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Singularities of the solution to a Monge--Amp\`ere equation on the boundary of the 3-simplex

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arxiv 2309.15263 v1 pith:WTM4U5SU submitted 2023-09-26 math.DG

Mattias Jonsson , Nicholas McCleerey , Neil Patram , Benjamin W. Scott This is my paper · ORCID
classification math.DG
keywords equationsolutionboundarydefinedmetricmonge-amppointssimplex
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abstract

We show that the metric defined by the solution to the tropical Monge-Amp\`ere equation, as defined by Hultgren, Mazzon, and the first two authors, on the boundary of the 3-simplex is asymptotic to the Gross-Wilson metric on $S^2$ near each of the 6 singular points. We deduce in addition that the solution is not $C^{1,1}$ across the singular points. Compared to previous works, our starting point is the real Monge-Amp\`ere equation, as opposed to the complex structure.

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  1. Boundary regularity of optimal transport maps on convex domains

    math.AP 2025-07 conditional novelty 8.0 of 10

    For convex domains with Hölder continuous densities, optimal transport potentials are C^{1,1-ε} and W^{2,p}; with C^{1,β} boundaries, they are C^{2,min(α,β)}.

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