Fine structure of the singular set of optimal transport maps for general costs and on Riemannian manifolds

De Philippis and Figalli (Publications Mathématiques de l'IHÉS, 2015) proved that optimal transport maps between densities bounded away from zero and infinity are smooth diffeomorphisms outside a relatively closed singular set $\Sigma$ of Lebesgue measure zero — for costs on $\mathbb{R}^n$ satisfying the structural conditions (C0)–(C3) and for the squared-distance cost $d^2/2$ on Riemannian manifolds. The theorem leaves $\Sigma$ otherwise uncharacterized. The fine structure of $\Sigma$ is a recognized open problem: it is conjectured to be countably $(n-1)$-rectifiable, with the same structure as the singular sets of gradients of convex functions (Goldman, 2019), but this is established only for the quadratic cost in the plane (Yu 2007; Figalli 2010; Chen–Li–Liu 2023 for polygonal targets) and in special multi-component target geometries (Kitagawa–McCann 2019; Chen–Liu 2022). No Hausdorff-dimension bound, rectifiability, or stratification of $\Sigma$ is known in dimension $n \ge 3$ for the quadratic cost, for general non-quadratic costs, or on manifolds; and since Loeper's theorem rules out cost conditions weaker than Ma–Trudinger–Wang (MTW) for uniform regularity, deciding emptiness is genuinely per-instance, and no non-perturbative characterization is known. The problem asks for the geometric and measure-theoretic characterization of $\Sigma$ in the full generality of the De Philippis–Figalli theorem — a structure theorem or a counterexample refuting the conjectured structure — or per-instance criteria deciding when $\Sigma$ is empty.

Background

Optimal transport asks for the cheapest way to move a probability measure with density $f$ on a domain $X$ onto a measure with density $g$ on a domain $Y$, given a transport cost $c(x,y)$. When the cost satisfies the twist condition (below), the optimizer is a single map $T$ pushing $f$ onto $g$, encoded by a potential: $T(x) = c\text{-}\mathrm{exp}_x(\nabla u(x))$ for a $c$-convex function $u$ (a supremum of functions of the form $-c(\cdot, y) + \lambda_y$), where $c\text{-}\mathrm{exp}_x$ is the inverse of the map $y \mapsto -D_xc(x,y)$. Via the change-of-variables formula, $u$ solves a Monge–Ampère-type equation (a fully nonlinear second-order PDE), so the regularity of $T$ is a question about this equation.

The regularity theory has three regimes. For the quadratic cost $|x-y|^2/2$ on $\mathbb{R}^n$, Caffarelli (1992) proved that $T$ is smooth when the target domain is convex, and that convexity cannot be dropped: for non-convex targets there are smooth, strictly positive densities for which $T$ is discontinuous. For general costs, Ma, Trudinger and Wang (2005) and Loeper (2009) identified the sharp cost-side hypothesis: the MTW condition, a fourth-order sign requirement on the cost (nonnegativity of a curvature-type tensor built from the mixed fourth derivatives of $c$), which for the manifold cost $c = d^2/2$ is strictly stronger than nonnegative sectional curvature. MTW plus convexity-type assumptions on the target yields smooth optimal maps; conversely, Loeper proved that if MTW fails at even one point, then some pair of smooth densities on well-shaped domains has a discontinuous optimal map — so no condition on the cost alone, strictly weaker than MTW, can guarantee regularity uniformly over all smooth density pairs.

Between full smoothness and discontinuity sits partial regularity. Figalli and Kim (2010) showed that for the quadratic cost, $T$ is always smooth outside a closed set of measure zero; De Philippis and Figalli (2015) proved the general statement: for costs satisfying (C0)–(C3) between bounded open sets $X, Y \subset \mathbb{R}^n$ with continuous densities bounded away from zero and infinity, and for $c = d^2/2$ on any smooth Riemannian manifold, the optimal map is a Hölder homeomorphism (a $C^{k+1,\alpha}$ diffeomorphism for smooth data) between the complements of relatively closed singular sets of Lebesgue measure zero. Here (C0) means $c \in C^2$ with bounded $C^2$-norm, (C1) and (C2) are the twist conditions (injectivity of $y \mapsto -D_xc(x,y)$ for each $x$ and of $x \mapsto -D_yc(x,y)$ for each $y$), and (C3) is nonvanishing of the determinant of the mixed Hessian $D^2_{xy}c$. The proof establishes local smoothness near points where the potential and its $c$-conjugate are twice differentiable, so the singular set is confined to the complement of that (measure-zero) region — but the argument yields no geometric information about the singular set $\Sigma$ beyond being relatively closed with measure zero: no Hausdorff-dimension bound, no rectifiability, no stratification, and no criteria deciding when $\Sigma$ is empty.

