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.
Hongcun DENG
. A BRAY-BRENDLE-NEVES TYPE INEQUALITY FOR A RIEMANNIAN MANIFOLD[J]. Acta mathematica scientia, Series B, 2021
, 41(2)
: 487
-492
.
DOI: 10.1007/s10473-021-0212-3
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