Entire solutions of the degenerateMonge-Ampere equation with a finite number of singularities

TL;DR

This paper classifies all solutions to the degenerate Monge-Ampere equation with finitely many singularities, revealing geometric structures and constraints on solutions in punctured planes.

Contribution

It provides a comprehensive classification of solutions with finite singularities, including geometric descriptions and conditions for analyticity.

Findings

Solutions with up to two singularities are fully classified.

For more than two singularities, solutions form cone-like structures outside convex polyhedra.

Analytic solutions have at most one singularity, being either a cylinder or a cone.

Abstract

We determine the global behavior of every C^2-solution to the two-dimensional degenerate Monge-Ampere equation, u_{xx}u_{yy}-u_{xy}^2=0, over the finitely punctured plane. With this, we classify every solution in the once or twice punctured plane. Moreover, when we have more than two singularities, if the solution u is not linear in a half-strip, we obtain that the singularities are placed at the vertices of a convex polyhedron P and the graph of u is made by pieces of cones outside of P which are suitably glued along the sides of the polyhedron. Finally, if we look for analytic solutions, then there is at most one singularity and the graph of $u$ is either a cylinder (no singularity) or a cone (one singularity).