In this article we study the two-dimensional completeλ-translators immersed in the Euclidean space R^3 and Minkovski space R1^ 3.We obtain two classification theorems:one for two-dimensional completeλ-translators x:...In this article we study the two-dimensional completeλ-translators immersed in the Euclidean space R^3 and Minkovski space R1^ 3.We obtain two classification theorems:one for two-dimensional completeλ-translators x:M 2→R^3 and one for two-dimensional complete space-likeλ-translators x:M 2→R1^3,with a second fundamental form of constant length.展开更多
In this paper,we study the pinching problem for a hypersurface with constant mean curvature in space forms to be totally umbilical by osing the relationship between the square of the length of the second fundamental f...In this paper,we study the pinching problem for a hypersurface with constant mean curvature in space forms to be totally umbilical by osing the relationship between the square of the length of the second fundamental form and the mean curvature. We obtained a best pinching interval and decided the complete classification of hypersurfaces at the terminal of the interval.This improved the relative results of M. Okumura,Shen Yibihg and Sun Ziqi,etc.展开更多
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 obtain a formula for submanifolds in Sn+p by calculating the Laplacian of the Moebius second fundamental form. Using this formula, we obtain some pinching theorems about the minimal eigenvalue of the...In this paper, we obtain a formula for submanifolds in Sn+p by calculating the Laplacian of the Moebius second fundamental form. Using this formula, we obtain some pinching theorems about the minimal eigenvalue of the Blaschke tensor.展开更多
Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which...Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n > 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.展开更多
We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of...We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of the second fundamental form of M, and fk = ∑λi^k and λi（1 ≤ i ≤ n） are the principal curvatures of M. We prove that there exists a positive constant δ（n）（≥ n/2） depending only on n such that if n ≤ S ≤ n ＋δ（n）, then S ≡ n, i.e., M is one of the Clifford torus S^K （√k/n） × S^n-k （V√n-k/n） for 1≤ k ≤ n - i. Moreover, we prove that if S is a constant, then there exists a positive constant T（n）（≥ n -2/3） depending only on n such that ifn ≤ S 〈 n ＋ τ（n）, then S ≡n, i.e.. M is a Clifford torus.展开更多
This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.
A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For ...A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.展开更多
We first establish a integral inequality for compact maximal space-like subman ifolds in pseudo-Riemannian manifolds Np(n+p). Then, we investigate compact space-like sub manifolds and hupersurfaces with parallel secon...We first establish a integral inequality for compact maximal space-like subman ifolds in pseudo-Riemannian manifolds Np(n+p). Then, we investigate compact space-like sub manifolds and hupersurfaces with parallel second fundamental form in Np(n+p) and some other ambient spaces. We obtain some distribution theorems for the square norm of the second fundamental form.展开更多
In this paper,we study the complete space-like submanifold Mn with constant scalar curvature R≤c in the de Sitter space Spn+p(c) and obtain a pinching condition for Mn to be totally umbilical ones.The result generali...In this paper,we study the complete space-like submanifold Mn with constant scalar curvature R≤c in the de Sitter space Spn+p(c) and obtain a pinching condition for Mn to be totally umbilical ones.The result generalizes that in [5,Main Theorem] to higher codimension and give a complement for n=2 there.展开更多
Let E_(s)^(m+p+1) ?R_(s+1)^(m+p+2)(m≥ 2,p≥ 1,0≤s≤p) be the standard(punched)light-cone in the Lorentzian space R_(s+1)^(m+p+2),and let Y:M^(m)→E_(s)^(m+p+1) be a space-like immersed submanifold of dimension m.The...Let E_(s)^(m+p+1) ?R_(s+1)^(m+p+2)(m≥ 2,p≥ 1,0≤s≤p) be the standard(punched)light-cone in the Lorentzian space R_(s+1)^(m+p+2),and let Y:M^(m)→E_(s)^(m+p+1) be a space-like immersed submanifold of dimension m.Then,in addition to the induced metric g on Mm,there are three other important invariants of Y:the Blaschke tensor A,the conic second fundamental form B,and the conic Mobius form C;these are naturally defined by Y and are all invariant under the group of rigid motions on E_(s)^(m+p+1).In particular,g,A,B,C form a complete invariant system for Y,as was originally shown by C.P.Wang for the case in which s=0.The submanifold Y is said to be Blaschke isoparametric if its conic Mobius form C vanishes identically and all of its Blaschke eigenvalues are constant.In this paper,we study the space-like Blaschke isoparametric submanifolds of a general codimension in the light-cone E_(s)^(m+p+1) for the extremal case in which s=p.We obtain a complete classification theorem for all the m-dimensional space-like Blaschke isoparametric submanifolds in Epm+p+1of constant scalar curvature,and of two distinct Blaschke eigenvalues.展开更多
The geometry of hypersurfaces of a Kaehler manifold are studied. Some well-known formulas and theorems in theory of surfaces of Euclidean 3-space are generalized to the hypersurfaces in a Kaehler manifold, such as Gau...The geometry of hypersurfaces of a Kaehler manifold are studied. Some well-known formulas and theorems in theory of surfaces of Euclidean 3-space are generalized to the hypersurfaces in a Kaehler manifold, such as Gauss's formulae, second fundamental form, the equation of Gauss and Codazzi and so forth.展开更多
The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the ...The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form （FFF） and the partial Noether operator＇s determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and fiat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.展开更多
The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We firs...The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized.展开更多
The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L1 introduced by S Y Cheng and S T Yau [3].
