Complete self-shrinkers with bounded the second fundamental form in $\mathbb{R}^{n+1}$

Author:

Yayun Chen, Tongzhu Li

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG)

journal:

date:

2023-06-16 16:00:00

Abstract

Let $X:M^n\to \mathbb{R}^{n+1}$ be a complete properly immersed self-shrinker. In this paper, we prove that if the squared norm of the second fundamental form $S$ satisfies $1\leq S< C$ for some constant $C$, then $S=1$. Further we classify the $n$-dimensional complete proper self-shrinkers with constant squared norm of the second fundamental form in $\mathbb{R}^{n+1}$, which solve the conjecture proposed by Q.M. Cheng and G. Wei when the self-shrinker is proper.

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