Stability properties and topology at infinity of f-minimal hypersurfaces

TL;DR

This paper investigates the stability and topological properties at infinity of f-minimal hypersurfaces in weighted manifolds with non-negative Bakry-Emery Ricci curvature, introducing new comparison results and Sobolev inequalities.

Contribution

It provides new stability criteria, a weighted version of Li-Tam theory, and a general weighted L^1-Sobolev inequality for hypersurfaces in Cartan-Hadamard weighted manifolds.

Findings

Established a new comparison result in weighted geometry.

Proved a weighted L^1-Sobolev inequality for hypersurfaces.

Analyzed the topology at infinity of f-minimal hypersurfaces.

Abstract

We study stability properties of $f$-minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the topology at infinity of $f$-minimal hypersurfaces. On the way, we prove a new comparison result in weighted geometry and we provide a general weighted $L^1$-Sobolev inequality for hypersurfaces in Cartan-Hadamard weighted manifolds, satisfying suitable restrictions on the weight function.