New examples of Calabi-Bernstein problems for some nonlinear equations.
type: Item
instance of: scholarly article
title: New examples of Calabi-Bernstein problems for some nonlinear equations.
zbMATH Open document ID: 1041.53039
DOI: 10.1016/S0926-2245(01)00057-2
author: José M. Latorre
author: Alfonso Romero
published in: Differential Geometry and its Applications
publication date: 2003-08-06T00:00:00Z
review text: A Robertson-Walker space is the product manifold \(M=I\times \mathbb{R}^2\) with the (warped) Lorentzian metric \[ ds^2=-dt^2+f(t)g, \] where \(t\in I\), \(f\) is a positive function in \(I\), and \(g\) is the usual metric of \(\mathbb{R}^2\). In the case \(f\equiv1\), we fall back in the usual Lorentz-Minkowski metric \(L^3\). It is assumed that \(M\) satisfies the null convergence condition (NCC), i.e., that \(\text{Ricc}(Z,Z)\geq 0\), for all vectors \(Z\) with \(\langle Z,Z\rangle=0\). This paper studies the Calabi-Bernstein problem for complete maximal surfaces in a RW-space satisfying NCC. One of its results is as follows: A Robertson-Walker space with fiber \(\mathbb{R}^2\) and not locally constant warping function \(f\) with \(\sup(f)<\infty\), which obeys NCC, admits at most one complete maximal surface which is a totally geodesic spacelike slice.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 1959884
zbMATH Keywords: Robertson-Walker space
zbMATH Keywords: Calabi-Bernstein problem
zbMATH Keywords: maximal surfaces
reviewed by: Manfredo P. do Carmo
MaRDI profile type: Publication
Wikidata QID: Q115338256
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full work available at URL: s0926-2245(01)00057-2
OpenAlex ID: W2010082612
description: scientific article; zbMATH DE number 1959884