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

TL;DR

This paper classifies surfaces in Euclidean 3-space with prescribed linear Weingarten curvature relations, extending classical surface theory to more general nonlinear PDEs and providing a comprehensive understanding of their geometric properties.

Contribution

It introduces a classification of rotational surfaces satisfying a generalized curvature relation, broadening the scope of classical Weingarten surface theory.

Findings

Classification of rotational solutions under elliptic and hyperbolic PDE conditions

Extension of classical constant curvature surface results

Identification of conditions for existence of such surfaces

Abstract

Given $a,b\in\mathbb{R}$ and $\Phi\in C^1(\mathbb{S}^2)$, 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.