Singular structures in solutions to the Monge-Ampère equation with point masses

Abstract

We construct new examples of Monge-Ampère 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.

Keywords

• monge-ampère equation
• optimal transport
• syz conjecture
• regularity and stability
• singular solutions