The conjecture [1] asserts that any biharmonic submanifold in sphere has constant mean curvature. In this paper, we first prove that this conjecture is true for pseudo-umbilical biharmonic submanifolds M n in constant...The conjecture [1] asserts that any biharmonic submanifold in sphere has constant mean curvature. In this paper, we first prove that this conjecture is true for pseudo-umbilical biharmonic submanifolds M n in constant curvature spaces S n+p (c)(c > 0), generalizing the result in [1]. Secondly, some sufficient conditions for pseudo-umbilical proper biharmonic submanifolds M n to be totally umbilical ones are obtained.展开更多
In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4(-1)(H5(-1)) with constant mean curvature H satisfying H2 6634 (H2 114785 respectively) must be compact. Speciall...In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4(-1)(H5(-1)) with constant mean curvature H satisfying H2 6634 (H2 114785 respectively) must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4(-1) (resp. H5(-1)) with constant mean curvature H satisfying H2 6643 (resp. H2 114785 ) must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].展开更多
In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces' geometric invariants. This extends the previous similar result on compact m...In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces' geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.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
Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n + 1)-dimensional Riemannian manifold N^n+1 whose (n - 1)th Ricci curvature satisfying Ric^N(...Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n + 1)-dimensional Riemannian manifold N^n+1 whose (n - 1)th Ricci curvature satisfying Ric^N(n-1) (n - 1)c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If |φ|^2 - (n- 2)√n(n- 1)|H||φ|+ n(2n - 1)(H^2+ c) 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H^2 〉 0, then M must have only one end.展开更多
In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons' type integral inequality of compact submanifolds as well ...In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons' type integral inequality of compact submanifolds as well as some pinching theorems on the second fundamental form.展开更多
<span style="line-height:1.5;">For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric </span><img src="Edit_7bad0ce2-ecaa-4318-b3c...<span style="line-height:1.5;">For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric </span><img src="Edit_7bad0ce2-ecaa-4318-b3c9-5bbcfa7c087e.png" alt="" style="line-height:1.5;" /><span style="line-height:1.5;"></span><span "="" style="line-height:1.5;"><span> and the momentum </span><img src="Edit_c86b710a-9b65-4220-a4e2-cff8eeab9642.png" alt="" /></span><span style="line-height:1.5;"></span><span style="line-height:1.5;">. Canonical quantization requires a proper promotion of these classical variables to quantum operators, which, according to Dirac, the favored operators should be those arising from classical variables that formed Cartesian coordinates;sadly, in this case, that is not possible. However, an affine quantization feature</span><span style="line-height:1.5;">s</span><span "="" style="line-height:1.5;"><span> promoting the metric </span><img src="Edit_d0035f64-c366-4510-9cc7-d1053f755369.png" alt="" /></span><span "="" style="line-height:1.5;"><span> and the momentric </span><img src="Edit_60c18bb8-525b-4896-ae8f-2cd6456eb6f7.png" alt="" /></span><span "="" style="line-height:1.5;"><span> to operators. Instead of these classical variables belonging to a constant zero curvature space (</span><i><span>i.e.</span></i><span>, instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.展开更多
In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piece...In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.展开更多
In this paper, we study a special class of two-dimensional Finsler metrics defined by a Riemannian metric and 1-form. We classify those which are locally projectively flat with constant flag curvature.
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
Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v(k ≥ 1) of a submanifol...Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v(k ≥ 1) of a submanifold M^n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k^v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.展开更多
By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stabl...By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1+1 are considered.展开更多
Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where...Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖<sub>2</sub> denotes the L<sup>2</sup>-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S<sup>3</sup> with constant mean curvature.展开更多
For a Riemannian manifold(Mn,g) with curvature tensor R,the Jacobi operator J(X) is give.In this paper,the flat Riemannian manifolds are characterized in terms of special commutation properties of their Jacobi ope...For a Riemannian manifold(Mn,g) with curvature tensor R,the Jacobi operator J(X) is give.In this paper,the flat Riemannian manifolds are characterized in terms of special commutation properties of their Jacobi operators.展开更多
In this paper we obtain some formulas for totally umbilical submanifolds in a localiy symmetric manifold, and dcrivc some local rcsults on the submanifolds from these formulas.
In this paper,we study hypersurfaces of H^(2)×H^(2).We first classify the hypersurfaces with constant principal curvatures and constant product angle functions.Then we classify homogeneous hypersurfaces and isopa...In this paper,we study hypersurfaces of H^(2)×H^(2).We first classify the hypersurfaces with constant principal curvatures and constant product angle functions.Then we classify homogeneous hypersurfaces and isoparametric hypersurfaces,respectively.Finally,we classify the hypersurfaces with at most two distinct constant principal curvatures,as well as those with three distinct constant principal curvatures under some additional conditions.展开更多
It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12(4ε) of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is "Besides tota...It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12(4ε) of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is "Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12(46) are there?" In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12(4ε) with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12(4ε) is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.展开更多
基金Supported by the NNSF of China(71061012)Supported by the Young Talents Project of Dingxi Teacher's College(2012-2017)
文摘The conjecture [1] asserts that any biharmonic submanifold in sphere has constant mean curvature. In this paper, we first prove that this conjecture is true for pseudo-umbilical biharmonic submanifolds M n in constant curvature spaces S n+p (c)(c > 0), generalizing the result in [1]. Secondly, some sufficient conditions for pseudo-umbilical proper biharmonic submanifolds M n to be totally umbilical ones are obtained.