The geometric properties for Gaussian image of submanifolds in a sphere are investigated. The computation formula, geometric equalities and inequalities for the volume of Gaussian image of certain submanifolds in a sp...The geometric properties for Gaussian image of submanifolds in a sphere are investigated. The computation formula, geometric equalities and inequalities for the volume of Gaussian image of certain submanifolds in a sphere are obtained.展开更多
Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, M...Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.展开更多
Let M be a complete Riemannian manifold possibly with a boundary?M.For any C^1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizatio...Let M be a complete Riemannian manifold possibly with a boundary?M.For any C^1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.展开更多
基金Foundation of Natural Sciences of China(11671121,11871197 and 11971153)。
文摘In this article we study the two-dimensional completeλ-translators immersed in the Euclidean space R^3 and Minkovski space R1^ 3.We obtain two classification theorems:one for two-dimensional completeλ-translators x:M 2→R^3 and one for two-dimensional complete space-likeλ-translators x:M 2→R1^3,with a second fundamental form of constant length.
文摘In this paper,we study the pinching problem for a hypersurface with constant mean curvature in space forms to be totally umbilical by osing the relationship between the square of the length of the second fundamental form and the mean curvature. We obtained a best pinching interval and decided the complete classification of hypersurfaces at the terminal of the interval.This improved the relative results of M. Okumura,Shen Yibihg and Sun Ziqi,etc.
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.
文摘In this paper, we obtain a formula for submanifolds in Sn+p by calculating the Laplacian of the Moebius second fundamental form. Using this formula, we obtain some pinching theorems about the minimal eigenvalue of the Blaschke tensor.
基金Supported by the Department of Education of Hubei Province(B2014281)
文摘Let x : M → R n be an umbilical free hypersurface with non-zero principal curvatures, then x is associated with a Laguerre metric g, a Laguerre tensor L, a Laguerre form C, a Laguerre second fundamental form B, which are invariants of x under Laguerre transformation group. A classical theorem of Laguerre geometry states that M(n > 3) is characterized by g and B up to Laguerre equivalence. A Laguerre isopararmetric hypersurface is defined by satisfying the conditions that C = 0 and all the eigenvalues of B with respect to g are constant. It is easy to see that all Laguerre isopararmetric hypersurfaces are Dupin hypersurfaces. In this paper, we established a complete classification for all Laguerre isopararmetric hypersurfaces with three distinct principal curvatures in R7.
基金Supported by the National Natural Science Foundation of China (11071211)
文摘We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S^n＋1 satisfying S f4 - f^2 3 ≤1/nS^3 where S is the squared norm of the second fundamental form of M, and fk = ∑λi^k and λi（1 ≤ i ≤ n） are the principal curvatures of M. We prove that there exists a positive constant δ（n）（≥ n/2） depending only on n such that if n ≤ S ≤ n ＋δ（n）, then S ≡ n, i.e., M is one of the Clifford torus S^K （√k/n） × S^n-k （V√n-k/n） for 1≤ k ≤ n - i. Moreover, we prove that if S is a constant, then there exists a positive constant T（n）（≥ n -2/3） depending only on n such that ifn ≤ S 〈 n ＋ τ（n）, then S ≡n, i.e.. M is a Clifford torus.
基金Supported by the NNSF of China(10231010)the Trans-Century Training Programme Foundation for Talents by the Ministry of Education of China+1 种基金the Natural Science Foundation of Zhejiang Province(101037) Fudan Postgraduate Students Innovation Project(CQH5928002)
文摘This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.
