Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of ...Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.展开更多
This paper proves that if the energy density of a harmonic map to a unit sphere varies between two successive half eigenvalues, then it must be one of them. Applying this result to the Gaussian maps of some submanifol...This paper proves that if the energy density of a harmonic map to a unit sphere varies between two successive half eigenvalues, then it must be one of them. Applying this result to the Gaussian maps of some submanifolds, the quantum phenomena of the square length of the second fundamental forms of these submanifolds is obtained. Some related topics are discussed in this note.展开更多
In this paper, we are concerned with the convergence behavior of a sequence of conformal immersions {fn} from long cylinders Pn with Pn｜Afn｜2＋ μ（fn（Pn）） 〈 Λ. We show that if {fn} does not converge to a point...In this paper, we are concerned with the convergence behavior of a sequence of conformal immersions {fn} from long cylinders Pn with Pn｜Afn｜2＋ μ（fn（Pn）） 〈 Λ. We show that if {fn} does not converge to a point, the total Gauss curvatures and the measures of the images of {fn} will not lose on the necks and each neck consists of a point.展开更多
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
Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constan...Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).展开更多
In this paper,the authors give a characterization theorem for the standard tori S^(1)(a)×S^(1)(b),a,b>0,as the compact Lagrangianξ-submanifolds in the two-dimensional complex Euclidean space C^(2),and obtain ...In this paper,the authors give a characterization theorem for the standard tori S^(1)(a)×S^(1)(b),a,b>0,as the compact Lagrangianξ-submanifolds in the two-dimensional complex Euclidean space C^(2),and obtain the best version of a former rigidity theorem for compact Lagrangianξ-submanifold in C^(2).Furthermore,their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in C^(2).展开更多
基金Project supported by the Stress Supporting Subject Foundation of Zhejiang Province, China
文摘Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.
基金Research supported by the NNSF of China (10071021)
文摘This paper proves that if the energy density of a harmonic map to a unit sphere varies between two successive half eigenvalues, then it must be one of them. Applying this result to the Gaussian maps of some submanifolds, the quantum phenomena of the square length of the second fundamental forms of these submanifolds is obtained. Some related topics are discussed in this note.
基金Supported by National Natural Science Foundation of China(Grant No.11201131)
文摘In this paper, we are concerned with the convergence behavior of a sequence of conformal immersions {fn} from long cylinders Pn with Pn｜Afn｜2＋ μ（fn（Pn）） 〈 Λ. We show that if {fn} does not converge to a point, the total Gauss curvatures and the measures of the images of {fn} will not lose on the necks and each neck consists of a point.
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 National Natural Science Foundation of China(Grant No.11531012)China Postdoctoral Science Foundation(Grant No.BX20180274)Natural Science Foundation of Zhejiang Province(Grant No.LY20A010024)。
文摘Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constantγ(n)depending only on n such that if|H|≤γ(n)andβ(n,H)≤S≤β(n,H)+n/18,then S≡β(n,H)and M is a Clifford torus.Here,β(n,H)=n+n^(3)/2(n-1)H^(2)+n(n-2)/2(n-1)(1/2)n^(2)H^(4)+4(n-1)H^(2).
基金supported by the National Natural Science Foundation of China(Nos.11671121,11871197)
文摘In this paper,the authors give a characterization theorem for the standard tori S^(1)(a)×S^(1)(b),a,b>0,as the compact Lagrangianξ-submanifolds in the two-dimensional complex Euclidean space C^(2),and obtain the best version of a former rigidity theorem for compact Lagrangianξ-submanifold in C^(2).Furthermore,their argument in this paper also proves a new rigidity theorem which is a direct generalization of a rigidity theorem by Li and Wang for Lagrangian self-shrinkers in C^(2).
