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