This problem concerns the three open questions posed in the concluding comments of Figalli–Rifford–Villani's characterization of the continuity of optimal transport maps on compact Riemannian manifolds (Tohoku Math. J. 63 (2011) 855–876). For the cost given by half the squared geodesic distance, the transport continuity property (TCP) — continuity of the unique optimal map for all pairs of measures with densities bounded away from zero and infinity — was there shown to imply convexity of injectivity domains (CI) together with nonnegativity of the Ma–Trudinger–Wang (MTW) curvature tensor on orthogonal vector pairs, and to follow from the strict variants (SCI)+(MTW+) in every dimension; on surfaces, TCP is equivalent to (CI)+(MTW). The surviving open components are: (i) whether this equivalence extends to dimension three and higher; (ii) whether Riemannian products of small C4C^4 perturbations of the round 2-sphere satisfy TCP — such products satisfy the orthogonality-free condition (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) by results of Delanoë–Ge and Kim–McCann, yet fall outside all known sufficient conditions; and (iii) whether a Riemannian manifold can satisfy (MTW) without (MTW⊥̸)(\mathrm{MTW}^{\not\perp}), no such example being known.

Background

The Monge–Kantorovich optimal transport problem asks, for two probability measures μ\mu and ν\nu on a common space, for a measurable map TT pushing μ\mu forward onto ν\nu (that is, T#μ=νT_\#\mu = \nu) which minimizes the total transport cost ∫c(x,T(x)) dμ(x)\int c(x, T(x))\,d\mu(x). On a smooth compact connected Riemannian manifold (M,g)(M,g) with geodesic distance dd, the natural cost is c(x,y)=d(x,y)2/2c(x,y) = d(x,y)^2/2, half the squared distance. McCann (2001) proved that whenever μ\mu is absolutely continuous with respect to the Riemannian volume, the problem has a unique optimal map, of the form T(x)=exp⁡x(∇ψ(x))T(x) = \exp_x(\nabla\psi(x)) for a cc-convex potential ψ\psi (a notion replacing ordinary convexity, adapted to the cost cc). Whether this map is continuous — the analogue on manifolds of the Euclidean regularity theory of Brenier, Caffarelli and Urbas — depends on the curvature of (M,g)(M,g), and characterizing that dependence is the subject of this problem.

Two geometric structures govern the regularity. First, for x∈Mx \in M the injectivity domain I(x)⊂TxMI(x) \subset T_xM is the set of tangent vectors vv such that the geodesic t↦exp⁡x(tv)t \mapsto \exp_x(tv), t∈[0,1]t \in [0,1], is the unique minimizing geodesic from xx to exp⁡x(v)\exp_x(v); (M,g)(M,g) satisfies (CI) (respectively (SCI)) if I(x)I(x) is convex (respectively strictly convex) for every xx. Second, the Ma–Trudinger–Wang (MTW) tensor, a fourth-order curvature quantity introduced in the regularity theory of the Monge–Ampère-type equation for optimal transport, is defined for v∈I(x)v \in I(x) and tangent vectors ξ,η∈TxM\xi, \eta \in T_xM by

S(x,v)(ξ,η)  =  −32 ∂2∂s2∣s=0∂2∂t2∣t=0  c(exp⁡x(t ξ), exp⁡x(v+s η)).\mathfrak{S}_{(x,v)}(\xi,\eta) \;=\; -\frac{3}{2}\,\frac{\partial^2}{\partial s^2}\bigg|_{s=0}\frac{\partial^2}{\partial t^2}\bigg|_{t=0}\; c\big(\exp_x(t\,\xi),\, \exp_x(v+s\,\eta)\big).

The manifold satisfies (MTW) if S(x,v)(ξ,η)≥0\mathfrak{S}_{(x,v)}(\xi,\eta) \ge 0 whenever ⟨ξ,η⟩x=0\langle \xi, \eta \rangle_x = 0 (and (MTW+) if equality then forces ξ=0\xi = 0 or η=0\eta = 0), and satisfies the a priori stronger (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) if S(x,v)(ξ,η)≥0\mathfrak{S}_{(x,v)}(\xi,\eta) \ge 0 for all (ξ,η)(\xi,\eta) with no orthogonality restriction — in Kim–McCann's terminology, nonnegative cross-curvature (NNCC). At v=0v = 0 the tensor reduces to a positive multiple of the sectional curvature of the plane spanned by ξ,η\xi,\eta (with the sign convention in which the round sphere has positive curvature), so (MTW) already forces nonnegative sectional curvature — Loeper's maximum principle, which also shows that (MTW) is necessary for any continuity theory: if the optimal map is continuous for all smooth positive densities, then (MTW) must hold.

