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arXiv · 1411.2904

Isolated singularities of graphs in warped products and Monge-Ampère equations

José A. Gálvez·Asun Jiménez·Pablo Mira
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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-Ampère 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 $\mathbb{S^2}$.

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José A. Gálvez, Asun Jiménez, Pablo Mira. 2014-11-11. Isolated singularities of graphs in warped products and Monge-Ampère equations. https://arxiv.org/abs/1411.2904

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