基金Research supported by the NSFC (10231010)Trans-Century Training Programme Foundation for Talents by the Ministry of Education of ChinaNatural Science Foundation of Zhejiang Province (101037).
文摘A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.
文摘We first establish a integral inequality for compact maximal space-like subman ifolds in pseudo-Riemannian manifolds Np(n+p). Then, we investigate compact space-like sub manifolds and hupersurfaces with parallel second fundamental form in Np(n+p) and some other ambient spaces. We obtain some distribution theorems for the square norm of the second fundamental form.
文摘In this paper,we study the complete space-like submanifold Mn with constant scalar curvature R≤c in the de Sitter space Spn+p(c) and obtain a pinching condition for Mn to be totally umbilical ones.The result generalizes that in [5,Main Theorem] to higher codimension and give a complement for n=2 there.
基金supported by Foundation of Natural Sciences of China(11671121,11871197 and 11431009)。
文摘Let E_(s)^(m+p+1) ?R_(s+1)^(m+p+2)(m≥ 2,p≥ 1,0≤s≤p) be the standard(punched)light-cone in the Lorentzian space R_(s+1)^(m+p+2),and let Y:M^(m)→E_(s)^(m+p+1) be a space-like immersed submanifold of dimension m.Then,in addition to the induced metric g on Mm,there are three other important invariants of Y:the Blaschke tensor A,the conic second fundamental form B,and the conic Mobius form C;these are naturally defined by Y and are all invariant under the group of rigid motions on E_(s)^(m+p+1).In particular,g,A,B,C form a complete invariant system for Y,as was originally shown by C.P.Wang for the case in which s=0.The submanifold Y is said to be Blaschke isoparametric if its conic Mobius form C vanishes identically and all of its Blaschke eigenvalues are constant.In this paper,we study the space-like Blaschke isoparametric submanifolds of a general codimension in the light-cone E_(s)^(m+p+1) for the extremal case in which s=p.We obtain a complete classification theorem for all the m-dimensional space-like Blaschke isoparametric submanifolds in Epm+p+1of constant scalar curvature,and of two distinct Blaschke eigenvalues.
基金The Project (No.19771068) Supported by the National Science Foundation of China.
文摘The geometry of hypersurfaces of a Kaehler manifold are studied. Some well-known formulas and theorems in theory of surfaces of Euclidean 3-space are generalized to the hypersurfaces in a Kaehler manifold, such as Gauss's formulae, second fundamental form, the equation of Gauss and Codazzi and so forth.
文摘The derivation of conservation laws for the wave equation on sphere, cone and flat space is considered. The partial Noether approach is applied for wave equation on curved surfaces in terms of the coefficients of the first fundamental form （FFF） and the partial Noether operator＇s determining equations are derived. These determining equations are then used to construct the partial Noether operators and conserved vectors for the wave equation on different surfaces. The conserved vectors for the wave equation on the sphere, cone and fiat space are simplified using the Lie point symmetry generators of the equation and conserved vectors with the help of the symmetry conservation laws relation.
文摘The purpose of this paper is to study complete space-like submanifolds with parallel mean curvature vector and flat normal bundle in a locally symmetric semi-defnite space satisfying some curvature conditions. We first give an optimal estimate of the Laplacian of the squared norm of the second fundamental form for such submanifold. Furthermore, the totally umbilical submanifolds are characterized.
基金Supported the NSF of the Education Department of Jiangsu Province(04KJD110192)
文摘The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L1 introduced by S Y Cheng and S T Yau [3].
基金Supported by the National Natural Science Foundation of China(10231010)Trans-Century Training Programme Foundation for Talents by the Ministry of Education of Chinathe Natural Science Foundation of Zhejiang Province(101037).
文摘The geometric properties for Gaussian image of submanifolds in a sphere are investigated. The computation formula, geometric equalities and inequalities for the volume of Gaussian image of certain submanifolds in a sphere are obtained.
文摘Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.
基金supported by National Natural Science Foundation of China(Grant Nos.11771326 and 11431014)
文摘Let M be a complete Riemannian manifold possibly with a boundary?M.For any C^1-vector field Z,by using gradient/functional inequalities of the(reflecting)diffusion process generated by L:=?+Z,pointwise characterizations are presented for the Bakry-Emery curvature of L and the second fundamental form of?M if it exists.These characterizations extend and strengthen the recent results derived by Naber for the uniform norm‖RicZ‖∞on manifolds without boundaries.A key point of the present study is to apply the asymptotic formulas for these two tensors found by the first author,such that the proofs are significantly simplified.
