Surfaces of prescribed linear Weingarten curvature in ${\mathbb{R}}^3$

Abstract

Given $a,b\in \mathbb{R}$ and $\hat{I}¦\in C^1({\mathbb{S}}^2)$, we study immersed oriented surfaces $\hat{I}\pounds$ in the Euclidean 3-space ${\mathbb{R}}^3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\hat{I}¦(N)$, where $N:\hat{I}\pounds \to {\mathbb{S}}^2$ 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 $\hat{I}¦$, 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.

21 pages, 9 figures