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

TL;DR

This paper establishes lower bounds on the Morse index of $f$-minimal hypersurfaces and self-shrinkers in weighted Euclidean spaces, relating it to topological invariants like the first Betti number and genus.

Contribution

It provides new index estimates for $f$-minimal hypersurfaces and self-shrinkers, improving previous bounds by linking the index to topological quantities.

Findings

Index is bounded below by an affine function of the first Betti number for compact hypersurfaces.

In non-compact cases, the index relates to the dimension of weighted $f$-harmonic 1-forms.

For surfaces in dimension 2, the index estimate depends on the genus.

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.