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Surfaces of prescribed linear Weingarten curvature in $\mathbb{R}^3$ (2201.07480v1)

Published 19 Jan 2022 in math.DG

Abstract: Given $a,b\in\mathbb{R}$ and $\Phi\in C1(\mathbb{S}2)$, we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb{R}3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi(N)$, where $N:\Sigma\rightarrow\mathbb{S}2$ is the Gauss map. This theory widely generalize some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.

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