José A. Gálvez, Asun Jiménez, Pablo Mira
Published 2014-11-11Version 1
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 $\mathbb{S^2}$.
On Totally umbilical surfaces in the warped product $\mathbb{M}(κ)_f\times\mathbb{R}$
Gradient Yamabe Solitons on Warped Products
On a generalization of Monge--Ampère equations and Monge--Ampère systems
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