Area maximizing hypersurfaces in Minkowski space having an isolated singularity
type: Item
instance of: scholarly article
title: Area maximizing hypersurfaces in Minkowski space having an isolated singularity
zbMATH Open document ID: 0594.58023
DOI: 10.1007/BF01168501
author: Klaus Ecker
published in: Manuscripta Mathematica
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155177
review text: We consider area maximizing hypersurfaces in Minkowski space \(L^{n+1}\), i.e. spacelike hypersurfaces given by \(u\in C^{0,1}({\bar \Omega})\), \(\Omega \subset {\mathbb{R}}^ n\), which solve the variational problem \[ \int_{\Omega}\sqrt{1-| Dv|^ 2}\to \max \quad in\quad K=\{v\in C^{0,1}({\bar \Omega})| | Dv| \leq 1\quad a.e.\quad in\quad \Omega,\quad v|_{\partial \Omega}=u\}. \] The corresponding Euler- Lagrange-equation is given by \[ -\sum^{n}_{i=1}D_ i(D_ iu/\sqrt{1-| Du|^ 2})=0,\quad | Du| <1\quad in\quad \Omega; \] solutions of this equation are called maximal hypersurfaces. We prove that area maximizing hypersurfaces are asymptotic to an upper resp. lower light cone near an isolated singularity. Furthermore we show that entire maximal hypersurfaces having an isolated singularity are radially symmetric up to Lorentz transformations.
Mathematics Subject Classification ID: 58E12
Mathematics Subject Classification ID: 49Q05
zbMATH DE Number: 3956038
zbMATH Keywords: area maximizing hypersurfaces
zbMATH Keywords: Minkowski space
zbMATH Keywords: spacelike hypersurfaces
zbMATH Keywords: elliptic variational problem
zbMATH Keywords: quasilinear elliptic second order equations
zbMATH Keywords: light cone
zbMATH Keywords: isolated singularity
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3956038