becomes tricky. We shall see that there, the curvature of the tangent focal locus plays a crucial role. In this section we compute this curvature... Local behavior of TFL. Let us define the function α ρα = t F σ α = t F x, σ α, so that the tangent focal locus of M at x is given by the equation {ρ = ρα} in polar coordinates. The function ρ is the first nonzero solution of the implicit equation. α, ρα =. The identity.6 implies that f does not vanish in a neighborhood of { = }, so by the implicit function theorem ρ is a smooth function of α. As before, for a function f = fα, τ we write f = f/ α, f = f/ τ. Differentiating. with respect to α and using.6 again yields. ρ α = f α, ρα = f f α, ρα. A second differentiation, combined with., yields.3 ρ = + ρ f + ρ f = f f + f f, where the right-hand side is evaluated at α, ρα. The term has disappeared because = k =.

10 A. FIGALLI, L. RIFFORD, AND C. VILLANI Now we can apply classical formulas to compute the signed curvature κα of TFLx at α, ρα: with v = ρασ α, [ ] v det v v v κα = v + v = ρ + ρ ρ ρ 3/ ρ + ρ 3/.4 = ρ + f + ρ f ρf f + ρf f [ ρ + f ], 3/ evaluated at α, ρα. In the last formula we have used again f f = and f f = f f f, both deduced from.6. So the nonfocal domain is convex resp. uniformly convex around v = tσ α if and only.4 is nonnegative resp. positive for any α in a neighborhood of α... Local behavior of TFCL. Now we particularize the preceding computation by considering a velocity which is not only focal, but also a cut velocity. So let again v = tσ α TFCL, and let ρ = t C v = t F v = ρα. By Proposition A.6 in the Appendix,.5 ρ α =. By.6 we have f at α, ρ, so.5 is equivalent to.6 f α, ρ =. Then the expressions obtained in Subsection. simplify as follows:.7 ρ α = f, κα = + f, ρ ρ where f and f are evaluated at α, ρ. 3. MTW conditions at the edge From the expression of the MTW tensor given in.8, one can easily see that S x,v varies smoothly with respect the metric, as long as is bounded away from, i.e. as long as v NFx is far away from TFLx. Hence, when one is interested in the stability of the Ma Trudinger Wang condition inside IM, the critical part is to understand this condition near the natural boundary of its domain of validity, which is not the tangent cut locus, but rather the tangent focal cut locus. The main theme of this section is that there is almost equivalence of the three following conditions:

11 MA TRUDINGER WANG CONDITIONS a the MTW tensor is nonnegative near TFCL b the MTW tensor is bounded below near TFCL c TFCL is locally convex. Condition c really means that for any x, the nonfocal domain NFx is convex in the neighborhood of any focal cut velocity. Since the description of the tangent cut locus is easy away from focalization, condition c allows to prove a rather strong geometric property, not known for general manifolds: injectivity domains are semiconvex, i.e. smooth deformations of convex sets see [3]. Compare with the open problem stated in [6, Problem 3.4]. We do not know how far one can push the equivalence between a, b and c. For the moment we shall only establish certain partial implications between variants of these conditions. Proposition 3.. Let v = ρσ α TFCLx and let κ be the signed curvature of TFLx at v. Then: i if κ >, there are δ, K, C >, depending only on upper bounds on f i, f i, f i i =,, and on a lower bound on κ and on the injectivity radius of M, such that for all α α δ, α + δ and t t F σ α δ, t F σ α, v = tσ α, 3. ξ, η T x M T x M, S x,v ξ, η K ξ + ξ η C ξ, η. f ii if κ <, then { } 3. lim inf S x,tσαξ, ξ ; t t F σ α, α α, ξ T x M, ξ = =. The above proposition says the following remarkable thing: whenever TFLx is uniformly convex near a point v TFCLx, then MTWK, C holds in a neighborhood of v in T x M. As already observed in [,, 4], this analytic property allows to deduce strong geometric consequences on the injectivity domains. Remark 3.. As can be easily seen from the proof of point i, the constant K actually depends only on a lower bound on κ and on the injectivity radius of M, provided δ is chosen sufficiently small. Moreover, as can be immediately seen from the proof, the assumption that v TFCLx is needed only to ensure that f α, t F α is sufficiently small in a neighborhood of α, t F α. Hence all the results in Proposition 3. hold true for any v = t F ασ α TFLx such that f α, t F α η, for some universal constant η >.

