Hypersurfaces with prescribed affine Gauss-Kronecker curvature
type: Item
instance of: scholarly article
title: Hypersurfaces with prescribed affine Gauss-Kronecker curvature
zbMATH Open document ID: 0972.53013
DOI: 10.1023/A:1005232207442
author: Udo Simon
author: An-Min Li
author: Guosong Zhao
published in: Geometriae Dedicata
publication date: 2000-11-22T00:00:00Z
review text: It is well known that on a nondegenerate hypersurface in \(\mathbb{R}^{n+1}\), it is possible to introduce a canonical transversal vector field, called the affine normal. Using this affine normal, it is then possible, similar as in the Euclidean case, to introduce a connection (the induced affine connection), a symmetric bilinear form \(h\) (the affine metric) and a 1-1 tensor field \(S\) called the affine shape operator. In the paper under review the authors study hypersurfaces for which \(h(X,Y)\) is positive definite and \(h(SX, Y)\) is negative definite. Such hypersurfaces are called hyperbolic hypersurfaces. The affine Gauss-Kronecker curvature \(S_n\) is defined by \(S_n= (-1)^n\det S\). The authors show that given a bounded convex domain \(\Omega\) and a differentiable function \(\varphi\) on the boundary of \(\Omega\) and a strictly positive function \(S_n\) defined on \(\overline\Omega\) there exists a hyperbolic hypersurface with the following properties: (1) the equiaffine Gauss-Kronecker curvature is the given function \(S_n\), (2) \(M\) is Euclidean complete and also the metric \(h(SX,Y)\) is complete (Weingarten complete) (3) if \(\Omega\) is the ball then \(M\) is also complete with respect to the affine metric. One of the main ideas of the proof is to split the fourth-order PDE into 2 second order PDEs of Monge Ampère type. This was also used in [\textit{A. Li}, \textit{U. Simon} and \textit{B. Chen}, J. Reine Angew. Math. 487, 179-200 (1997; Zbl 0886.35065)].
reviewed by: Luc Vrancken
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 1505833
zbMATH Keywords: affine normal
zbMATH Keywords: affine shape operator
zbMATH Keywords: affine Gauss-Kronecker curvature
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1505833