Continuity of manifold transport under MTW and convex injectivity domains
Let $(M,g)$ be a compact connected smooth Riemannian manifold without boundary, of dimension $n$, with distance $d_g$, and let $c(x,y) = d_g(x,y)^2/2$. For $x \in M$ let $I_x \subset T_xM$ be the open set of vectors $v$ along which $\exp_x$ is minimizing beyond time one, that is, $t \mapsto \exp_x(tv)$ is minimizing on $[0, s]$ for some $s > 1$. The weak Ma–Trudinger–Wang (MTW) condition is $$-\frac32 \, \partial_s^2 \partial_t^2 \, c\bigl(\exp_x(t\xi), \exp_x(v + s\eta)\bigr)\Big|_{s=t=0} \ge 0$$ for all $x \in M$, $v \in I_x$ and $\xi, \eta \in T_xM$ with $\xi \perp \eta$. Figalli, Rifford and Villani showed that weak MTW and convexity of every $I_x$ are necessary for the optimal transport maps between all pairs of densities bounded away from zero and infinity to be continuous, and that on surfaces ($n = 2$) they are also sufficient.
Problem. Let $n \ge 3$, and assume that every $I_x$ is convex and that the weak MTW condition holds. For every pair of probability densities $f, g_1$ with respect to Riemannian volume with $0 < a \le f, g_1 \le b < \infty$ almost everywhere, must the almost everywhere unique optimal map transporting $f$ to $g_1$ for the cost $c$ have a continuous representative on $M$?
Source: De Philippis and Figalli (2014), §5.2, Open problem (7), in every dimension; the case of surfaces is Figalli, Rifford and Villani (2011). Stated by AIM — Open Applied Problems for dimensions n ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DG
- Keywords
- AIM Open Applied Problems, Ma–Trudinger–Wang condition, optimal transport, Riemannian manifolds
References
@article{dephilippis2014,
author = {De Philippis, Guido and Figalli, Alessio},
title = {The {M}onge--{A}mp{\`e}re equation and its link to optimal transportation},
journal = {Bull. Amer. Math. Soc. (N.S.)},
volume = {51},
number = {4},
year = {2014},
pages = {527--580},
doi = {10.1090/S0273-0979-2014-01459-4},
eprint = {1310.6167},
archivePrefix = {arXiv},
}
@article{figalli2011,
author = {Figalli, Alessio and Rifford, Ludovic and Villani, C{\'e}dric},
title = {Necessary and sufficient conditions for continuity of optimal transport maps on {R}iemannian manifolds},
journal = {Tohoku Math. J. (2)},
volume = {63},
number = {4},
year = {2011},
pages = {855--876},
doi = {10.2748/tmj/1325886291},
}
@misc{aim-mtw-convex-domains-transport-continuity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Continuity of manifold transport under MTW and convex injectivity domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/401-mtw-convex-domains-transport-continuity.md},
note = {Entry 401 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We prove that the weak Ma--Trudinger--Wang condition on a fixed smooth connected compact boundaryless Riemannian manifold of dimension at least two implies a common Hölder estimate for optimal transport maps and their inverses over the entire class of probability densities with fixed positive upper and lower bounds. The maps have homeomorphic representatives, conjugate cut points are allowed, and no density …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: math @ 1581d4c, build/main.tex. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00698, "Continuity of manifold transport under MTW and convex injectivity domains", clopen.solutions.