# Stability of weighted minimal hypersurfaces under a lower $1$-weighted Ricci curvature bound

## TL;DR

This paper investigates the geometric properties and stability of weighted minimal hypersurfaces in manifolds with a lower bound on the 1-weighted Ricci curvature, deriving new criteria, structure theorems, and non-existence results.

## Contribution

It introduces new stability criteria and structure theorems for weighted minimal hypersurfaces under 1-weighted Ricci curvature bounds, extending classical results to weighted manifolds.

## Key findings

- Established a Schoen-Yau type criterion for stability
- Proved a structure theorem for 3D weighted manifolds with non-negative 1-weighted Ricci curvature
- Derived non-existence results under volume growth conditions

## Abstract

We will study the $1$-weighted Ricci curvature in view of the extrinsic geometric analysis. We derive several geometric consequences concerning stable weighted minimal hypersurfaces in weighted manifolds under a lower $1$-weighted Ricci curvature bound. We prove a Schoen-Yau type criterion, and conclude a structure theorem for three-dimensional weighted manifolds of non-negative $1$-weighted Ricci curvature. We also show non-existence results under volume growth conditions, and conclude smooth compactness theorems.