Simons' type equation for $f$-minimal hypersurfaces and applications

Abstract

We derive the Simons' type equation for $f$-minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed $f$-minimal hypersurfaces immersed in the product manifold ${\mathbb{S}}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with $f=\frac{t^2}{4}$. Also we classify closed $f$-minimal hypersurfaces with $L_f$-index one immersed in ${\mathbb{S}}^n(\sqrt{2(n-1)})\times \mathbb{R}$ with the same $f$ as above.

22 pages