Communications in Analysis and Geometry

Volume 17 (2009)

Number 1

Manifolds with weighted Poincaré inequality and uniqueness of minimal hypersurfaces

Pages: 139 – 154

DOI: https://dx.doi.org/10.4310/CAG.2009.v17.n1.a6

Authors

Xu Cheng (Insitituto de Matemática, Universidade Federal Fluminense, Niteró, RJ 24020-140, Brazil)

Detang Zhou (Insitituto de Matemática, Universidade Federal Fluminense, Niteró, RJ 24020-140, Brazil)

Abstract

In this paper, we obtain results on rigidity of complete Riemannianmanifolds with weighted Poincar\'e inequality. As an application, weprove that if $M$ is a complete $\frac{n-2}{n}$-stable minimalhypersurface in $\mathbb{R}^{n+1}$ with $n\geq 3$ and has boundednorm of the second fundamental form, then $M$ must either have onlyone end or be a catenoid.

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Published 1 January 2009