Flat maximal immersions
type: Item
instance of: scholarly article
title: Flat maximal immersions
zbMATH Open document ID: 0863.53042
DOI: 10.1016/0393-0440(95)00005-4
author: Jens Chr. Larsen
published in: Journal of Geometry and Physics
publication date: 1997-05-28T00:00:00Z
review text: The author studies Lorentzian immersed surfaces with zero mean curvature in the three-dimensional Minkowski space \(\mathbb{R}^3_1\). Paralleling the classical terminology of minimal surfaces in Euclidean space, the author refers to this kind of surfaces as maximal surfaces. First, it is shown that a geodesically complete maximal surface in \(\mathbb{R}^3_1\) is flat if and only if it is the graph over a timelike plane of a function of one variable. On the other hand, it is well-known that zero mean curvature surfaces arise naturally as critical points of the area functional. In this respect, the author considers the stability of maximal Lorentzian surfaces in \(\mathbb{R}^3_1\), and he shows that they are always unstable, so that these surfaces neither locally maximize nor locally minimize the area. Finally, the author studies maximal Lorentzian surfaces of revolution. To the reviewer, the fact that Lorentzian surfaces with zero mean curvature in Minkowski 3-space are always unstable means that it does not seem very appropriate to refer to them as maximal or minimal surfaces. Maybe they should just be called stationary or critical surfaces.
reviewed by: Luis J. Alías
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A10
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 866112
zbMATH Keywords: maximal Lorentzian immersion
zbMATH Keywords: metric singularities
zbMATH Keywords: surfaces of revolution
MaRDI profile type: Publication
full work available at URL: 0393-0440(95)00005-4
OpenAlex ID: W2030221465
cites work: Complex analytic curves and maximal surfaces
cites work: Existence of maximal surfaces in asymptotically flat spacetimes
cites work: Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces
cites work: Area maximizing hypersurfaces in Minkowski space having an isolated singularity
cites work: Maximal surfaces with conelike singularities
cites work: Complex analysis, maximal immersions and metric singularities
cites work: Surfaces in Minkowski 3-space on which H and K are linearly related
cites work: Q3881472
description: scientific article; zbMATH DE number 866112