The Bernstein problem for affine maximal hypersurfaces
type: Item
instance of: scholarly article
title: The Bernstein problem for affine maximal hypersurfaces
zbMATH Open document ID: 0978.53021
author: Neil S. Trudinger
author: Xu-Jia Wang
published in: Inventiones Mathematicae
publication date: 2002-02-07T00:00:00Z
review text: The Bernstein problem for hypersurfaces with vanishing equiaffine mean curvature (called ``affine minimal'' in the terminology of Chern) was first proposed, for dimension \(n=2\), i.e., for surfaces, in the article [\textit{S. S. Chern}, Proc. Japan-United States Sem., Tokyo, 1977, 17-30 (1979; Zbl 0439.53008)]. Its statement is the following: ``A Euclidean complete, affine minimal, locally strongly convex surface in \(\mathbb{R}^3\) must be an elliptic paraboloid''. Subsequently the problem was considered by Calabi, who showed, first, that if a hypersurface \(M\) is a critical point of the area functional \(A\), the second variation of \(A\) at \(M\) is non-positive, that is, the affine area \(A(M)\) reaches a maximum under smooth interior perturbations. Accordingly he proposed that such an \(M\) be called, more properly, an affine maximal hypersurface. Then, he verified Chern's conjecture under the hypothesis that the affine metric of \(M\) be complete [\textit{E. Calabi}, Am. J. Math. 104, 91-126 (1982; Zbl 0501.53037)]. In the present article, the authors prove the validity of the conjecture in the two-dimensional case. More generally, they consider the \(n\)-dimensional case, showing that the corresponding result holds in higher dimensions provided that a uniform, ``strict convexity'' condition holds. They also extend the notion of ``affine maximal'' to non-smooth convex graphs and produce a counterexample showing that the Bernstein result does not hold in this generality for dimension \(d\geq 10\).
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 1463434
zbMATH Keywords: affine completeness
zbMATH Keywords: uniform strict convexity
zbMATH Keywords: Bernstein problem
zbMATH Keywords: equiaffine mean curvature
zbMATH Keywords: affine maximal hypersurface
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
full work available at URL: s002220000059
OpenAlex ID: W1988312779
DOI: 10.1007/S002220000059
Recommended article: Q4489581
Recommended article: Q4781385
Recommended article: A Bernstein property of affine maximal hypersurfaces
Recommended article: Affine differential geometry and partial differential equations of fourth order
Recommended article: Q4543032
Recommended article: Q3822786
Recommended article: The Calabi conjecture on affine maximal surfaces
Recommended article: On the affine Bernstein problem
Recommended article: Q5714411
Recommended article: Some theorems in affine differential geometry
description: scientific article; zbMATH DE number 1463434