A compact manifold (M,g)(M,g) satisfies the transport continuity property (TCP) if for every pair of probability measures with densities bounded above and below by positive constants, the optimal map TT is continuous. Figalli–Rifford–Villani (2011) proved, in every dimension: (Theorem 1.2(i)) TCP implies (CI) and (MTW); and (Theorem 1.2(ii)) the strict variants (SCI) and (MTW+) together imply TCP. On surfaces they further proved (Theorem 1.3) that TCP is equivalent to (CI) and (MTW). The two-dimensional argument combines two facts with no known higher-dimensional analogue: the Figalli–Loeper theorem that in dimension two continuity follows from (MTW), and the fact that, on a surface, the tangent focal locus of a point — the set of tangent vectors at which the differential of the exponential map first degenerates along each direction — is a one-dimensional curve whose curvature must be nonnegative when (MTW) holds. The catalogue of manifolds known to satisfy TCP is correspondingly narrow — flat tori, round spheres and their small C4C^4 deformations, Riemannian submersions of round spheres (a Riemannian submersion being a submersion π:M~→M\pi: \tilde M \to M whose differential restricts to a linear isometry on the directions orthogonal to its kernel), products of round spheres and their quotients by discrete isometry groups, compact quotients of products of spheres with Rn\mathbb{R}^n whose cut locus is nonfocal, that is, contains no focal point of the exponential map (the result for these quotients uses strict extended MTW conditions), and compact surfaces with (CI) and (MTW) — every example being either strict, symmetric, two-dimensional, or focal-free.

Two later developments sharpen the picture without closing the gap. Lebedeva–Petrunin–Zolotov (2019) recast the conditions in synthetic comparison geometry: their quadruple-point “4(1)-tree comparison” is equivalent to convexity of tangent injectivity domains together with (MTW⊥̸)(\mathrm{MTW}^{\not\perp}), and a strict “spherical” strengthening of it implies TCP — a sufficiency condition of the same strength as (SCI)+(MTW+). Kim–McCann (2012) proved that nonnegative cross-curvature is stable under Riemannian products and submersions. This yields a natural family of manifolds — products of small C4C^4 perturbations of the round sphere S2S^2 — which satisfy (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) (hence (MTW)) and (CI), yet are not covered by any known sufficient condition; whether they satisfy TCP is open, and is the concrete test instance of the general sufficiency question. At the same time, no Riemannian manifold is known to satisfy (MTW) without (MTW⊥̸)(\mathrm{MTW}^{\not\perp}), so the exact strength of the orthogonality restriction in the definition of (MTW) is itself undetermined. The problem below is the conjunction of these three open components.

Problem Statement

The problem is the set of three open questions posed by Figalli–Rifford–Villani in the concluding comments (Section 2.3) of the source paper. All three share a single objective — closing the gap between the necessary and sufficient conditions for transport continuity on compact Riemannian manifolds — and they are scientifically linked: a resolution of the problem addresses all three.

(1) Higher-dimensional characterization. Let (M,g)(M,g) be a smooth compact connected Riemannian manifold of dimension n≥3n \ge 3. Does (CI) together with (MTW) imply TCP? Since TCP implies (CI) and (MTW) in every dimension (Theorem 1.2(i) of the source), an affirmative answer is exactly the extension to n≥3n \ge 3 of the surface equivalence TCP   ⟺  \iff (CI)+(MTW) (Theorem 1.3 of the source). A negative answer consists of a compact manifold of dimension n≥3n \ge 3 satisfying (CI) and (MTW), together with two densities bounded away from zero and infinity whose optimal transport map is discontinuous. The question is also posed with (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) in place of (MTW) (Lebedeva–Petrunin–Zolotov, Question 8.7); since (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) is the stronger hypothesis, that variant is a weaker implication unless the two conditions are equivalent — the subject of component (3). In either form, any candidate must by Loeper's maximum principle have nonnegative sectional curvature.

(2) Transport continuity on products of small perturbations of the sphere. Let M=M1×⋯×MkM = M_1 \times \cdots \times M_k be a Riemannian product, each factor (Mi,gi)(M_i, g_i) a small C4C^4 perturbation of the round sphere S2S^2 (that is, gig_i is a smooth metric on S2S^2 lying within a sufficiently small C4C^4-neighbourhood of the round metric — the regime in which the results of Figalli–Rifford and Delanoë–Ge apply). Each factor satisfies (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) (Delanoë–Ge), hence so does the product (Kim–McCann); each factor satisfies TCP and (CI) (Figalli–Rifford; Figalli–Rifford–Villani), and since the injectivity domain of a Riemannian product is the product of the injectivity domains of its factors, the product satisfies (CI). Does MM satisfy TCP? This is the canonical test case of component (1) in the (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) regime: the simplest class beyond products of round spheres (where TCP holds, by Figalli–Kim–McCann), obtained by deforming the factors rather than the product metric. The strict condition (MTW+) of the known sufficient theorem is not available here, so the known theorems do not apply.

