Estimates for stable hypersurfaces of prescribed \(F\)-mean curvature
type: Item
instance of: scholarly article
title: Estimates for stable hypersurfaces of prescribed \(F\)-mean curvature
zbMATH Open document ID: 1094.53059
author: Sven Winklmann
published in: Manuscripta Mathematica
publication date: 2006-01-23T00:00:00Z
review text: Curvature estimates for stable minimal hypersurfaces \(x:M\to N^{n+1}\) have been proved by \textit{R. Schoen, L. Simon, S.-T. Yau} [Acta Math. 134, 275--288 (1975; Zbl 0323.53039)]. It turns out that the method of Schoen-Simon-Yau is in fact general enough to be applicable to other important situations. In the paper under review the author considers immersed hypersurfaces \(X:M\to \mathbb R^{n+1}\) with prescribed anisotropic mean curvature \(H_F={\mathcal H}_F(X)\). Such hypersurfaces can be characterized as critical points of parametric functions of the type \[ {\mathcal F}^0(X)=\int_MF(N)\,d\mu+\int_M<Q(X),N>d\mu, \] with an elliptic Lagrangian \(F\) depending on normal directions and a smooth vector field \(Q\) satisfying \(\text{ div}_{\mathbb R^{n+1}}Q={\mathcal H}_F\). If \(F\) is \(C^3\)-close to the area integrand, the author establishes curvature estimates for stable hypersurfaces of dimension \(n\leq 5\). The case \(Q\equiv 0\) was studied by the author in two previous papers see [Calc. Var. Partial Differ. Equ. 23, 391--414 (2005; Zbl 1075.53053); Ann. Inst. H. Ponicaré, Anal. Non Linéaire 22, 543--555 (2005; Zbl 1088.53042)].
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 49Q10
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 5000076
zbMATH Keywords: stable hypersurfaces
zbMATH Keywords: prescribed anisotropic mean curvature
zbMATH Keywords: curvature estimate
reviewed by: Haizhong Li
MaRDI profile type: Publication
full work available at URL: s00229-005-0600-3
OpenAlex ID: W2050955336
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DOI: 10.1007/S00229-005-0600-3
description: scientific article; zbMATH DE number 5000076