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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
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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 Weingarten surface, extrinsic curvature, intrinsic curvature, height estimate, rotational Weingarten surface
MSC Classifications: 53C42, 53C30 show english descriptions Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]
Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]
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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