Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2014; 51(2): 567-577

Printed March 31, 2014

https://doi.org/10.4134/BKMS.2014.51.2.567

Copyright © The Korean Mathematical Society.

Linear Weingarten hypersurfaces in Riemannian space forms

Xiaoli Chao and Peijun Wang

Southeast University, Southeast University

Abstract

In this note, we generalize the weak maximum principle in \cite{A1} to the case of complete {\it linear Weingarten hypersurface} in Riemannian space form $\mathbb{M}^{n+1}(c)$ $(c=1,0,-1)$, and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in $\mathbb{S}^{n+1}(1)$ with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei \cite{Li2} and Shu \cite{S} to the case of {\it complete hypersurface}.

Keywords: linear Weingarten hypersurface, maximum principle, space form, Clifford torus, circular cylinder, hyperbolic cylinder

MSC numbers: 53C40, 53C42