Bernstein theorems for complete \(\alpha\)-relative extremal hypersurfaces
type: Item
instance of: scholarly article
title: Bernstein theorems for complete \(\alpha\)-relative extremal hypersurfaces
zbMATH Open document ID: 1264.53014
author: Ruiwei Xu
author: Min Xiong
author: Li Sheng
published in: Annals of Global Analysis and Geometry
publication date: 2013-03-21T00:00:00Z
review text: The following main theorem is proved. { Theorem.} Let \(y: M\to \mathbb R^{n+1}\) be a locally strongly convex \(\alpha\) relative extremal hypersurface, complete with respect to the \(\alpha\)-metric \(G^{(\alpha)}\), which is given by a locally strongly convex function: \(x_{n+1}=f(x_1,\cdots,x_n)\). Then there is a positive constant \(K(n)\) depending only on the dimension \(n\), such that \(|\alpha|>K(n)\) implies that \(M\) is an elliptic paraboloid.
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 6147896
zbMATH Keywords: Bernstein theorem
zbMATH Keywords: relative extremal hypersurface
zbMATH Keywords: blow-up analysis
reviewed by: Huili Liu
MaRDI profile type: Publication
full work available at URL: s10455-012-9338-9
OpenAlex ID: W2085685507
cites work: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens
cites work: Interior estimates for solutions of a fourth order nonlinear partial differential equation
cites work: The Calabi conjecture on affine maximal surfaces
cites work: A Bernstein property of some fourth order partial differential equations
cites work: Q5320680
cites work: Q4169389
cites work: Affine complete locally convex hypersurfaces
cites work: Hypersurfaces with Li-normalization and prescribed Gauss-Kronecker curvature
cites work: Q3644623
cites work: A Bernstein theorem for the Abreu equation
cites work: Hyperbolic relative hyperspheres with Li-normalization
cites work: Bernstein properties for some relative parabolic affine hyperspheres
cites work: Q3570995
cites work: Bending of simply supported incompressible saturated poroelastic beams with axial diffusion
DOI: 10.1007/S10455-012-9338-9
description: scientific article; zbMATH DE number 6147896