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.
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Cheng, X., Zhou, D. Stability properties and gap theorem for complete f-minimal hypersurfaces. Bull Braz Math Soc, New Series 46, 251–274 (2015). https://doi.org/10.1007/s00574-015-0092-z
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DOI: https://doi.org/10.1007/s00574-015-0092-z