Villani's conjecture: convexity of injectivity domains on compact Riemannian manifolds satisfying the Ma–Trudinger–Wang condition

Let $(M,g)$ be a smooth compact Riemannian manifold and let $I(x)\subset T_xM$ denote its injectivity domain at $x$, the open star-shaped set of tangent vectors whose geodesics minimize up to time one. Villani conjectured that the Ma–Trudinger–Wang (MTW) sign condition alone — a fourth-order positivity condition on the squared-distance cost, necessary for continuity of optimal transport maps — forces all injectivity domains of $M$ to be convex, with no nonfocality assumption. Figalli–Gallouët–Rifford (SIAM J. Math. Anal. 2015) proved the conjecture under the additional hypothesis that the manifold is nonfocal (every tangent cut velocity is itself nonfocal) and located the entire remaining difficulty at purely focal velocities, where the exponential map degenerates, the extended MTW tensor can change sign, and the known estimates break down. The conjecture remains open: prove or refute that (MTW) implies convexity of all injectivity domains without nonfocality.

Background

Optimal transport and regularity. Let $(M,g)$ be a smooth compact connected Riemannian manifold with geodesic distance $d$, and let $c(x,y)=d(x,y)^2/2$ be the quadratic geodesic cost. The Monge optimal transport problem asks, for two probability measures $\mu,\nu$ on $M$ that are absolutely continuous with respect to the volume measure with densities bounded above and below, for a measurable map $T:M\to M$ pushing $\mu$ forward to $\nu$ (i.e. $\nu(A)=\mu(T^{-1}(A))$ for every measurable $A$) and minimizing $\int_M c(x,T(x))\,d\mu(x)$. Such a map exists and is unique up to sets of $\mu$-measure zero, and the basic regularity question is whether $T$ is continuous. Injectivity domains and the cut locus. For $x\in M$, the injectivity domain $I(x)\subset T_xM$ is the open star-shaped set of tangent vectors $v$ such that the geodesic $t\mapsto\exp_x(tv)$ is the unique minimizing geodesic from $x$ to $\exp_x(v)$ on $[0,1]$; here $\exp_x$ is the exponential map, sending a tangent vector to the point reached by the geodesic it generates. The boundary of $I(x)$ is the tangent cut locus $\mathrm{TCL}(x)$, and its image under $\exp_x$ is the cut locus of $x$, the classical object recording where geodesics from $x$ stop minimizing. On the unit tangent bundle $UM$ (the bundle of tangent vectors of norm one), the tangent cut time $t_{\mathrm{cut}}(x,v)$ records the first time at which the geodesic from $x$ with initial unit velocity $v$ stops being minimizing, so $I(x)=\{t\,v\,:\,v\in U_xM,\ 0\le t<t_{\mathrm{cut}}(x,v)\}$. The MTW tensor. Studying the regularity of the potential function of the transport (a solution of a Monge–Ampère type partial differential equation), Ma, Trudinger and Wang (2005) introduced a fourth-order curvature-type tensor associated with the cost: for $x\in M$, $v\in I(x)$ and tangent vectors $\xi,\eta\in T_xM$, the quantity $S_{(x,v)}(\xi,\eta)$ is built from the mixed fourth derivatives of $c$ along the pair of geodesic variations $t\mapsto\exp_x(t\xi)$ and $s\mapsto\exp_x(v+s\eta)$. The (MTW) sign condition requires $S_{(x,v)}(\xi,\eta)\ \ge\ 0$ whenever $x\in M$, $v\in I(x)$ and $\langle\xi,\eta\rangle_x=0$. Loeper (2009) observed that for unit orthogonal $\xi,\eta$ the value $S_{(x,0)}(\xi,\eta)$ equals the sectional curvature of the plane spanned by $\xi$ and $\eta$; hence (MTW) implies nonnegative sectional curvature, so manifolds satisfying it are quite rigid. Loeper also proved that (MTW) is necessary for regularity: if it fails, there are smooth densities $\mu,\nu$ for which the optimal map is discontinuous. The transport continuity property (TCP) is said to hold if the optimal map is continuous for every admissible pair $\mu,\nu$. Figalli–Rifford–Villani (2011) proved that TCP implies both (MTW) and convexity of all injectivity domains, that strict versions of the two conditions imply TCP, and that in dimension two TCP is equivalent to (MTW) together with convexity of injectivity domains. Convexity of injectivity domains is thus a necessary condition for transport regularity and, modulo strictness and dimension, a sufficient one; merging the two conditions into one is exactly Villani's conjecture. Focal points and nonfocality. A tangent vector $v\in T_xM$ is focal (also called conjugate) if the differential of the exponential map $\exp_x$ degenerates at $tv$ for some $t\in[0,1]$; the nonfocal domain $\mathrm{NF}(x)$ collects the $v$ with no such degeneracy, and $M$ is nonfocal if for every $x$ the tangent cut locus $\mathrm{TCL}(x)$ is contained in $\mathrm{NF}(x)$, i.e. every tangent cut velocity is itself nonfocal. Loeper–Villani (2010) proved that on a nonfocal manifold satisfying the strict inequality $S_{(x,v)}(\xi,\eta)\ge K|\xi|_x^2|\eta|_x^2$ for some $K>0$, all injectivity domains are uniformly convex (convex with a uniform quantitative bound on their boundaries). Figalli–Gallouët–Rifford (2015) dropped the strictness: on a nonfocal manifold, (MTW) alone implies convexity of all injectivity domains (Theorem 1.6 there). The proof bootstraps local convexity from radial semi-convexity estimates (semi-convexity meaning convex up to the addition of a fixed quadratic), using the extended MTW tensor — the same tensor evaluated for velocities $v$ ranging beyond $I(x)$, a device introduced by Figalli–Rifford (2009) in their study of small deformations of the round sphere. Theorem 4.1 of the 2015 paper isolates what is needed in general, without nonfocality: if for every radius $r$ one can construct a compact set $Z\subset TM$ with radial fibers, containing $I(x)\cap B_x(r+\beta)$ and contained in the closure of $\mathrm{NF}(x)$, on which (i) the extended tensor satisfies a one-sided bound $S\ge -C|\langle\xi,\eta\rangle_x||\xi|_x|\eta|_x-D\,\rho_x(v,I(x))\,|\xi|_x^2|\eta|_x^2$, where $\rho_x(v,I(x))$ is the distance from $v$ to $I(x)$, together with (ii) a quantitative focal-distance bound, and which is (iii) stable under convex combinations, then all injectivity domains are convex. The remaining obstruction. The construction of such sets $Z$ fails precisely at purely focal velocities. For $v\in\mathrm{TCL}(x)$ set $\delta(v)=\max\{|v-w|_x\,:\,w\in\mathrm{TCL}(x),\ \exp_x w=\exp_x v\}$; a cut vector $v$ is purely focal when $\delta(v)=0$, meaning the cut phenomenon at $\exp_x v$ is not caused by the meeting of several minimizing geodesics but by a focal degeneracy of the exponential map itself, and cut vectors with small $\delta(v)$ are near such points. The authors state the two outstanding tasks: giving a sign to the extended MTW tensor near purely focal points, where it can change sign, and isolating those points; both require understanding the distribution of purely focal points and the behavior of the tensor near them, which is not currently available. The same degeneracy governs the regularity of the cut time: the 2015 paper proves that $t_{\mathrm{cut}}$ is Lipschitz continuous in geodesic directions (in the line of work of Itoh–Tanaka, Li–Nirenberg and Castelpietra–Rifford on the distance function to the cut locus), globally on $UM$ under the condition $\delta(TM)>0$ (a condition that rules out purely focal velocities but is weaker than nonfocality), and in full generality in dimension two; whether $t_{\mathrm{cut}}$ is Lipschitz on $UM$ in every dimension is left open there. Villani's conjecture — that (MTW) alone implies convexity of all injectivity domains, without any nonfocality — remains the open target. Every manifold on which (MTW) is known to hold — the round spheres and their small $C^4$ deformations treated by Figalli–Rifford (2009) and Figalli–Rifford–Villani (2012), with ellipsoids of revolution analyzed separately by Bonnard–Caillau–Rifford — is treated in the literature together with nonfocality or has convexity verified directly, so the conjecture is untested by the known examples.

