Convexity of injectivity domains from the Ma–Trudinger–Wang condition in the focal case, and the attendant stability and second-order structure questions

On a smooth compact Riemannian manifold, the Ma–Trudinger–Wang (MTW) condition is a fourth-order curvature inequality: writing $\mathfrak S(x,v)(\xi,\eta)$ for the MTW tensor, a covariant tensor built from the mixed fourth derivative of the quadratic cost $c(x,y)=d(x,y)^2/2$ along geodesics, the condition MTW(0) requires $\mathfrak S(x,v)(\xi,\eta)\ge 0$ for every point $x$, every vector $v$ in the injectivity domain $I(x)\subset T_xM$ (the largest star-shaped open set around the origin on which the exponential map $\exp_x$ is a diffeomorphism), and every pair $\xi,\eta$ of orthogonal tangent vectors at $x$. Villani conjectured that every injectivity domain of a smooth compact manifold satisfying MTW(0) is convex. The nonfocal branch is resolved: Figalli–Gallouët–Rifford (2015) proved that MTW(0) plus nonfocality (no tangent cut point is focal) implies convexity of all injectivity domains. The open core concerns the focal case — whether convexity persists when the tangent cut locus contains focal points, where the change-of-variables and convexity machinery degenerates because $D\exp_x$ becomes singular (on the round sphere the entire tangent cut locus is focal) — together with the quantitative ($\kappa$-uniform) version, the general focal-case stability of MTW under $C^4$ perturbations of the metric, and the second-order structure (semiconvexity of $I(x)$, semiconcavity of the cut-time function on the unit tangent bundle).

Background

Optimal transport on a Riemannian manifold $M$ asks to move one probability measure to another while minimizing a cost; for the quadratic cost $c(x,y)=\frac{1}{2}d(x,y)^2$ (with $d$ the geodesic distance), the regularity of the optimal map is governed by the interaction between curvature and the global geometry of geodesics. The Ma–Trudinger–Wang (MTW) tensor is the covariant tensor $\mathfrak S(x,v)(\xi,\eta)$ obtained from the mixed fourth derivative of the cost along geodesics: for $v\in T_xM$ it is built from the curvature tensor along the geodesic $t\mapsto\exp_x(tv)$, and at $v=0$ it reduces, on orthogonal unit pairs, to the sectional curvature. The condition MTW($\kappa$) requires $\mathfrak S(x,v)(\xi,\eta)\ge\kappa|\xi|^2|\eta|^2$ for all $x\in M$, all $v\in I(x)$, and all $\xi,\eta\in T_xM$ with $\langle\xi,\eta\rangle_x=0$; Loeper proved that MTW(0) is necessary for continuity of optimal maps, in the strong sense that wherever $\mathfrak S(x,v)(\xi,\eta)<0$ at an admissible $(x,v)$ on an orthogonal pair, one can construct smooth positive densities between which the optimal map is discontinuous. A second, geometric ingredient controls the domain on which the cost is smooth: the injectivity domain $I(x)\subset T_xM$ is the largest star-shaped open set containing the origin on which $\exp_x$ is a diffeomorphism, equivalently $v\in I(x)$ precisely when there exists $t>1$ such that $d(x,\exp_x(tv))=|tv|_x$; its boundary is the tangent cut locus $\mathrm{TCL}(x)$, and points where $D\exp_x$ becomes singular are called focal (conjugate) points; the manifold is nonfocal when $\mathrm{TCL}(x)$ is contained in the nonfocal domain $\mathrm{NF}(x)$, the set of vectors $v$ such that $D\exp_x$ is nonsingular along the whole segment $[0,1]v$. Loeper–Villani (Duke Math. J. 2010) proved that a strict MTW condition (MTW($K$) with $K>0$) together with nonfocality implies that all injectivity domains are uniformly convex, and they posed in their final section and appendices the natural open problems: does plain MTW(0) imply convexity of injectivity domains; what happens when focal points are present; how stable is the MTW class under $C^4$ perturbations of the metric; and what second-order regularity (semiconvexity of $I(x)$ in the sense of sets whose boundary is locally a graph of semiconvex functions, semiconcavity of the cut-time function on the unit tangent bundle) the cut locus enjoys. Figalli–Rifford–Villani introduced the transport continuity property (TCP: optimal maps between absolutely continuous measures with densities bounded away from zero and infinity are continuous) and proved that TCP implies MTW(0) together with convexity of all injectivity domains in any dimension, and that on surfaces the three properties are equivalent; in dimension at least 3 whether MTW(0) plus convexity of injectivity domains implies TCP remains open, making the convexity of $I(x)$ the key structural unknown. The round sphere satisfies a strong form of MTW and has uniformly convex injectivity domains, yet its entire tangent cut locus consists of focal points — showing that focal behavior is not pathological but generic at the boundary, and that any proof must handle the degeneracy of the change of variables $y\mapsto\nabla_x c(x,y)$ where $D\exp_x$ is singular.

Problem Statement

Let $M$ be a smooth compact Riemannian manifold satisfying MTW(0), i.e. $\mathfrak S(x,v)(\xi,\eta)\ge 0$ for every $x\in M$, every $v\in I(x)$, and every $\xi,\eta\in T_xM$ with $\langle\xi,\eta\rangle_x=0$, where $I(x)\subset T_xM$ is the injectivity domain of $x$ (the largest star-shaped open set around the origin on which $\exp_x$ is a diffeomorphism; equivalently $v\in I(x)$ precisely when there exists $t>1$ such that $d(x,\exp_x(tv))=|tv|_x$). Villani's conjecture, formulated in his 2011 survey and posed in focal form by Loeper–Villani (2010), asks: are all injectivity domains $I(x)$ convex? The resolved nonfocal branch aside, the open core consists of: (1) the focal case — prove or disprove that MTW(0) implies convexity of every injectivity domain when the tangent cut locus contains focal points, i.e. without any nonfocality assumption; (2) the quantitative version — determine whether, when the tangent cut locus contains focal points, MTW(0) alone yields $\kappa$-uniform convexity (a uniform interior-ball condition) of the injectivity domains, or whether the known strict-MTW/round/nearly-round hypotheses are genuinely necessary; (3) the general focal-case stability of the MTW class under $C^4$ perturbations of the metric — decide whether the strict MTW condition is stable, i.e. whether a metric satisfying MTW($\kappa$) with $\kappa>0$ continues to satisfy MTW($\kappa'$) for a comparable $\kappa'>0$ after any sufficiently small $C^4$ perturbation of the metric; and (4) the second-order structure — decide whether injectivity domains are semiconvex sets and whether the cut-time function on the unit tangent bundle is semiconcave in the presence of focal points. Partial (nonfocal, nearly-round, or conditional) resolutions of these components exist, and no complete answer to any of them is known. The four components are independently sufficient: a complete answer to any one of them — a proof resolving it or an explicit counterexample refuting it — resolves the task.

References

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