12 A. FIGALLI, L. RIFFORD, AND C. VILLANI Proof of Proposition 3.. We start by rewriting.8. After repeated use of.6, we obtain [ f 3 S x,vξ, η = f f ξ η + t ] 3.3 ξ η + ξ η t f + t ξ η + 4 t + f ξ ξ η η f + + t f ξ η f [ f f f f f ξ η η t + f t t f + f + f t f f ] ξ η f. Note that the coefficient f /t is positive near the edge near focalization. This follows from.6, observing that f < near the edge. Let us examine the behavior of the various coefficients: as we approach the focal locus. Since f α, t F α =, we may choose δ small enough that we can impose f φ, with φ arbitrarily small. In the coefficient of ξ η the highest order terms ±f /tf 3 cancel each other; in the end this coefficient is γ = [ + f f ] + O. f t t By.7, if the curvature κ of TFLx at α, t F α is nonzero and if φ is small enough, then for δ sufficiently small γ = f where ω = ωα, t satisfies ω /4. t κ + ω,

13 MA TRUDINGER WANG CONDITIONS 3 Then we choose very small with respect to the other parameters. In the end [ f 3 S x,vξ, η = f f ξ η + t ] ξ η + ξ η t f t ξ η + t κ + ω ξ f η + O ξ ξ η η + O f ξ η + O ξ f η η. Next, we write O f O ξ η + O ξ η η λ f f ξ ξ η η λ ξ η + O f ξ η, ξ η, where λ > is arbitrarily small but fixed. We conclude that for and δ small enough, S x,vξ, η = f t where, say, ω /. [ f f f + t + ω ξ η + t κ f ξ η + t f ] ξ η + ξ η After these preparations we can prove Proposition ω ξ η + O ξ η, I First we assume κ < and we wish to prove instability. From 3.5, S x,vξ, ξ = f t [ f f f + t + ω ξ4 + t κ f ξ ξ t f ] ξ + ξ f + ω ξ ξ + O ξ, 4 From the definition of t F we have α, t F α, so that we deduce = [ f α, t F α ] = f α α, t F α + f α, t F α t F α, f

14 4 A. FIGALLI, L. RIFFORD, AND C. VILLANI and since t F α = this implies = [ f α α, t F α ] + f α, t F α t F α, α=α whence, recalling.7, [ f α α, t F α ] = f α, t F α t F α α=α = f f α, t F α = f α, t F α t F α κ t F α. The latter quantity is positive since f α, t F α < and κ < by assumption. Recalling that α, t F α =, we conclude that α, t F α is a small positive number for α > α, α α. We fix such an α. Then for t < t F α, t t F α, we have α, t, f α, t, and we can choose a unit vector ξ = ξα, t such that f f f + f + 4t ξ ξ = t α, t f α, t F α as t t F α. Then the first term in the right-hand side of 3.6 vanishes, and as t t F α we have 3 S x,tσ αξ, ξ = + ω + t F α κ t F α α, t F α + ω + Of. This expression takes arbitrarily large negative values as α α, which proves 3.. II Next we assume κ > and we wish to prove stability. The first three terms in the right-hand side of 3.5 have the right sign, so the issue is to control the remaining one. We introduce a small constant ε >, and distinguish three cases: If η ε η, then the dangerous term is obviously controlled by the term involving κ, as soon as ε is small enough. Moreover η η, and so S x,v ξ, η t + ω ξ η + t κ + ω ξ f η K ξ + ξ η, f for K > sufficiently small.

15 MA TRUDINGER WANG CONDITIONS 5 If η ε η and t f ξ η ε ξ η or t f ξ η ε ξ η, then choosing δ small enough we have f f ξ η f 4 t ξ η, and so f t [ f f f ξ η + t f ] ξ η + ξ η f t t f f 3 ξ η + ξ η which easily dominates the dangerous term for ε small enough. Hence since η ε η S x,v ξ, η f t f ξ t f 3 η + ξ η + t + ω ξ η + t κ + ω ξ f η c ξ η + ξ η + ξ η ξ + c η, f for some small constant c >. Thanks to the inequality a b + a + b, we deduce that ξ η ξ η + ξ, η, which easily implies S x,v ξ, η K ξ + f ξ η C ξ, η for some positive constants K, C. If η ε η and ε ξ η t f ξ η ε ξ η, then ξ η Of ε ξ η note that is strictly negative near =, so for ξ = η = we have, 3.7 S x,v ξ, η C ε, for some constant C.