The conjectured answer, stated for instance in Goldman's 2019 lecture notes, is that $\Sigma$ should be 'much smaller': countably $(n-1)$-rectifiable — that is, covered up to sets of vanishing $(n-1)$-dimensional Hausdorff measure $\mathcal{H}^{n-1}$ by countably many Lipschitz (equivalently $C^1$) hypersurfaces — with the same structure as the singular sets of gradients of convex functions, whose non-differentiability sets are classically understood to be rectifiable of dimension at most $n-1$. Kitagawa and McCann (2019) obtained a result in this direction for MTW costs and multi-component targets. For the quadratic cost in the plane the structure is largely understood: Yifeng Yu (2007) proved the singular set is a one-dimensional $C^1$ manifold (a countable union of curves) away from a countable set when the source is convex and the target has $C^1$ boundary; Figalli (2010) extended this to general non-convex planar domains; Chen, Li and Liu (2023) proved that for non-convex polygonal targets the singular set is locally a smooth curve away from finitely many points and formulated the corresponding higher-dimensional conjecture for polytopal targets. In the opposite direction, Savin and Yu (2020) and Mooney and Rakshit (2024) prove global Sobolev ($W^{2,p}$) estimates for the potential on convex and special non-convex planar domains, which quantify the regular regime but do not describe the singular set. Nothing comparable is known in dimension $n \ge 3$, for general non-quadratic costs, or for the manifold cost $d^2/2$, where the cut locus (the set beyond which geodesics from a point stop minimizing, so that $d^2/2$ loses smoothness) and the failure of MTW add new phenomena.

Problem Statement

Setting (that of the De Philippis–Figalli theorem). Let $X, Y \subset \mathbb{R}^n$ be bounded open sets; let $f, g$ be continuous probability densities bounded away from zero and infinity on $X$ and $Y$; let $c : X \times Y \to \mathbb{R}$ satisfy (C0)–(C3) — $c \in C^2$ with bounded $C^2$-norm, the twist conditions (injectivity of $y \mapsto -D_xc(x,y)$ for every $x$ and of $x \mapsto -D_yc(x,y)$ for every $y$), and nonvanishing determinant of the mixed Hessian $D^2_{xy}c$. Alternatively, let $M$ be a smooth Riemannian manifold with cost $c = d^2/2$, where $d$ is the Riemannian distance on $M$, and densities locally bounded away from zero and infinity. Let $T$ be the unique optimal transport map, and define the singular set $\Sigma_X$ as the complement in $X$ of the largest open set on which $T$ is a local diffeomorphism; $\Sigma_X$ is relatively closed and has Lebesgue measure zero (De Philippis–Figalli).

Objective. Characterize the geometry and measure-theoretic structure of $\Sigma_X$ in this generality, and determine when it is empty. Specifically:

(a) Prove or disprove that $\Sigma_X$ is countably $(n-1)$-rectifiable — covered, up to sets of vanishing $(n-1)$-dimensional Hausdorff measure, by countably many $C^1$ hypersurfaces — or at least that its Hausdorff dimension is at most $n-1$; a stronger result would be the finer stratification conjectured by Chen, Li and Liu for polytopal targets (locally a smooth hypersurface away from a set of Hausdorff dimension at most $n-2$), in the general setting of the theorem.

(b) Give a sharp per-instance characterization of when $\Sigma_X$ is empty: necessary and sufficient conditions on the cost, the domains, and the density pair for the optimal map to be globally smooth — recognizing, by Loeper's necessity theorem, that no condition on the cost alone strictly weaker than the Ma–Trudinger–Wang (MTW) condition can guarantee emptiness for all smooth density pairs.

