Complete surfaces of finite total curvature
type: Item
instance of: scholarly article
title: Complete surfaces of finite total curvature
zbMATH Open document ID: 0631.53007
author: Brian White
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: It is well known that the total curvature of a complete minimal surface in \(E^ n\) is an integral multiple of \(2\pi\) (or of \(4\pi\) if \(n=3)\). Furthermore the Gauss map extends continuously to the end compactification \(\bar M\) where \(M=\bar M\setminus \{p_ 1,...,p_ k\}\). The proof relies heavily on complex analysis. In the present paper the author assumes the weaker condition that the total norm square of the second fundamental form is finite. Again he obtains that the total curvature is an integral multiple of \(2\pi\) (or of \(4\pi\) if \(n=3)\). Furthermore he gives a sufficient condition for the continuous extension of the Gauss map to \(\bar M.\) An example shows that this is not true in general.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C65
zbMATH DE Number: 4026254
zbMATH Keywords: rectifiable current
zbMATH Keywords: Gauss map at infinity
zbMATH Keywords: minimal surface
zbMATH Keywords: second fundamental form
zbMATH Keywords: total curvature
Wikidata QID: Q115184504
reviewed by: Wolfgang Kühnel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441372
OpenAlex ID: W1594292678
DOI: 10.4310/JDG/1214441372
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description: scientific article; zbMATH DE number 4026254