Stability properties and gap theorem for complete f-minimal hypersurfaces

TLDR: In this article, complete oriented f -minimal hypersurfaces properly immersed in a cylinder shrinking soliton were studied and a pinching theorem for them was proved, where the sphere in the hypersurface is a sphere of the same radius.

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 $${S}^{n - 1}$$ denotes the sphere in $$\mathbb{S}^n$$ of the same radius. Also we prove a pinching theorem for them.