Components (a) and (b) are two readings of one objective — a complete description of the exceptional set of the De Philippis–Figalli partial-regularity theorem. The problem asks for: a structure theorem for $\Sigma_X$ in the theorem's generality (a proof of rectifiability, an optimal dimension bound, or a stratification); or a counterexample within the admissible-data class whose singular set violates the conjectured structure (Hausdorff dimension strictly larger than $n-1$, non-rectifiability, or non-σ-finite $(n-1)$-dimensional Hausdorff measure); or, for (b), the sharp per-instance criteria deciding when $\Sigma_X$ is empty.

References

1. Guido De Philippis and Alessio Figalli, 'Partial regularity for optimal transport maps', Publications Mathématiques de l'IHÉS 121 (2015), 81–112. DOI: 10.1007/s10240-014-0064-7, https://doi.org/10.1007/s10240-014-0064-7 (preprint: arXiv:1209.5640, https://arxiv.org/abs/1209.5640)
2. Luis A. Caffarelli, 'The regularity of mappings with a convex potential', Journal of the American Mathematical Society 5 (1992), 99–104. DOI: 10.1090/S0894-0347-1992-1124980-8, https://doi.org/10.1090/S0894-0347-1992-1124980-8
3. X.-N. Ma, N. S. Trudinger, and X.-J. Wang, 'Regularity of Potential Functions of the Optimal Transportation Problem', Archive for Rational Mechanics and Analysis 177 (2005), 151–183. DOI: 10.1007/s00205-005-0362-9, https://doi.org/10.1007/s00205-005-0362-9
4. Grégoire Loeper, 'On the regularity of solutions of optimal transportation problems', Acta Mathematica 202 (2009), 241–283. DOI: 10.1007/s11511-009-0037-8, https://doi.org/10.1007/s11511-009-0037-8
5. Alessio Figalli and Young-Heon Kim, 'Partial regularity of Brenier solutions of the Monge–Ampère equation', Discrete and Continuous Dynamical Systems 28 (2010), no. 2, 559–565. DOI: 10.3934/dcds.2010.28.559, https://doi.org/10.3934/dcds.2010.28.559
6. Yifeng Yu, 'Singular Set of a Convex Potential in Two Dimensions', Communications in Partial Differential Equations 32 (2007), 1883–1894. DOI: 10.1080/03605300701318757, https://doi.org/10.1080/03605300701318757
7. Alessio Figalli, 'Regularity Properties of Optimal Maps Between Nonconvex Domains in the Plane', Communications in Partial Differential Equations 35 (2010), 465–479. DOI: 10.1080/03605300903307673, https://doi.org/10.1080/03605300903307673
8. Shibing Chen and Alessio Figalli, 'Partial $W^{2,p}$ regularity for optimal transport maps', Journal of Functional Analysis 272 (2017), 4588–4605. DOI: 10.1016/j.jfa.2017.02.025, https://doi.org/10.1016/j.jfa.2017.02.025 (preprint: arXiv:1606.05173, https://arxiv.org/abs/1606.05173)
9. Michael Goldman, 'A variational approach to regularity theory in optimal transportation', lecture notes, arXiv:1907.05627 (2019), https://arxiv.org/abs/1907.05627
10. Michael Goldman and Felix Otto, 'A variational proof of partial regularity for optimal transportation maps', Annales scientifiques de l'École normale supérieure (5) 53 (2020), no. 5, 1209–1233. DOI: 10.24033/ASENS.2444, https://doi.org/10.24033/ASENS.2444 (preprint: arXiv:1704.05339, https://arxiv.org/abs/1704.05339)
11. Jun Kitagawa and Robert McCann, 'Free discontinuities in optimal transport', Archive for Rational Mechanics and Analysis 232 (2019), 1505–1541. DOI: 10.1007/s00205-018-01348-3, https://doi.org/10.1007/s00205-018-01348-3 (preprint: arXiv:1708.04152, https://arxiv.org/abs/1708.04152)