16 6 A. FIGALLI, L. RIFFORD, AND C. VILLANI On the other hand, ξ = O ε ξ η / η = O ε is bounded above by ε/8 for small enough, and we get ξ, η ξ η ξ η ε 4. Also ξ /f = O. Combining this with 3.5 and 3.7, for δ and small enough we do have S x,v ξ, η K ξ η + ξ f η C ξ, η for K > small, and for some large enough constant C. This completes the proof of Proposition Stability If the condition MTW + is unstable, this can only be near TFCL. The main result of this section shows that a geometric condition on the focal locus near this dangerous set will prevent the instability. For any x M, let κx = inf α κα, where κα is the signed curvature of TFLx at α, ρα. See Subsection.. Then we define κm = inf x M κx. Theorem 4. Stability of MTW + on surfaces with convex nonfocal domain. Let M, g be a compact Riemannian surface satisfying MTW +. If κm > then any C 4 perturbation of g satisfies MTWK, C for some constants K, C >. This applies in particular to g itself. In other words, there is δ > such that for any other metric g on M, if g g C 4 < δ then M, g satisfies MTWK, C. Here the C 4 norm is measured by means of local charts on M. Remark 4.. Theorem 4. was proven in [] in the particular case when M is the sphere S. In that case however, there is a stronger statement according to which the extended condition MTWK, C survives perturbation. Proof of Theorem 4.. Let G be the set of Riemannian metrics on M, equipped with the C 4 topology. First we note that TFLM is continuous on G the C topology would be sufficient for that; and according to formula.4, the curvature of TFLx at v is a continuous function of x, v and also of g G. Next, TCLM is continuous on G also here the C topology would be sufficient. In particular, the injectivity radius of M is continuous on G; and TFCLM, g remains within an open neighborhood of TFCLM, g. This set can shrink drastically,

17 MA TRUDINGER WANG CONDITIONS 7 as the perturbation of the sphere shows. So if g g C 4 is small enough, we still have κm, g >. For each x, v TFCLM, g, the curvature of TFLx at v is positive; so Proposition 3. i implies the existence of a neighborhood U x,v of x, v in TM, and a neighborhood O x,v of g in G such that for any y, w U x,v and any g O x,v, ξ, η T y M \ {}, S eg y,wξ, η K ξ y η y C ξ, η y. By compactness, we can find an open neighborhood U of TFCLM, g and a neighborhood O of g in G on which this inequality holds. Outside of U,.8 shows that Sy,wξ, eg η is a uniformly continuous function of y, w IM, g, g G, and ξ, η in the unitary tangent bundle. Then one can conclude by the same compactness argument as in [, Section 6]. Example 4.3. Consider an ellipsoid E which is not too far from the sphere. If it satisfies a strict MTW condition and has uniformly convex nonfocal domains, then any C 4 perturbation of E will also satisfy the MTW condition. The point is that the MTW condition may be checked numerically on E, since geodesics and focal loci are given by known analytic expressions. 5. New counterexamples Following [8, Chapter ], let us agree that a manifold is regular if it satisfies the Ma Trudinger Wang condition and has convex injectivity domains. Examples of regular manifolds appear in [,, 3, 4, 9]. Regularity of the manifold is a necessary condition for the regularity theory of optimal transport [8, Chapter ]. In the class of positively curved surfaces, counterexamples were constructed in [7] and [4] in the last paper, this is essentially a cone touching a paraboloid. Here we shall construct new counterexamples, also with positive curvature. 5.. Surfaces of revolutions. In this subsection, we give simple formulas for the MTW curvature along certain well-chosen geodesics of surfaces with revolution symmetry. Let N and S respectively stand for the North and South Poles on S. We parameterize S \ {N, S} by polar coordinates θ, r from N, and define the Riemannian metric g = mr dθ + dr, where m is a positive smooth function. In the sequel, we shall identify points of S with their coordinates and denote by m k the k-th derivative of m.

