Solutions to the Monge-Amp\`{e}re equation with polyhedral and Y-shaped singularities

Abstract

We construct convex functions on $\mathbb{R}^3$ and $\mathbb{R}^4$ that are smooth solutions to the Monge-Amp\`{e}re equation $\det D^2u = 1$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a convex polytope. The examples solve the equation in the Alexandrov sense away from finitely many points. Our approach is based on solving an obstacle problem where the graph of the obstacle is a convex polytope.