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 ...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).展开更多
In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.
In this paper,we first construct compact embeddedλ-hypersurfaces with the topology of torus which are calledλ-torus in Euclidean spacesℝn^+1.Then,we give many compact immersedλ-hypersurfaces in Euclidean spacesℝn^+1.
基金National Natural Science Foundation of China (Grant Nos. 11531012, 11371315 and 11601478)the China Postdoctoral Science Foundation (Grant No. 2016M590530)。
文摘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).
基金supported by JSPS Grant-in-Aid for Scientific Research(B)(No.16H03937)the fund of Fukuoka University(No.225001)+2 种基金the National Natural Science Foundation of China(No.12171164)the Natural Science Foundation of Guangdong Province(No.2019A1515011451)GDUPS(2018)。
文摘In this paper, the authors give a survey about λ-hypersurfaces in Euclidean spaces. Especially, they focus on examples and rigidity of λ-hypersurfaces in Euclidean spaces.
基金supported by JSPS Grant-in-Aid for Scientific Research(B)(Grant No.16H03937)Challenging Exploratory Research+1 种基金supported by National Natural Science Foundation of China(Grant No.11771154)by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme(2018)。
文摘In this paper,we first construct compact embeddedλ-hypersurfaces with the topology of torus which are calledλ-torus in Euclidean spacesℝn^+1.Then,we give many compact immersedλ-hypersurfaces in Euclidean spacesℝn^+1.
