A Bernstein property of affine maximal hypersurfaces
type: Item
instance of: scholarly article
title: A Bernstein property of affine maximal hypersurfaces
zbMATH Open document ID: 1039.53017
DOI: 10.1023/A:1023059523458
author: An-Min Li
author: Fang Jia
published in: Annals of Global Analysis and Geometry
publication date: 2003-06-09T00:00:00Z
review text: The variational problem for the unimodular-affine area functional leads to a fourth order Euler-Lagrange equation, the vanishing of the affine mean curvature. For locally strongly convex surfaces, the second variation is negative, thus their extremals are ``affine maximal surfaces'' (Calabi). An affine Bernstein problem is the study of such Euler-Lagrange equations under completeness conditions. Proofs for the affine Bernstein conjectures in the versions of Calabi (complete Blaschke metric) and Chern (global graph representation), resp., in dimension \(n=2\), were given in 2000-2001. The authors state and solve an affine Bernstein problem for locally strongly convex hypersurfaces with another complete, affinely invariant metric, namely the metric defined by the Hessian of a graph representation over an \(n\)-dimensional domain \(G \in \mathbb R^{n+1}\); the assumptions imply that the hypersurface is an elliptic paraboloid. The proof works in dimensions \(n=2,3\). The very difficult and deep proof uses a splitting procedure for the solution of the fourth order Euler-Lagrange equation and Hofer's Lemma which is known from symplectic geometry.
reviewed by: Martin Wiehe
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1925041
zbMATH Keywords: Bernstein property
zbMATH Keywords: affine maximal hypersurface
zbMATH Keywords: Blaschke metric
zbMATH Keywords: Hofer's Lemma
zbMATH Keywords: symplectic geometry
MaRDI profile type: Publication
full work available at URL: a:1023059523458
OpenAlex ID: W1705259604
description: scientific article; zbMATH DE number 1925041