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.
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 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
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.
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).
