Manifolds with weighted Poincar\'e inequality and uniqueness of minimal hypersurfaces

Abstract

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