Problem Statement

Villani's conjecture (formulated by Villani in 2011 and stated as an open problem by Figalli–Gallouët–Rifford in 2015): Let $(M,g)$ be a smooth compact Riemannian manifold satisfying the (MTW) sign condition, $S_{(x,v)}(\xi,\eta)\ \ge\ 0\quad\text{for all }x\in M,\ v\in I(x),\ \xi,\eta\in T_xM\text{ with }\langle\xi,\eta\rangle_x=0.$ Then all injectivity domains of $M$ are convex: for every $x\in M$ the set $I(x)\subset T_xM$ — where $I(x)$ is the injectivity domain at $x$, the open star-shaped set of tangent vectors $v$ such that the geodesic $t\mapsto\exp_x(tv)$ is the unique minimizing geodesic from $x$ to $\exp_x(v)$ on $[0,1]$ — is convex. The problem is to settle this conjecture: either prove it in full generality — with no nonfocality hypothesis, no quantitative strengthening of (MTW), and no dimension, smoothness, analyticity, or symmetry restriction on the manifold, and no weakening of the conclusion (local or approximate convexity does not substitute) — or refute it by exhibiting a smooth compact Riemannian manifold satisfying (MTW) with a nonconvex injectivity domain, i.e. a point $x\in M$, velocities $v_0,v_1\in I(x)$, and $t\in(0,1)$ with $(1-t)v_0+t\,v_1\notin I(x)$. Any complete proof, or any valid counterexample, settles the problem.

References

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