Remarks on minimal hypersurfaces in shrinking gradient Ricci solitons

Abstract

In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in a shrinking gradient Ricci soliton $(M^{n},g,f)$ with scalar curvature $R\geq(n-1)λ$ must have vanished second fundamental form and vanished normal Ricci curvature. For shrinking gradient Ricci solitons with scalar curvature $R\geq(n-1)λ$, the existence of an area-minimizing hypersurface would imply that $M$ is splitting.