18 8 A. FIGALLI, L. RIFFORD, AND C. VILLANI Only two of the Christoffel symbols of g are nonzero: Γ r θθ = mr m r and Γ θ rθ = m r mr ; so the equation for geodesics is { θ + Γ θ rθṙ θ = r + Γ r θθ θ =. Further, the Gauss curvature of g at a point θ, r is equal to 5. kr = m r mr. We assume that k >, so that m is strictly concave. We define r as the unique value r such that m r =, and we assume that m 3 r =, so that k r =. We write k = kr. Let γ : R + S be the unit-speed geodesic starting at θ = in the θ-direction: γt = t, r. We shall study variations of γ. First of all, since the curvature is constant along γ, the functions, f introduced in. are given by 5. t = sin kt, f t = cos kt. k Next, let γ α be the geodesic starting at p = γ with velocity σ α = cos α, sin α. From 5. we deduce γ α α =, sin kt. α= k Then k α = α= k α = α= k γ α α, γ α α = m r m 4 r mr sin kt. k

19 MA TRUDINGER WANG CONDITIONS 9 Differentiating the Jacobi equation, we obtain fi fi + k = t α α f i f i k + k = f t α α α i. We deduce f i t α = f i t t t = Ks f α s f i s ds t Ks s f i s ds f t, where the function K is defined as sin ks 5.5 Ks := m 4 r m r m r. k Let us assume 5.6 mr =, so that the matrix of the metric at θ, r in the basis θ, r is the identity. Then we can apply. to compute the MTW curvature along γ. Recall that F = f. The expression simplifies when t = π k =: t, since then t = f t =. Hence 5.7 where 3 S p,tσ ξ, ξ = A ξ 4 + C ξ ξ + E ξ 4, A = t F t = t 5.8 C = 5 f 6 t + f F t t = 5 k 6 t F t E = f t f f 3 = k,

20 A. FIGALLI, L. RIFFORD, AND C. VILLANI with F t = f t 5.9 t = k m 4 r m r m r t sin ks ks cos ds. This expression is well suited to the construction of counterexamples: for that it is sufficient to devise the function m in such a way that F is very large near focalization. In practice, it is convenient to consider the metric as induced by a graph. Let F : [ a, a] R + be a smooth function satisfying F a = F a =, F > on a, a; then we may rotate the graph z = F x along the x-axis, thus sweeping a two-dimensional surface which is isometric to S, g with where m F is determined by the identity g = m F r dθ + dr, x [ a, a], m F rx = F x, rx = x a + F u du. We assume F =, F = F 3 =, and define r = a + F. Then m F r = m r = 5. and we can apply 5.8. F m F m 3 F m 4 F r = F r = r = F 4 4F 3, 5.. Ellipsoids of revolution. Let E ε be an ellipsoid of revolution of parameter ε > given in R 3 by the equation x 5. ε + y + z =. In the formalism of the previous subsection, this is the surface defined by rotating the graph of the function F ε : [ ε, ε] R + defined by x. 5. F ε x = ε

21 MA TRUDINGER WANG CONDITIONS Then Ellipsoid E ε with ε = F ε =, F ε =, F ε = ε, F 3 ε =, F 4 ε = 3 ε 4. So all the computations in the preceding subsection apply with m = m Fε. The curvature along γ is k = /ε, the focalization time along that geodesic is t = πε, and we can compute the various terms in 5.7 for t = πε/ : πε sin s/ε cos s/ε ds = π ε 6, m 4 r mr m r = 4 ε 4 ε, 5.4 A = 8 π ε C = 5 ε 4 π ε ε + 4 ε E = ε.

22 A. FIGALLI, L. RIFFORD, AND C. VILLANI Therefore the MTW condition is violated as soon as C > A E, or equivalently This is equivalent to which in turn holds for ε 4 ε 5 ε + 4 π ε > 8 π ε. ε < 48 π + 6 ε >, π π π Thanks to a classical result of Klingenberg on even-dimensional Riemannian manifolds [], we know that the injectivity and the conjugate radius coincide. Since along our geodesic the curvature is maximal this can be easily checked by an explicit computation, we easily deduce that t C γ = t F γ; in particular t < t C γ. Hence, invoking for instance [8, Theorem.44], we deduce an extremely strong negative result as regards the smoothness of optimal transport: Corollary 5.. If E ε is the ellipsoid of revolution defined by 5. with ε.9, then there are C positive probability densities f, g on E ε such that the solution T of the optimal transport between µdx = fx vol dx and νdy = gy vol dy, with transport cost d, is discontinuous Another counterexample to regularity. The previous subsection has shown that the MTW condition does not like variations of curvature. In this subsection we shall present another illustration of this phenomenon, by considering two half-balls joined by a cylinder. Set C = { x, y, z R 3 x + y =, z [, ] }, S + = { x, y, z R 3 x + y + z =, z } and S = { x, y, z R 3 x + y + z + =, z }. Let us denote by M the cylinder with boundary defined by M = C S + S.

