Locally strongly convex hypersurfaces with constant affine mean curvature
type: Item
instance of: scholarly article
title: Locally strongly convex hypersurfaces with constant affine mean curvature
zbMATH Open document ID: 1081.53007
author: Fang Jia
author: An-Min Li
published in: Differential Geometry and its Applications
publication date: 2005-05-04T00:00:00Z
review text: The authors study the Bernstein property of constant affine mean curvature surfaces in \(\mathbb R^3\). For any locally strongly convex surface given by a strictly convex function \(x_3= f(x_1,x_2)\) they consider the Riemannian metric defined by \(G^\#= \sum \frac{\partial^2f}{\partial x_i\partial x_j}\,dx_i\,dx_j\) on \(M=\{(x_1,x_2,f(x_1,x_2))\mid (x_1,x_2)\in \Omega\}\), where \(\Omega\) is a convex domain in \(\mathbb R^2\). Using a priori estimates on PDE they are able to show that if \(M\) is a constant affine mean curvature surface which is complete with respect to \(G^\#\), then \(M\) must be an elliptic paraboloid.
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 2165118
zbMATH Keywords: Bernstein property
zbMATH Keywords: affine mean curvature
zbMATH Keywords: affine maximal surface
reviewed by: Changping Wang
MaRDI profile type: Publication
full work available at URL: j.difgeo.2004.10.007
OpenAlex ID: W2025539402
Wikidata QID: Q115357992
cites work: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens
cites work: Euclidean complete affine surfaces with constant affine mean curvature
cites work: A Bernstein property of affine maximal hypersurfaces
cites work: Global affine differential geometry of hypersurfaces
cites work: Q4169389
DOI: 10.1016/J.DIFGEO.2004.10.007
description: scientific article; zbMATH DE number 2165118