12. Shibing Chen, Jiakun Liu, and Xu-Jia Wang, 'Global regularity of optimal mappings in non-convex domains', Science China Mathematics 62 (2019), 2057–2072. DOI: 10.1007/s11425-018-9465-8, https://doi.org/10.1007/s11425-018-9465-8 (preprint: arXiv:1901.10215, https://arxiv.org/abs/1901.10215)
13. Yash Jhaveri, 'Partial regularity of solutions to the second boundary value problem for generated Jacobian equations', Methods and Applications of Analysis 24 (2017), 445–476. DOI: 10.4310/MAA.2017.V24.N4.A2, https://doi.org/10.4310/MAA.2017.V24.N4.A2 (preprint: arXiv:1609.09680, https://arxiv.org/abs/1609.09680)
14. Ovidiu Savin and Hui Yu, 'Regularity of optimal transport between planar convex domains', Duke Mathematical Journal 169 (2020), 1305–1327. DOI: 10.1215/00127094-2019-0068, https://doi.org/10.1215/00127094-2019-0068 (preprint: arXiv:1806.06252, https://arxiv.org/abs/1806.06252)
15. Felix Otto, Maxime Prod'homme, and Tobias Ried, 'Variational approach to regularity of optimal transport maps: general cost functions', arXiv:2007.11549 (2020), https://arxiv.org/abs/2007.11549
16. Tatsuya Miura and Felix Otto, 'Sharp boundary ε-regularity of optimal transport maps', Advances in Mathematics 381 (2021), 107603. DOI: 10.1016/j.aim.2021.107603, https://doi.org/10.1016/j.aim.2021.107603
17. Yash Jhaveri and Ovidiu Savin, 'On the Regularity of Optimal Transports Between Degenerate Densities', Archive for Rational Mechanics and Analysis 245 (2022), 819–861. DOI: 10.1007/s00205-022-01796-y, https://doi.org/10.1007/s00205-022-01796-y (preprint: arXiv:2105.04312, https://arxiv.org/abs/2105.04312)
18. Connor Mooney and Arghya Rakshit, 'Sobolev Regularity for Optimal Transport Maps of Nonconvex Planar Domains', SIAM Journal on Mathematical Analysis 56 (2024), 4742–4758. DOI: 10.1137/23m1582436, https://doi.org/10.1137/23m1582436 (preprint: arXiv:2306.08139, https://arxiv.org/abs/2306.08139)
19. Shibing Chen and Jiakun Liu, 'Regularity of singular set in optimal transportation', preprint, arXiv:2210.13841 (2022, latest version July 2025), https://arxiv.org/abs/2210.13841
20. Shibing Chen, Yuanyuan Li, and Jiakun Liu, 'Optimal (partial) transport to non-convex polygonal domains', preprint, arXiv:2311.15655 (2023, latest version 2025), https://arxiv.org/abs/2311.15655
21. Michael Goldman and Lukas Koch, 'Partial regularity for optimal transport with p-cost away from fixed points', arXiv:2407.08846 (2024), https://arxiv.org/abs/2407.08846
22. Gero Friesecke and Tobias Ried, 'Partial regularity of optimal transport with Coulomb cost', Calculus of Variations and Partial Differential Equations 65 (2026), no. 7, article 220. DOI: 10.1007/s00526-026-03386-2, https://doi.org/10.1007/s00526-026-03386-2 (preprint: arXiv:2508.01756, https://arxiv.org/abs/2508.01756)
23. Micah Warren, 'A quantitative stability result for regularity of optimal transport on compact manifolds', Calculus of Variations and Partial Differential Equations 65 (2026), no. 7, article 214. DOI: 10.1007/s00526-026-03367-5, https://doi.org/10.1007/s00526-026-03367-5
24. Guido De Philippis and Alessio Figalli, 'The Monge–Ampère equation and its link to optimal transportation', Bulletin of the American Mathematical Society 51 (2014), 527–580. DOI: 10.1090/S0273-0979-2014-01459-4, https://doi.org/10.1090/S0273-0979-2014-01459-4
25. Ovidiu Savin and Hui Yu, 'Regularity of the singular set in the fully nonlinear obstacle problem', arXiv:1905.02308 (2019), https://arxiv.org/abs/1905.02308