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Regularity of the solution to a real Monge--Ampère equation on the boundary of a simplex

Rolf Andreasson, Jakob Hultgren, Mattias Jonsson, Enrica Mazzon, Nicholas McCleerey

TL;DR

This work studies the real Monge–Ampère equation on the boundary of a simplex through an optimal transport lens between symmetric measures on the boundary A and its dual B. By combining Caffarelli boundary regularity with the symmetry of the simplex, it establishes uniform C^{1,α} regularity (and smoothness under smooth data) for the transport potential φ and its c-transform ψ, yielding Hessian-type metrics g_φ and g_ψ that are isometric on the regular loci. It then analyzes the singular set, proving that the c-gradients are everywhere single-valued and Hölder continuous, mapping A_{IJ}^∘ to B_{JI}^∘ and preserving the reg/sing structure, with explicit behavior in dimension three. Finally, it studies metric completions, showing that the Euclidean identity maps extend to Hölder continuous surjections onto the metric completions with connected fibers, and that the reg-part maps are isometries, while discussing obstructions to global homeomorphisms in higher dimensions. These results connect real Monge–Ampère theory with Mirror Symmetry, the SYZ conjecture, and Kontsevich–Soibelman-type expectations for degenerating Calabi–Yau limits.

Abstract

Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Ampère 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.

Regularity of the solution to a real Monge--Ampère equation on the boundary of a simplex

TL;DR

This work studies the real Monge–Ampère equation on the boundary of a simplex through an optimal transport lens between symmetric measures on the boundary A and its dual B. By combining Caffarelli boundary regularity with the symmetry of the simplex, it establishes uniform C^{1,α} regularity (and smoothness under smooth data) for the transport potential φ and its c-transform ψ, yielding Hessian-type metrics g_φ and g_ψ that are isometric on the regular loci. It then analyzes the singular set, proving that the c-gradients are everywhere single-valued and Hölder continuous, mapping A_{IJ}^∘ to B_{JI}^∘ and preserving the reg/sing structure, with explicit behavior in dimension three. Finally, it studies metric completions, showing that the Euclidean identity maps extend to Hölder continuous surjections onto the metric completions with connected fibers, and that the reg-part maps are isometries, while discussing obstructions to global homeomorphisms in higher dimensions. These results connect real Monge–Ampère theory with Mirror Symmetry, the SYZ conjecture, and Kontsevich–Soibelman-type expectations for degenerating Calabi–Yau limits.

Abstract

Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Ampère 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.
Paper Structure (5 sections, 7 theorems, 40 equations, 1 figure)

This paper contains 5 sections, 7 theorems, 40 equations, 1 figure.

Table of Contents

  1. Setup
  2. An optimal transport problem
  3. Regularity on the singular set: proof of Theorem B
  4. Metric completions: proof of Theorem C
  5. $C^{1,\alpha}$-metrics with short curves

Key Result

Theorem 1.1

Let $U$ and $V$ be bounded convex domains in $\mathbb{R}^d$, and $\mu$, $\nu$ probability measures on $U$, $V$ with densities (with respect to Lebesgue measure) bounded away from zero and infinity (or more generally, densities that are doubling). Then there exist differentiable, strictly convex func Moreover, $u$ and $v$ are uniformly differentiable and strictly convex in the following sense. Ther

Figures (1)

  • Figure 1: The singular locus in dimension $d=3$. The left side (in $A$) depicts a negative vertex, i.e. three edges in the singular set meeting at an interior point of a 2-cell. The right side (in $B$) depicts a positive vertex, i.e. three edges in the singular set meeting at the midpoint of a 1-cell.

Theorems & Definitions (18)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem B
  • Remark 3.2
  • Lemma 4.1
  • ...and 8 more