A new pinching theorem for complete self-shrinkers and its generalization

Li Lei; Hongwei Xu; Zhiyuan Xu

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A new pinching theorem for complete self-shrinkers and its generalization

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In this paper,we firstly verify that if Mn is an n-dimensional complete self-shrinker with polynomial volume growth in Rn+1,and if the squared norm of the second fundamental form of M satisfies 0≤S-1≤1/18,then S≡1 and M is a round sphere or a cylinder.More generally,let M be a complete λ-hypersurface of codimension one with polynomial volume growth in Rn+1 with 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 of M satisfies0≤S-βλ≤1/18,then S≡βλ,λ> 0 and M is a cylinder.Here βλ=1/2(2+λ2+λ(λ2+4)1/2).

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来源
《中国科学》 |2020年第6期|P.1139-1152|共14页
作者
Li Lei; Hongwei Xu; Zhiyuan Xu;
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作者单位

Center of Mathem atical Sciences Zhejiang University Hangzhou 310027 China;

Center of Mathem atical Sciences Zhejiang University Hangzhou 310027 China;

Departm ent of Mathem atics Hangzhou Normal University Hangzhou 310036 China;

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原文格式 PDF
正文语种 chi
中图分类古典微分几何;
关键词
rigidity theorem; the second fundamental form; self-shrinker; λ-hypersurface;

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A new pinching theorem for complete self-shrinkers and its generalization

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