The affine complete hypersurfaces of constant Gauss-Kronecker curvature
type: Item
instance of: scholarly article
title: The affine complete hypersurfaces of constant Gauss-Kronecker curvature
zbMATH Open document ID: 1173.53008
author: Baofu Wang
publication date: 2009-09-03T00:00:00Z
review text: Many higher order PDEs that arise from geometric problems can be solved in a natural way by geometric methods. In [\textit{A. Li, U. Simon} and \textit{B. Chen}, J. Reine Angew. Math. 487, 179--200 (1997; Zbl 0886.35065)] the authors established a two-step Monge-Ampère procedure for the solution of certain non-linear fourth order PDEs. The geometric background lies in Blaschke's hypersurface theory (see. e.g. chapter 1 in [\textit{A.-M. Li, U. Simon} and \textit{G. Zhao}, Global affine differential geometry of hypersurfaces. Berlin: Walter de Gruyter (1993; Zbl 0808.53002)]). In affine hypersurface theory there are different, non-equivalent notions of ``completeness'' that play a role for global investigations (for details see e.g. section 2.3 in [Zbl 0808.53002]). Consider a locally strongly convex hypersurface in affine space \(\mathbb R^{n+1}\), given by an immersion \(x: M \rightarrow \mathbb R^{n+1}\) of an oriented, connected \(C^{\infty}\)-manifold \(M\). Any additional Euclidean structure on \(\mathbb R^{n+1}\) induces a Euclidean first fundamental form \(g\) on \(M\). The completeness of this metric \(g\) does not depend on the special choice of the Euclidean structure of the ambient space, and this completeness is an affine notion; it is called Euclidean completeness of the affine hypersurface, while the completeness of the Blaschke metric is called affine completeness, and the completeness of the affine spherical metric, defined by the Weingarten form, is called Weingarten completeness. Within these non-equivalent notions, S. Y. Cheng and S. T. Yau (see section 2.3 in [Zbl 0808.53002]) were the first to state sufficient conditions that the Euclidean completeness implies the affine completeness; meanwhile one knows different and more general results of this type. The author of the present paper proves the following Main Theorem, extending Theorem 3 from [Zbl 0886.35065]: Given a locally strongly convex, hyperbolic and Weingarten complete hypersurface \(x\)…
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 35G30
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 5601628
zbMATH Keywords: completeness in affine hypersurface theory
zbMATH Keywords: affine Gauss-Kronecker curvature
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: nonlinear fourth order PDE
reviewed by: Udo Simon
MaRDI profile type: Publication
full work available at URL: s10114-009-7063-5
OpenAlex ID: W2118516854
cites work: Global affine differential geometry of hypersurfaces
cites work: Spacelike hypersurfaces with constant Gauss-Kronecker curvature in the Minkowski space
cites work: Q4342826
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cites work: The dirichlet problem for nonlinear second-order elliptic equations I. Monge-ampégre equation
cites work: Centroaffine minimal hypersurfaces in \(\mathbb{R}^{n+1}\)
cites work: Centroaffine Bernstein problems
DOI: 10.1007/S10114-009-7063-5
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5601628