Fine regularity and the singular set of optimal transport maps for the squared-distance cost on Riemannian manifolds

Figalli's 2007 SIAM Journal on Mathematical Analysis paper 'Existence, Uniqueness, and Regularity of Optimal Transport Maps' proved existence, uniqueness, and almost-everywhere regularity of optimal transport maps for the squared-distance cost on possibly noncompact Riemannian manifolds, together with displacement convexity of internal energies under nonnegative Ricci curvature. Although the paper states no open problems, its actual limitations define a three-direction program: finer regularity beyond approximate differentiability; description of the exceptional cut-locus and conjugate-point sets; and displacement convexity beyond the Ricci-nonnegative case. The third direction is subsumed by the curvature-dimension theory (von Renesse-Sturm; Sturm; Lott-Villani; Figalli-Villani), and the first two are reshaped by the counterexamples of Loeper and Kim, which rule out global regularity under minimal hypotheses, and then substantially resolved for positive continuous densities by partial regularity theorems (Figalli-Gigli; De Philippis-Figalli; Chen-Figalli). The surviving open core - matching the open problems listed in the 2014 Bulletin of the AMS survey of De Philippis and Figalli - is the structure of the exceptional set: the conjectured Hausdorff-dimension bound $\dim_H\Sigma\le n-1$ and rectifiability of the singular sets; partial regularity under rough (unbounded or merely absolutely continuous) densities; the stay-away property asserting that the optimal map avoids the cut locus everywhere rather than almost everywhere; and the characterization of continuity of optimal maps in all dimensions by the MTW condition together with convexity of tangent cut-loci. The problem is open.

Background

The Monge-Kantorovich optimal transport problem asks to move one probability measure onto another at minimal total cost, where the cost of moving a unit of mass from $x$ to $y$ is a prescribed function $c(x,y)$. Kantorovich's relaxation allows mass splitting through transport plans (probability measures on the product space with the prescribed marginals), and its duality produces Kantorovich potentials - a pair of functions with $\varphi(x)+\psi(y)\le c(x,y)$ everywhere and equality on the support of an optimal plan. A set is $c$-cyclically monotone if no finite cycling of its points decreases the total cost; for regular costs, optimal plans are characterized by this property. On Euclidean space with the quadratic cost $c(x,y)=|x-y|^2/2$, Brenier's polar factorization theorem identifies the optimal plan, when the source measure is absolutely continuous (i.e. has a density), with the gradient map of a convex function; the potential then formally solves a Monge-Ampere equation - the nonlinear second-order PDE linking the source and target densities through the determinant of the potential's Hessian.

