On the structure of certain Weingarten surfaces with boundary a circle
type: Item
instance of: scholarly article
title: On the structure of certain Weingarten surfaces with boundary a circle
zbMATH Open document ID: 0901.53003
author: Ricardo Sa Earp
author: Fabiano G. B. Brito
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 1998-11-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/73418
review text: We call a surface \(M\) special Weingarten surface in \(\mathbb{R}^3\) if \(M\) satisfies the relation \(H=f (H^2-K)\), where \(H\) is the mean curvature of \(M\), \(K\) the Gaussian curvature of \(M\) and \(f\) a real smooth function such that \(4t (f(t))^2 <1\). The authors remark that they can apply maximum principle and Alexandrov reflection principle techniques to those surfaces as these were applied to constant mean curvature surfaces. They also look upon R. Bryant's construction of a global quadratic form \(Q\) on a surface \(M\) satisfying \(H= f(H^2-K)\), such that the zeros of \(Q\) are the umbilicial points of \(M\). So they characterize disk-type special Weingarten surfaces with boundary a round circle, extending some of their results for surfaces with constant mean curvature obtained in [\textit{F. Brito} and \textit{R. Sa Earp}, An. Acad. Bras. Cienc. 63, 5-19 (1991; Zbl 0803.53011)] and jointly with \textit{W. H. Meeks} and \textit{H. Rosenberg} in [Indiana Univ. Math. J. 40, 333-343 (1991; Zbl 0759.53003)].
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1139495
zbMATH Keywords: disk-type Weingarten surfaces
zbMATH Keywords: maximum principle
zbMATH Keywords: Alexandrov reflection principle
zbMATH Keywords: umbilicial points
reviewed by: Despina Papadopoulou-Florou
MaRDI profile type: Publication
OpenAlex ID: W2076093235
cites work: Q4890670
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DOI: 10.5802/AFST.863
description: scientific article; zbMATH DE number 1139495