23 MA TRUDINGER WANG CONDITIONS 3 The nonsmooth surface M This submanifold of R 3 is not C, but it is sufficiently smooth to define an exponential mapping and the concept of regular costs. We denote by d the geodesic distance on M and consider as usual the cost c = d. We set A =,,, and B =,,. If v = v,, v 3 is a unit vector in T A M with v 3, then the geodesic γ starting from A with initial speed v is given by γt = cos at π, sin at π, bt if t [, v3 ],

24 4 A. FIGALLI, L. RIFFORD, AND C. VILLANI and γt = cos t v3 cos π + v, sin π + v sin t v3 v sin v 3 π + v v 3, v cos v 3 π + v v 3, +, v 3 for t v 3 small enough. In the sequel, given a unit vector v = v,, v 3 T A M and l >, we denote by Bv, l the end-point of the geodesic starting from A with initial speed v and of length l. We set η =, V s = + sη, vs = s, ls = + s ls + s. + s Given s >, we set B = Bv s, l s and B + s = Bvs, ls. Then we have B = cos v s l s π, sin v s l s π, v 3 s l s and B + = cos ls v s sin + v s sin ls v s cos, v 3 s v 3 s v 3 s v 3 s cos ls v s cos + v s sin ls v s sin, v 3 s v 3 s v 3 s v 3 s sin ls v 3 s. v 3 s By construction, ca, B =, ca, B = l s = + s s, ca, B + = ls = + s + s. Let X M be a point given by θ, z in cylindrical coordinates, that is X = cosθ, sinθ, z.

25 MA TRUDINGER WANG CONDITIONS 5 Then we have Therefore, and cx, B = cx, B = θ + π + z, θ + π, v s l s + z + v 3 s l s = ca, B cx, B = θ + π z, = ca, B cx, B = l s θ + π v s l s z + v 3 s l s [ = θ + π z ] + s s v s l s + θ + π v s l s v 3 s l s z v 3 s l s = + θ + π v s l s + + z v 3 s l s. We compute easily ls = + s + s 3 7s 4 + 4s 5 + os 5 v s = s s + s 3 s 4 + s 5 + os 5 v 3 s = s 3 + 8s 4 8s 5 + os 5 v s/v 3 s = s s + s 3 s 4 + s 5 + os 5 cosv s/v 3 s = s / + s 3 35/4s 4 + s 5 /6 + os 5 sinv s/v 3 s = s s + 5/6s 3 s 4 / + 6s 5 / + os 5 ls /v 3 s = s + s 4 + 6s 5 + os 5 cosls /v 3 s = s / + s 4 /4 s 5 + os 5 sinls /v 3 s = s s 3 /6 + s s 5 + os 5 Let αs > be such that the point X α = cos α π, cos α π, α

26 6 A. FIGALLI, L. RIFFORD, AND C. VILLANI has the same first two coordinates as B +. In this way, we have cos α π cos ls v 3 sin v s s v 3 + v s s sin = cos ls v 3 s Also 5.5 cx α, B + = + α Since 5.5 implies and ls v 3 s + v s sin ls v 3 s cos v s v 3 s. + arcsin v 3 s sin ls. v 3 s αs = s + s 4 /3 + 3s 5 /6 + os 5, cx α, B + = + 4s + 4s + s 4 / s 5 + os 5, + = ca, B + cx α, B + = s s s 4 /3 + 8 s 5 + os 5. We check that 5 = s s s 4 / s os 5 > +, and =. Given s small enough and fixed, we can approximate X αs by X such that ca, B cx, B > max { ca, B cx, B, ca, B + cx, B + }. This inequality shows that the cost c is not regular [8, Chapter ]. Now let us regularize M into a smooth surface M ; this can be done in such a way that the Gauss curvature of M takes values in [, ]. Then by a classical result of Klingenberg [] we have t C π throughout the unit tangent bundle of M. Since da, B = < π, we can include A and X into a region Ω, and B, B +, B in an open set Λ, such that for any x Ω, the convex hull of log x Λ stays away from TCLx; and for any y Λ, the convex hull of log y Ω stays away from TCLy. For small values of the regularization parameter, the squared distance is not regular Ω Λ M M,