On a complete Riemannian manifold $M$ of dimension $n$ (a smooth space whose tangent spaces carry inner products and whose distances are measured along geodesics, its locally length-minimizing curves), with geodesic distance $d$, the natural analogue is the squared-distance cost $c(x,y)=d^2(x,y)/2$. Here the geometry interferes with smoothness: $d^2/2$ fails semiconcavity and semiconvexity (one-sided second-order bounds on a function) exactly at the cut locus - the set of pairs $(x,y)$ for which a minimizing geodesic from $x$ to $y$ stops being uniquely minimizing or infinitely extendable - and the exponential map (sending a tangent vector at $x$ to the point reached after unit time along the corresponding geodesic) becomes singular at conjugate points, where nearby geodesics refocus and the volume distortion coefficient $v_t(x,y)$ (the $t$-weighted comparison between geodesically interpolated and flat volume along the geodesic from $x$ to $y$) takes the value $+\infty$. Figalli's paper, 'Existence, Uniqueness, and Regularity of Optimal Transport Maps', SIAM J. Math. Anal. 39 (2007) 126-137, proves, on a possibly noncompact $M$ and without any integrability assumption on the cost: (a) from the $c$-cyclical monotonicity of a plan alone, assuming only that the source measure gives zero mass to sets of $\sigma$-finite $(n-1)$-dimensional Hausdorff measure (so it charges no hypersurface-like sets, the higher-dimensional analogue of a planar measure charging no curve), that the plan is concentrated on the graph of a unique measurable transport map $T$ (building on companion work with Fathi); (b) for $c=d^2/2$, uniqueness of the Wasserstein geodesic - the constant-speed shortest curve in the metric on probability measures defined by optimizing $\int d^2\,d\gamma$ over couplings $\gamma$ - between absolutely continuous measures, the interpolation being $T_t=\exp[-t\tilde\nabla\varphi]$, where $\tilde\nabla$ is the approximate gradient (a differential defined on a full-measure set through density points, in the sense of geometric measure theory); (c) almost-everywhere approximate differentiability of $T$ and an a.e. Jacobian identity relating the densities $f,g$ of the two measures through $\det(Y(H-\tilde\nabla^2\varphi))$, where $Y$ is the differential of the exponential map along the transport and $H=\frac12\nabla_x^2d^2_{T(x)}$; (d) a Brunn-Minkowski-type Jacobian inequality along the interpolation involving the volume ratios $v_t$; and (e) displacement convexity - convexity of functionals along Wasserstein geodesics - of the internal energies $U(\rho)=\int A(\rho)\,d\mathrm{vol}$ on manifolds with nonnegative Ricci curvature ($\mathrm{Ric}\ge0$; the trace of the curvature tensor, controlling infinitesimal volume distortion), for integrands $A$ satisfying the condition that $s\mapsto s^nA(s^{-n})$ is convex and nonincreasing, introduced by McCann. These results extended the compact-manifold theory of Cordero-Erausquin, McCann and Schmuckenschlager to noncompact manifolds.

The paper itself states no open problems; later work synthesized its actual limitations into a three-direction program: (i) finer regularity of the potential and of $T$ beyond approximate differentiability and a.e. second-order Taylor expansions; (ii) description of the exceptional sets where the transport meets the cut locus or the conjugate locus; (iii) displacement convexity beyond $\mathrm{Ric}\ge0$ and beyond McCann's integrand class. The subsequent literature settled (iii) and reshaped (i)-(ii): global regularity is impossible under minimal hypotheses, so the well-posed targets are partial regularity and the structure of the exceptional (singular) set.

Direction (iii) was subsumed by the curvature-dimension theory. von Renesse and Sturm characterized $\mathrm{Ric}\ge K$ through convexity of the entropy along Wasserstein geodesics; Sturm and Lott-Villani built the synthetic curvature-dimension condition $\mathrm{CD}(K,N)$ (a lower Ricci-curvature bound $K$ combined with an upper dimension bound $N$, expressed purely through displacement convexity of entropy-type functionals on metric-measure spaces) for every real $K$ and on possibly noncompact spaces; the displacement-convexity classes $DC_N$ of Lott-Villani coincide exactly with McCann's condition, i.e. with condition (9) of the source paper, and their appendix treats the noncompact case. Figalli and Villani upgraded the resulting weak (single-geodesic) convexity to strong convexity, closing a question Lott-Villani had left open. The standard synthesis is Villani's book 'Optimal Transport: Old and New'.

For directions (i)-(ii), Loeper proved that continuity of the optimal map for all smooth positive data requires a structural condition on the cost - nonnegativity of its cost-sectional curvature, the MTW condition (named after Ma, Trudinger, and Wang: a positivity hypothesis on a curvature-like quantity built from the fourth-order derivatives of the cost; for $c=d^2/2$ it implies nonnegative sectional curvature) - so that global continuity fails on every manifold carrying a negatively curved tangent plane; Kim moreover constructed discontinuous optimal maps even on manifolds of everywhere positive sectional curvature. The corrected target is therefore partial regularity: Figalli and Gigli proved local semiconvexity of Kantorovich potentials on noncompact manifolds without curvature assumptions; De Philippis and Figalli proved that for continuous densities locally bounded away from zero and infinity the optimal map is a local homeomorphism (a diffeomorphism for smooth densities) outside two closed singular sets $\Sigma_X,\Sigma_Y$ of measure zero; Chen and Figalli obtained partial $W^{2,p}$ regularity of the potential. The measure-theoretic smallness of the cut and conjugate loci themselves is classical (semiconcavity of $d^2/2$; integrality of the Hausdorff dimension of a cut locus, Hausdorff dimension being the covering-based notion of dimension that allows non-integer values, due to Itoh and Tanaka). What remains open - explicitly listed as open problems in the 2014 Bulletin of the AMS survey of De Philippis and Figalli - is the fine structure of the singular set, the stay-away property of the transport from the cut locus, the all-dimensional characterization of continuity, and partial regularity under minimal density hypotheses.

