关键词: Riemannian manifold, area-minimizing hypersurface, Yamabe invariant

Abstract: In this paper, for any local area-minimizing closed hypersurface $\Sigma$ with $Rc_{\Sigma}=\frac{R_\Sigma}{n}g_{\Sigma}$, immersed in a $(n+1)$-dimension Riemannian manifold $M$ which has positive scalar curvature and nonnegative Ricci curvature, we obtain an upper bound for the area of $\Sigma$. In particular, when $\Sigma$ saturates the corresponding upper bound, $\Sigma$ is isometric to $\mathbb{S}^n$ and $M$ splits in a neighborhood of $\Sigma$. At the end of the paper, we also give the global version of this result.

Key words: Riemannian manifold, area-minimizing hypersurface, Yamabe invariant