Blaschke hypersurfaces with constant negative affine mean curvature
type: Item
instance of: scholarly article
title: Blaschke hypersurfaces with constant negative affine mean curvature
zbMATH Open document ID: 1319.53011
author: An-Min Li
author: Li Sheng
author: Udo Simon
published in: Annals of Global Analysis and Geometry
publication date: 2015-03-23T00:00:00Z
review text: The authors prove the following main theorems. Theorem 1. Let \(x: M\to\mathbb R^{n+1}\) be a locally strongly convex, affine complete Blaschke hypersurface, satisfying the following conditions: {\parindent=8mm\begin{itemize}\item[(i)] The affine Weingarten operator \(B\) is bounded below: \(B\geq k\cdot \mathrm{id}\) for \(0>k\in R;\) \item[(ii)] \(H=\) const. \(<0\); \item[(iii)] the affine support function \(\Lambda(x_0)\) is bounded for some \(x_o\in R^{n+1}\). \end{itemize}} Then \(x(M)\) is an affine complete hyperbolic affine sphere. Theorem 2. A locally strongly convex affine complete hypersphere, satisfying the following condition \(\chi\geq 0\) on the normalized scalar curvature, is either a quadratic hypersurface or a hyperbolic hypersphere of type \(Q(c,n)\).
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 6418169
zbMATH Keywords: negative affine mean curvature
zbMATH Keywords: affine sphere
zbMATH Keywords: scalar curvature of the Blaschke metric
zbMATH Keywords: maximum principal of Omori-Yau
reviewed by: Huili Liu
MaRDI profile type: Publication
full work available at URL: s10455-014-9443-z
OpenAlex ID: W2085595105
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DOI: 10.1007/S10455-014-9443-Z
description: scientific article; zbMATH DE number 6418169