Problem Statement

Let $M$ be a smooth complete Riemannian manifold of dimension $n$ (possibly noncompact) with geodesic distance $d$ and volume measure $\mathrm{vol}$, and let $T$ be the optimal transport map for the cost $c(x,y)=d^2(x,y)/2$ between two Borel probability measures absolutely continuous with respect to $\mathrm{vol}$, with densities $f$ and $g$. Terminology: the singular set is the complement of the set where $T$ is a local diffeomorphism; the Hausdorff dimension $\dim_H$ is the covering-based notion of dimension allowing non-integer values; a set is $(n-1)$-rectifiable if it is contained, up to a null set, in a countable union of Lipschitz images of subsets of $\mathbb{R}^{n-1}$; $C^{k+1,\alpha}$ means $(k+1)$ times continuously differentiable with $\alpha$-Hölder continuous highest derivatives; $W^{2,p}$ is the Sobolev space of functions with two weak derivatives in $L^p$; the tangent cut-locus at a point is the set of tangent vectors whose geodesics stop being minimizing immediately beyond them.

The common objective is to complete the fine regularity theory of $T$ and of its Kantorovich potential beyond the almost-everywhere baseline of approximate differentiability: a full description of the exceptional set where regularity fails, and of the minimal hypotheses under which regularity holds. The interlinked open components are:

(1) Size and structure of the singular set. For the closed singular sets $\Sigma_X,\Sigma_Y$ of measure zero produced by the partial regularity theorem of De Philippis and Figalli, prove or disprove the conjecture $\dim_H\Sigma_X\le n-1$ and $\dim_H\Sigma_Y\le n-1$, and determine whether the singular sets are $(n-1)$-rectifiable (or piecewise smooth) under reasonable hypotheses on $f,g$. This is open already for the quadratic cost $|x-y|^2/2$ on $\mathbb{R}^n$ when the target support is nonconvex.

(2) Minimal density hypotheses. Extend the partial regularity theorem - $T$ a homeomorphism/diffeomorphism outside a closed null singular set, with the corresponding $C^{k+1,\alpha}$ and $W^{2,p}$ statements - from continuous densities locally bounded away from zero and infinity to rougher data (unbounded densities, or merely absolutely continuous measures), or exhibit by explicit construction data for which the singular set has positive measure or the regularity conclusions fail.

(3) Stay-away from the cut locus. Decide whether $T(x)\notin\mathrm{cut}(x)$ for every $x$ (not merely for almost every $x$), i.e. whether the optimal transport keeps away from the cut locus; this is known only in special cases, and it is the property that would allow the Euclidean interior regularity theory for the Monge-Ampere equation to be localized to Riemannian manifolds.

(4) Characterization of continuity in all dimensions. Prove or disprove that the MTW condition together with convexity of the tangent cut-loci is necessary and sufficient for continuity of optimal transport maps (for all smooth positive densities) on every Riemannian manifold; sufficiency is currently known only in dimension two, by Figalli, Rifford and Villani.

The four components are logically independent open problems, each with its own refutation condition, and together they constitute the common objective. Within component (1) an $(n-1)$-rectifiable set has Hausdorff dimension at most $n-1$, so the rectifiability demand is the stronger of the two.

The displacement-convexity direction of the original program (extension to $\mathrm{Ric}\ge K$ with $K\ne0$ and to integrands beyond McCann's class) is excluded from the open core, having been subsumed by the curvature-dimension theory of von Renesse-Sturm, Sturm, and Lott-Villani.

References

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