(3) Strength of the orthogonality restriction. Does there exist a Riemannian manifold (compact or not) satisfying (MTW) but not (MTW⊥̸)(\mathrm{MTW}^{\not\perp})? Or are (MTW) and (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) equivalent for the squared-distance cost d2/2d^2/2, so that the orthogonality restriction in the definition of (MTW) is immaterial? For general (non-Riemannian) cost functions the analogous conditions are known to differ; the question concerns precisely the compatibility of the two sign conditions with Riemannian structure.

Dependencies among components: a positive resolution of (1) implies a positive resolution of (2), as products of small C4C^4 perturbations of S2S^2 satisfy (CI) and (MTW). A positive resolution of the existence question in (3) — a manifold satisfying (MTW) but not (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) — would show that the class of manifolds to which the sufficiency question (1) meaningfully applies is strictly larger than the (MTW⊥̸)(\mathrm{MTW}^{\not\perp}) class, and would recalibrate the curvature hypothesis in (2). A complete resolution of the problem addresses all three components; partial resolutions of individual components are meaningful intermediate results.

References

  1. A. Figalli, L. Rifford, C. Villani, “Necessary and sufficient conditions for continuity of optimal transport maps on Riemannian manifolds”, Tohoku Mathematical Journal 63 (2011), no. 4, 855–876. DOI: 10.2748/tmj/1325886291. URL: https://hal.science/hal-00923320.
  2. A. Figalli, L. Rifford, C. Villani, “Nearly round spheres look convex”, American Journal of Mathematics 134 (2012), no. 1, 109–139. DOI: 10.1353/ajm.2012.0000. URL: https://doi.org/10.1353/ajm.2012.0000.
  3. A. Figalli, L. Rifford, C. Villani, “On the Ma–Trudinger–Wang curvature on surfaces”, Calculus of Variations and Partial Differential Equations 39 (2010), no. 3–4, 307–332. DOI: 10.1007/s00526-010-0311-9. URL: https://doi.org/10.1007/s00526-010-0311-9.
  4. A. Figalli, L. Rifford, “Continuity of optimal transport maps and convexity of injectivity domains on small deformations of S²”, Communications on Pure and Applied Mathematics 62 (2009), no. 12, 1670–1706. DOI: 10.1002/cpa.20293. URL: https://doi.org/10.1002/cpa.20293.
  5. A. Figalli, G. Loeper, “C¹ regularity of solutions of the Monge–Ampère equation for optimal transport in dimension two”, Calculus of Variations and Partial Differential Equations 35 (2009), no. 4, 537–550. DOI: 10.1007/s00526-009-0222-9. URL: https://doi.org/10.1007/s00526-009-0222-9.
  6. A. Figalli, Y.-H. Kim, R. J. McCann, “Regularity of optimal transport maps on multiple products of spheres”, Journal of the European Mathematical Society 15 (2013), no. 4, 1131–1166. DOI: 10.4171/jems/388. URL: https://doi.org/10.4171/jems/388.
  7. G. Loeper, “On the regularity of solutions of optimal transportation problems”, Acta Mathematica 202 (2009), no. 2, 241–283. DOI: 10.1007/s11511-009-0037-8. URL: https://doi.org/10.1007/s11511-009-0037-8.
  8. X.-N. Ma, N. S. Trudinger, X.-J. Wang, “Regularity of potential functions of the optimal transportation problem”, Archive for Rational Mechanics and Analysis 177 (2005), no. 2, 151–183. DOI: 10.1007/s00205-005-0362-9. URL: https://doi.org/10.1007/s00205-005-0362-9.
  9. R. J. McCann, “Polar factorization of maps on Riemannian manifolds”, Geometric and Functional Analysis 11 (2001), no. 3, 589–608. DOI: 10.1007/pl00001679. URL: https://doi.org/10.1007/pl00001679.
  10. G. Loeper, C. Villani, “Regularity of optimal transport in curved geometry: the nonfocal case”, Duke Mathematical Journal 151 (2010), no. 3, 431–485. DOI: 10.1215/00127094-2010-003. URL: https://doi.org/10.1215/00127094-2010-003.
