A new pinching theorem for complete self-shrinkers and its generalization
A new pinching theorem for complete self-shrinkers and its generalization
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完全自收缩器的新箍缩定理及其推广
DOI：
10.1007/s11425-018-9397-y
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通讯作者：
Zhiyuan Xu
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作者：
Li Lei;Hongwei Xu;Zhiyuan Xu
本文首先证明了：如果M是n+1中多项式体积增长的n维完全自收缩体，且M的第二基本形式的平方范数满足，则S ∈ n +1和M是圆球或圆柱.更一般地说，设M是余维为1的完备λ-超曲面，其体积在λ n+1中多项式增长，其中λ ≠ 0。然后我们证明存在一个正的常数γ，使得如果|λ| <$γ和M的第二基本形式的平方范数满足，则S <$βλ，λ > 0且M是圆柱。这里.
In this paper, we firstly verify that ifMnis ann-dimensional complete self-shrinker with polynomial volume growth in ℝn+1, and if the squared norm of the second fundamental form ofMsatisfies, thenS≡ 1 andMis a round sphere or a cylinder. More generally, letMbe a complete λ-hypersurface of codimension one with polynomial volume growth in ℝn+1with λ ≠ 0. Then we prove that there exists a positive constant γ, such that if |λ| ⩽ γ and the squared norm of the second fundamental form ofMsatisfies, thenS≡βλ, λ > 0 andMis a cylinder. Here.