Remarks on minimal hypersurfaces in gradient shrinking Ricci solitons

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