  11. P. Delanoë, Y. Ge, “Locally nearly spherical surfaces are almost-positively c-curved”, Methods and Applications of Analysis 18 (2011), no. 3, 269–302. DOI: 10.4310/maa.2011.v18.n3.a2. URL: https://doi.org/10.4310/maa.2011.v18.n3.a2.
  12. P. Delanoë, Y. Ge, “Regularity of optimal transport on compact, locally nearly spherical, manifolds”, Journal für die reine und angewandte Mathematik 646 (2010), 65–115. DOI: 10.1515/crelle.2010.066. URL: https://doi.org/10.1515/crelle.2010.066.
  13. Y.-H. Kim, R. J. McCann, “Towards the smoothness of optimal maps on Riemannian submersions and Riemannian products (of round spheres in particular)”, Journal für die reine und angewandte Mathematik 664 (2012), 1–27. DOI: 10.1515/crelle.2011.105. URL: https://doi.org/10.1515/crelle.2011.105.
  14. Y. Ge, J. Ye, “Regularity of optimal transport maps on locally nearly spherical manifolds”, Annales de la Faculté des Sciences de Toulouse (6) 30 (2021), no. 2, 353–409. DOI: 10.5802/afst.1678. URL: https://afst.centre-mersenne.org/item/10.5802/afst.1678/.
  15. J. Ye, “Regularity of optimal transport map on compact Riemannian manifolds” (French title: “Régularité de l'application du transport optimal sur des variétés riemanniennes compactes”), PhD thesis, Université Toulouse III – Paul Sabatier (cotutelle: University of Science and Technology of China), defended 23 July 2018, NNT 2018TOU30354. DOI: 10.70675/92863850zf500z4057z83b5z7944554e7a71. URL: https://theses.hal.science/tel-02927159.
  16. N. Lebedeva, A. Petrunin, V. Zolotov, “Bipolar comparison”, Geometric and Functional Analysis 29 (2019), no. 1, 258–282. DOI: 10.1007/s00039-019-00481-9. URL: https://arxiv.org/abs/1711.09423 (arXiv version of the published article).
  17. Du Nguyen, “Families of costs with zero and nonnegative MTW tensor in optimal transport and the c-divergences”, arXiv:2401.00953 (v1 January 2024, v2 December 2025). URL: https://arxiv.org/abs/2401.00953.
  18. S.-Z. Du, Q.-R. Li, “Positivity of Ma–Trudinger–Wang curvature on Riemannian surfaces”, Calculus of Variations and Partial Differential Equations 51 (2014), no. 3–4, 495–523. DOI: 10.1007/s00526-013-0684-7. URL: https://doi.org/10.1007/s00526-013-0684-7.
  19. P. W. Y. Lee, J. Li, “New Examples Satisfying Ma–Trudinger–Wang Conditions”, SIAM Journal on Mathematical Analysis 44 (2012), no. 1, 61–73. DOI: 10.1137/110820543. URL: https://doi.org/10.1137/110820543. (Preprint, under the title “New examples on spaces of negative sectional curvature satisfying Ma–Trudinger–Wang conditions”: arXiv:0911.3978, URL: https://arxiv.org/abs/0911.3978).
  20. J. Vétois, “Continuity and injectivity of optimal maps”, Calculus of Variations and Partial Differential Equations 52 (2015), no. 3–4, 587–607. DOI: 10.1007/s00526-014-0725-x. URL: https://doi.org/10.1007/s00526-014-0725-x.
  21. J. Kitagawa, C. Letrouit, Q. Mérigot, “Stability of optimal transport maps on Riemannian manifolds”, arXiv:2504.05412 (2025). URL: https://arxiv.org/abs/2504.05412.
  22. M. Warren, “A quantitative stability result for regularity of optimal transport on compact manifolds”, arXiv:2507.09035 (2025). URL: https://arxiv.org/abs/2507.09035.
  23. Y. Ge, J. Serres, “A generalization of Caffarelli's contraction theorem to nearly spherical manifolds”, arXiv:2512.01496 (2025). URL: https://arxiv.org/abs/2512.01496.
  24. S. Shivakumar, G. A. Bondar, G. Khan, A. Halder, “Sum-of-squares programming for Ma–Trudinger–Wang regularity of optimal transport maps”, Journal of Optimization Theory and Applications 209 (2026), no. 3. DOI: 10.1007/s10957-026-03015-0. URL: https://doi.org/10.1007/s10957-026-03015-0. (Preprint: arXiv:2412.13372, December 2024, URL: https://arxiv.org/abs/2412.13372).