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# Weingarten Type Surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$

Published:2017-03-16
Printed: Dec 2017
• Abigail Folha,
Universidade Federal Fluminense, Instituto de Matemática e Estatística, Departamento de Geometria, Niterói, RJ - Brasil
• Carlos Peñafiel,
Universidade Federal de Rio de Janeiro, Instituto de Matemática e Estatística, Departamento de Métodos Matemáticos, Rio de Janeiro, RJ - Brasil
 Format: LaTeX MathJax PDF

## Abstract

In this article, we study complete surfaces $\Sigma$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b\gt 0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.
 Keywords: Weingarten surface, extrinsic curvature, intrinsic curvature, height estimate, rotational Weingarten surface
 MSC Classifications: 53C42 - Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42] 53C30 - Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

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