Singular structures in solutions to the Monge-Amp\`{e}re equation with point masses

TL;DR

This paper constructs new Monge-Ampère metrics with polyhedral singularities, motivated by optimal transport and mirror symmetry, and studies their stability under data perturbations.

Contribution

It introduces novel examples of singular Monge-Ampère solutions and analyzes their stability, advancing understanding of geometric structures in optimal transport and mirror symmetry.

Findings

Constructed new polyhedral singular solutions

Analyzed stability under data perturbations

Linked singular structures to mirror symmetry applications

Abstract

We construct new examples of Monge-Amp\`{e}re metrics with polyhedral singular structures, motivated by problems related to the optimal transport of point masses and to mirror symmetry. We also analyze the stability of the singular structures under small perturbations of the data given in the problem under consideration.