27 MA TRUDINGER WANG CONDITIONS 7 otherwise we could pass to the limit as in [9] to deduce the same property for the limit M. Then we can apply the method of Loeper [] [8, Proof of Theorem.36] to show that M does not satisfy the MTW condition. Appendix: On the Riemannian cut locus of surfaces A.. Generalities. Let M be a smooth, compact, connected n-dimensional Riemannian manifold equipped with a Riemannian metric, and a geodesic distance d. We recall that d : M M R is defined by { } dx, y = inf γt dt γ Lip[, ]; M, γ = x, γ = y. Let x M be fixed, we denote by d x = dx, the distance to the point x. The function d x is locally semiconcave on M \ {x}. We denote by Σ x the singular set of d x or equivalently of d x in M \ {x}, that is Σ x = { y M ; d x not differentiable at y }. Since the function d x is locally semiconcave, thanks to Rademacher s Theorem, it is differentiable almost everywhere. For every y M, we call limiting gradient of d x at y the subset of T y M defined as D d x y = { w T y M ; w k w s.t. w k = d x y k, y k y }. For every y M \ {x}, there is a one-to-one correspondence between D d x y and the set of minimizing geodesics joining x to y: for every w D d x y, there is a minimizing geodesic γ : [, ] R joining x to y such that γ = dx, yw. Then we have { } Σ x = y M; v v TCLx s.t. exp x v = exp x v = y. We denote by J x the set J x = { exp x v; v TCLx s.t. d v exp x is singular } = fcutx. We notice that if y Σ x is such that the set D d x y is infinite, then it belongs to J x. The following result holds []: Proposition A.. If M is a compact Riemannian manifold, then for any x M we have cutx = Σ x = Σ x J x.

28 8 A. FIGALLI, L. RIFFORD, AND C. VILLANI In the sequel, we shall say that a point y cutx or by abuse of language that the tangent vector v = exp x y is purely focal if y does not belong to Σ x, that is if the function d x is differentiable at y. We denote by Jx the set of purely focal points. In particular, we have cutx = Σ x = Σ x Jx. A.. Cut loci on surfaces. Let M be a smooth, compact, connected Riemannian surface equipped with a Riemannian metric, and x M be fixed. For every y M, we call generalized gradient of d x at y, the convex subset of T y M defined by d x y = conv D d x y. The set d x y being convex, it has dimension, or. In fact, given y x, the function d x is differentiable at y if and only d x y has dimension. We set for i =,, { } Σ i x = y M \ {x} ; dim d x y = i. Proposition A.3. The set Σ x is discrete, and Σ x is countably -rectifiable. We stress that Σ x is not necessarily a closed set, as it may have accumulation points which do not belong to Σ x. The following proposition on the propagation of singularities will be useful: Proposition A.4. Let y Σ x, p d x y \ D d x y, and q T y M \ {} such that q, p p y, p d x y. Then there exists a Lipschitz arc y : [, ε] M such that ẏ = q and ys Σ x, The above two results can be found in []. s [, ε]. A.3. On focal velocities in dimension. We assume from now on that, is a given Riemannian metric on a smooth compact surface M. Let x M be fixed, we denote by S x the unit sphere in T x M, that is S x = {v T x M; v x = }. We define the focal function at x and the cut function at x by t F = t F x, : v S x t F x, v and t C = t C x, : v S x t C x, v The function t F is smooth on S x its domain, while t C is Lipschitz. For every pair v v S x with v close to v, we denote by Iv, v the shortest of the two curves in S x which joins v to v.

29 MA TRUDINGER WANG CONDITIONS 9 Lemma A.5. There is ɛ > such that for every pair v v Sx satisfying v v x < ɛ and exp x t C vv = exp x t C v v, there is v Iv, v such that either t C v v is purely focal that is D d x exp x t C v v is a singleton, or v is focal and t F v =. Proof of Lemma A.5. There is ε > such that for any v v S x with v v x < ε, if we denote by γ, γ the two minimizing geodesics with constant speed t C v = t C v joining x to the point y = exp x t C vv = exp x t C v v, then the set C = γ[, ] γ [, ] separates M into two connected components. Moreover, we can also assume that for each velocity w in the small interval Iv, v, any point exp x tw with t, t C w belongs to the smallest component O. Let v, v S x with v v x < ε be fixed. Denote by v a speed in Iv, v such that t C w t C v w Iv, v. We claim that either v is purely focal, or that v is focal and satisfies t F v =. Indeed, assume that v is not purely focal. Set y = exp x t C v v, and note that y belongs to O Σ x. By Proposition A.4, there is no p d x y \ D d x y and q such that q, p p, p d x y, and such that the Lipschitz curve given by Proposition A.4 makes the function d x strictly decreasing. This means that there is necessarily a non-constant curve w : [, ] D d x y such that w = w, where w is the velocity at time t C v of the minimizing geodesic starting at x with speed v. For every t close to, denote by v t the speed in Sx such that exp x t C v t v t = exp x t C v and such that the speed of the minimizing geodesic starting at x with speed v t has the velocity wt at time s = t C v. Any v t is focal and satisfies t F v t = t F v = t C v. This shows that v is focal and that t F v has to be zero. To our knowledge, the following result and its corollary are new. Proposition A.6. Let v Sx be such that t C v = t F v. Then t F v =. For the proof we shall use the following lemma from []. Here, will denote the Euclidean scalar product in R n. Lemma A.7. Let Ω be an open subset of R n and F : Ω R n a map of class C. Let z Ω be such that DF z has rank n. Set ȳ = F z, let θ be a generator of

