Isolated singularities of graphs in warped products and Monge-Amp\`ere equations

TL;DR

This paper investigates the behavior of graphs with positive extrinsic curvature near isolated singularities in warped product spaces, classifies such surfaces using Monge-Ampère equations, and establishes a correspondence with convex curves on the sphere.

Contribution

It provides a classification of surfaces with prescribed curvature in warped products and space forms, linking geometric singularities to convex curves on the sphere.

Findings

Classification of surfaces with isolated singularities using Monge-Ampère equations

Establishment of a one-to-one correspondence with convex Jordan curves on the sphere

Description of surface behavior at non-removable singularities in warped product spaces

Abstract

We study graphs of positive extrinsic curvature with a non-removable isolated singularity in 3-dimensional warped product spaces, and describe their behavior at the singularity in several natural situations. We use Monge-Amp\`ere equations to give a classification of the surfaces in 3-dimensional space forms which are embedded around a non-removable isolated singularity and have a prescribed, real analytic, positive extrinsic curvature function at every point. Specifically, we prove that this space is in one-to-one correspondence with the space of regular, analytic, strictly convex Jordan curves in the 2-dimensional sphere $\mathbb{S^2}$.