Maximal surfaces with conelike singularities
type: Item
instance of: scholarly article
title: Maximal surfaces with conelike singularities
zbMATH Open document ID: 0548.53006
author: Osamu Kobayashi
published in: Journal of the Mathematical Society of Japan
publication date: 1984-01-01T00:00:00Z
review text: A space-like surface S in a 3-dimensional Minkowski space \(L^ 3=({\mathbb{R}}^ 3,dx^ 2+dy^ 2-dz^ 2)\) is said to be maximal if the mean curvature vanishes identically. Any such surface is represented locally by \(Z=u(x,y)\) and S is maximal if u satisfies the equation: \((1- u^ 2_ x)u_{yy}+2u_ xu_ yu_{xy}+(1-u^ 2_ y)u_{xx}=0\). Maximal surfaces S in \(L^ 3\) have singularities which are of different kinds from those appearing in minimal surfaces in the Euclidean space. In this paper a conelike singularity is defined and the following theorem is proved: Let S be a complete maximal surface in \(L^ 3\), with at least one conelike singularity. Suppose that the Gauss map of S is one-to-one. Then S is congruent to the surface defined by \(\sqrt{x^ 2+y^ 2}+a \sinh (z/a)=0,\) where a is a nonzero real constant.
reviewed by: R. Rosca
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 3874113
zbMATH Keywords: Maximal surfaces
zbMATH Keywords: singularities
zbMATH Keywords: conelike singularity
zbMATH Keywords: Gauss map
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03640609
OpenAlex ID: W2160651616
DOI: 10.2969/JMSJ/03640609
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description: scientific article; zbMATH DE number 3874113