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arXiv · 1307.5099

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

Xu Cheng·Detang Zhou
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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.

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Xu Cheng, Detang Zhou. 2013-07-18. Stability properties and gap theorem for complete $f$-minimal hypersurfaces. https://arxiv.org/abs/1307.5099

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