Regularity of the solution to a real Monge--Amp\`ere equation on the boundary of a simplex
Analysis of PDEs
2025-01-14 v2 Differential Geometry
Abstract
Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex.
Cite
@article{arxiv.2403.01620,
title = {Regularity of the solution to a real Monge--Amp\`ere equation on the boundary of a simplex},
author = {Rolf Andreasson and Jakob Hultgren and Mattias Jonsson and Enrica Mazzon and Nicholas McCleerey},
journal= {arXiv preprint arXiv:2403.01620},
year = {2025}
}
Comments
13 pages; to appear in International Mathematics Research Notices