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

TL;DR

This paper investigates the singularities of solutions to a tropical Monge-Ampère equation on the boundary of a 3-simplex, showing asymptotic behavior to the Gross-Wilson metric and non-smoothness at singular points.

Contribution

It provides the first analysis of the real Monge-Ampère equation's singularities on the boundary of a 3-simplex, linking tropical geometry to metric asymptotics.

Findings

Solution is asymptotic to the Gross-Wilson metric near singular points

The solution is not $C^{1,1}$ across singularities

Establishes a connection between tropical Monge-Ampère and real Monge-Ampère equations

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.