Stability properties and gap theorem for complete $f$-minimal hypersurfaces

TL;DR

This paper investigates the stability and geometric properties of complete $f$-minimal hypersurfaces in a shrinking soliton, establishing classification results for those with minimal index and a pinching theorem.

Contribution

It classifies complete $f$-minimal hypersurfaces with $L_f$-index one in a shrinking soliton and proves a related pinching theorem, advancing understanding of their stability properties.

Findings

Hypersurfaces with $L_f$-index one are either $ ext{S}^n imes ext{0}$ or $ ext{S}^{n-1} imes ext{R}$.

A pinching theorem constrains the geometry of such hypersurfaces.

The work enhances classification and stability theory for $f$-minimal hypersurfaces.

Abstract

In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1}$ denotes the sphere in $\mathbb{S}^{n}$ of the same radius. Also we prove a pinching theorem for them.