The $f$-Stability Index of the Constant Weighted Mean Curvature Hypersurfaces in Gradient Ricci Solitons

TL;DR

This paper investigates the properties and stability of hypersurfaces with constant weighted mean curvature in gradient Ricci solitons, providing conditions for properness and estimates for their $f$-stability index.

Contribution

It establishes properness criteria for hypersurfaces with finite weighted volume and derives an estimate for the $f$-stability index in shrinking gradient Ricci solitons.

Findings

Properness of hypersurfaces with finite weighted volume is proven.

An estimate for the $f$-stability index is obtained.

A necessary condition for equality in the stability estimate is provided.

Abstract

In this paper, we prove that a noncompact complete hypersurface with finite weighted volume, weighted mean curvature vector bounded in norm, and isometrically immersed in a complete weighted manifold is proper. In addition, we obtain an estimate for $f$-stability index of a constant weighted mean curvature hypersurface with finite weighted volume and isometrically immersed in a shrinking gradient Ricci soliton that admits at least one parallel field globally defined. For such hypersurface, we still give a necessary condition for equality to be achieved in the estimate obtained.