Stability properties and gap theorem for complete $f$-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.

23 pages

References in corpus (4)

• $f$-minimal surface and manifold with positive $m$-Bakry-Émery Ricci curvature
• Volume growth, eigenvalue and compactness for self-shrinkers
• Eigenvalues of the drifted Laplacian on complete metric measure spaces
• Stability and compactness for complete $f$-minimal surfaces

Cited by in corpus (1)

• Smooth compactness of $f$-minimal hypersurfaces with bounded $f$-index