30 3 A. FIGALLI, L. RIFFORD, AND C. VILLANI Ker DF z and w be a nonzero vector orthogonal to Im DF z. Suppose that F z w >. θ Then there exist ρ, σ > such that the equation has no solution if σ < s <. F z = ȳ + sw, z B z, ρ, Proof of Proposition A.6. Without loss of generality, we may assume that the metric, in T x M is given by the Euclidean metric. In this way, we can see a speed v S x as an angle on S. Define F : R S by F θ, r = exp x r cos θ, r sin θ. Up to a change of coordinates, we may indeed assume that F is valued in R. By construction of the focal function t F, for all θ we have therefore F θ θ, t F θ =, F θ θ, t F θ + t F θ F θ r θ, t F θ = Since F θ, t θ r F θ never vanishes, we have to show that F θ θ, t F θ =, for every θ such that t C θ = t F θ. Let θ be fixed such that t C θ = t F θ. Argue by contradiction and assume that F θ θ, t F θ, Set z = θ, t C θ and ȳ = F z. Two cases may appear. Case : ȳ J x By Lemma A.7, there is a small ball B z, ρ such that the equation F z = ȳ + sw has no solution for small negative s. Therefore, for each s = /k with k N, there is a minimizing geodesic γ k : [, ] M joining x to the point ȳ + s k w whose initial speed v k does not belong to B z, ρ. Taking the limit as k +, we obtain

31 MA TRUDINGER WANG CONDITIONS 3 a minimizing geodesic γ : [, ] M joining x to ȳ with initial speed v / B z, ρ. This contradicts the fact that ȳ is purely focal. Case : ȳ J x \ J x Denote by V the set of θ S such that F θ, t C θ = ȳ. Two cases may appear. Subcase.: θ is isolated in V. Thus the minimizing geodesic γ : [, ] M joining x to z is isolated among the set of minimizing geodesics joining x to ȳ. Therefore, we can modify the Riemannian metric outside a neighborhood of γ[, ] in such a way that γ becomes the unique minimizing geodesic between x and ȳ. Since the modification of the metric has been done far from γ, the function t F and the new focal function t F coincide in a neighborhood of θ which is purely conjugate. By Case, we obtain that t F θ =. Subcase.: θ is not isolated in V. Thus there is a sequence {θ k } in V converging to θ. By Lemma A.5, this yields a sequence {θ k } converging to θ such that each θ k is either purely focal or such that t F θ k =. In any case, thanks to Case above and the continuity of t F, one has t F θ =. We deduce as a consequence of Proposition A.6 an improvement of the classical result H J x =, as follows: Corollary A.8. The set J x has zero Hausdorff dimension. Proof. Consider the smooth map Ψ : v Sx expt F vv. Any v TFCLx is a critical point of Ψ. We conclude by Sard s Theorem. Remark A.9. In fact, a generalization of Corollary A.8 holds in any dimension: the set J x has Hausdorff dimension at most n ; see []. References [] P. Cannarsa and C. Sinestrari. Semiconcave functions, Hamilton Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, 4. [] M. Castelpietra and L. Rifford. Regularity properties of the distance function to conjugate and cut loci for viscosity solutions of Hamilton Jacobi equations and applications in Riemannian geometry. ESAIM Control Optim. Calc. Var., to appear. [3] J. Cheeger and D. Ebin. Comparisons theorems in Riemannian geometry. North-Holland, Amsterdam, and Elsevier, New-York, 975.

