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

Part of: Qualitative theory Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  22 July 2022

Antonio Bueno  and
Irene Ortiz
Antonio Bueno
Affiliation:
Departamento de Ciencias, Centro Universitario de la Defensa de San Javier, E-30729 Santiago de la Ribera, Spain (antonio.bueno@cud.upct.es, irene.ortiz@cud.upct.es)
Irene Ortiz
Affiliation:
Departamento de Ciencias, Centro Universitario de la Defensa de San Javier, E-30729 Santiago de la Ribera, Spain (antonio.bueno@cud.upct.es, irene.ortiz@cud.upct.es)
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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 generalizes 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.

Keywords

Prescribed Weingarten curvaturelinear Weingarten surfacerotational surfacenonlinear autonomous systemphase plane analysis

MSC classification

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 34C05: Location of integral curves, singular points, limit cycles 34C40: Equations and systems on manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 4 , August 2023 , pp. 1347 - 1370
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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Footnotes

This paper is dedicated to the first author's mother, whose daily fight, strength and spirit of overcoming adversity prove that there are way more difficult and important issues in life than a mathematical problem

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