Index estimates for free boundary $f$-minimal hypersurfaces

TL;DR

This paper establishes a lower bound on the index of compact free boundary $f$-minimal hypersurfaces in positively curved weighted manifolds, relating it linearly to the first Betti number.

Contribution

It provides a new linear lower bound on the index for free boundary $f$-minimal hypersurfaces in specific curved weighted manifolds.

Findings

Index is bounded below by a linear function of the first Betti number.

Applicable to compact free boundary $f$-minimal hypersurfaces in certain curved weighted manifolds.

Abstract

We prove that the index is bounded from below by a linear function of its first Betti number for any compact free boundary $f$-minimal hypersurface in certain positively curved weighted manifolds.