32 3 A. FIGALLI, L. RIFFORD, AND C. VILLANI [4] P. Delanoë and Y. Ge. Regularity of optimal transportation maps on compact, locally nearly spherical, manifolds. J. Reine Angew. Math., to appear. [5] P. Delanoë and Y. Ge. Positivity of c-curvature on surfaces locally near to sphere and its application. In preparation. [6] A. Figalli. Regularity of optimal transport maps after Ma Trudinger Wang and Loeper. Séminaire Bourbaki. Vol. 8/9, Exp. No. 9. Available online at [7] A. Figalli, Y. H. Kim and R. J. McCann. Continuity and injectivity of optimal maps for non-negatively cross-curved costs. Preprint, 9. [8] A. Figalli, Y. H. Kim and R. J. McCann. Regularity of optimal transport maps on multiple products of spheres. In preparation. [9] A. Figalli and G. Loeper. C regularity of solutions of the Monge Ampère equation for optimal transport in dimension two. Calc. Var. Partial Differential Equations, 35 9, no. 4, [] A. Figalli and L. Rifford. Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S. Comm. Pure Appl. Math., 6 9, no., [] A. Figalli, L. Rifford and C. Villani. Nearly round spheres look convex. Preprint, 9. [] A. Figalli, L. Rifford and C. Villani. Continuity of optimal transport on Riemannian manifolds in presence of focalization. In preparation. [3] A. Figalli, L. Rifford and C. Villani. On the semiconvexity of tangent cut loci on surfaces. In preparation. [4] A. Figalli and C. Villani. An approximation lemma about the cut locus, with applications in optimal transport theory. Meth. Appl. Anal., 5 8, no., [5] S. Gallot, D. Hulin and J. Lafontaine. Riemannian geometry, second ed. Universitext. Springer- Verlag, Berlin, 99. [6] J. Itoh and M. Tanaka. The Lipschitz continuity of the distance function to the cut locus. Trans. Amer. Math. Soc., 353, no., 4. [7] Y. H. Kim. Counterexamples to continuity of optimal transportation on positively curved Riemannian manifolds. Int. Math. Res. Not. IMRN, 8, Art. ID rnn, 5 pp. [8] Y. H. Kim and R. J. McCann. Continuity, curvature, and the general covariance of optimal transportation. J. Eur. Math. Soc., to appear. [9] Y. H. Kim and R. J. McCann. Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products of round spheres in particular. J. Reine Angew. Math., to appear. [] W. Klingenberg. Contributions to Riemannian geometry in the large Ann. of Math., , [] W. Klingenberg. Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung. Comment. Math. Helv., 35 96, [] G. Loeper. On the regularity of solutions of optimal transportation problems. Acta Math., to appear. [3] G. Loeper. Regularity of optimal maps on the sphere: The quadratic cost and the reflector antenna. Arch. Ration. Mech. Anal., to appear.

33 MA TRUDINGER WANG CONDITIONS 33 [4] G. Loeper and C. Villani. Regularity of optimal transport in curved geometry: the nonfocal case. Duke Math. J., to appear. [5] X. N. Ma, N. S. Trudinger and X. J. Wang. Regularity of potential functions of the optimal transportation problem. Arch. Ration. Mech. Anal. 77, 5, [6] N. S. Trudinger and X. J. Wang. On the second boundary value problem for Monge Ampère type equations and optimal transportation. Ann. Sc. Norm. Super. Pisa Cl. Sci. 5, 8 9, no., [7] N. S. Trudinger and X. J. Wang. On strict convexity and continuous differentiability of potential functions in optimal transportation. Arch. Ration. Mech. Anal., 9 9, no. 3, [8] C. Villani. Optimal transport, old and new. Grundlehren des mathematischen Wissenschaften, Vol Springer-Verlag, Berlin, 9. [9] C. Villani. Stability of a 4th-order curvature condition arising in optimal transport theory. J. Funct. Anal., 55 8, no. 9, Alessio Figalli Department of Mathematics The University of Texas at Austin Postal address: University Station, C Street address: 55 Speedway, RLM.48 Austin, Texas 787-8, USA figalli@math.utexas.edu Ludovic Rifford Université de Nice Sophia Antipolis Labo. J.-A. Dieudonné, UMR 66 Parc Valrose, 68 Nice Cedex, FRANCE rifford@unice.fr Cédric Villani ENS Lyon & Institut Universitaire de France UMPA, UMR CNRS allée d Italie Lyon Cedex 7, FRANCE cvillani@umpa.ens-lyon.fr