Index and first Betti number of $f$-minimal hypersurfaces and self-shrinkers

Abstract: We study the Morse index of self-shrinkers for the mean curvature flow and, more generally, of $f$-minimal hypersurfaces in a weighted Euclidean space endowed with a convex weight. When the hypersurface is compact, we show that the index is bounded from below by an affine function of its first Betti number. When the first Betti number is large, this improves index estimates known in literature. In the complete non-compact case, the lower bound is in terms of the dimension of the space of weighted square summable $f$-harmonic $1$-forms; in particular, in dimension $2$, the procedure gives an index estimate in terms of the genus of the surface.