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.
Xiaoli Chao and Peijun Wang
Southeast University, Southeast University
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
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