基金supported by NSFC (10901067)partially supported by NSFC (10801058) and Hubei Key Laboratory of Mathematical Sciences
文摘In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4(-1)(H5(-1)) with constant mean curvature H satisfying H2 6634 (H2 114785 respectively) must be compact. Specially, we verify that any complete and stable hypersurface in the hyperbolic space H4(-1) (resp. H5(-1)) with constant mean curvature H satisfying H2 6643 (resp. H2 114785 ) must be compact. It shows that there is no manifold satisfying the conditions of some theorems in [7, 9].
文摘In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces' geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.
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Supported by the National Natural Science Foundation of China (10771187 10671087)+1 种基金Trans-Century Training Programme Foundation for Talents by the Ministry of Education of ChinaJiangxi Province Natural Science Foundation (2008GZS0060)
文摘Let M^n be an n-dimensional complete noncompact oriented weakly stable constant mean curvature hypersurface in an (n + 1)-dimensional Riemannian manifold N^n+1 whose (n - 1)th Ricci curvature satisfying Ric^N(n-1) (n - 1)c. Denote by H and φ the mean curvature and the trace-free second fundamental form of M respectively. If |φ|^2 - (n- 2)√n(n- 1)|H||φ|+ n(2n - 1)(H^2+ c) 〉 0, then M does not admit nonconstant bounded harmonic functions with finite Dirichlet integral. In particular, if N has bounded geometry and c + H^2 〉 0, then M must have only one end.
基金Foundation item: Supported by the National Natural Science Foundation of China(ll071005) Supported by the Natural Science Foundation of Anhui Province Education Department(KJ2008A05zC)
文摘In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons' type integral inequality of compact submanifolds as well as some pinching theorems on the second fundamental form.
文摘<span style="line-height:1.5;">For purposes of quantization, classical gravity is normally expressed by canonical variables, namely the metric </span><img src="Edit_7bad0ce2-ecaa-4318-b3c9-5bbcfa7c087e.png" alt="" style="line-height:1.5;" /><span style="line-height:1.5;"></span><span "="" style="line-height:1.5;"><span> and the momentum </span><img src="Edit_c86b710a-9b65-4220-a4e2-cff8eeab9642.png" alt="" /></span><span style="line-height:1.5;"></span><span style="line-height:1.5;">. Canonical quantization requires a proper promotion of these classical variables to quantum operators, which, according to Dirac, the favored operators should be those arising from classical variables that formed Cartesian coordinates;sadly, in this case, that is not possible. However, an affine quantization feature</span><span style="line-height:1.5;">s</span><span "="" style="line-height:1.5;"><span> promoting the metric </span><img src="Edit_d0035f64-c366-4510-9cc7-d1053f755369.png" alt="" /></span><span "="" style="line-height:1.5;"><span> and the momentric </span><img src="Edit_60c18bb8-525b-4896-ae8f-2cd6456eb6f7.png" alt="" /></span><span "="" style="line-height:1.5;"><span> to operators. Instead of these classical variables belonging to a constant zero curvature space (</span><i><span>i.e.</span></i><span>, instead of a flat space), they belong to a space of constant negative curvatures. This feature may even have its appearance in black holes, which could strongly point toward an affine quantization approach to quantize gravity.
基金supported in part by the NSFC(11801496,11926352)the Fok Ying-Tung Education Foundation(China)the Hubei Key Laboratory of Applied Mathematics(Hubei University).
文摘In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.
基金Supported by the Fundamental Research Funds for the Central Universities
文摘In this paper, we study a special class of two-dimensional Finsler metrics defined by a Riemannian metric and 1-form. We classify those which are locally projectively flat with constant flag curvature.
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partially supported by NSFC(Grant No.11331002)the Fundamental Research Funds for the Central Universities
文摘Motivated by the theory of isoparametric hypersurfaces,we study submanifolds whose tubular hypersurfaces have some constant higher order mean curvatures.Here a k-th order mean curvature Q_k^v(k ≥ 1) of a submanifold M^n-is defined as the k-th power sum of the principal curvatures,or equivalently,of the shape operator with respect to the unit normal vector v.We show that if all nearby tubular hypersurfaces of M have some constant higher order mean curvatures,then the submanifold M itself has some constant higher order mean curvatures Q_k^v independent of the choice of v.Many identities involving higher order mean curvatures and Jacobi operators on such submanifolds are also obtained.In particular,we generalize several classical results in isoparametric theory given by E.Cartan,K.Nomizu,H.F.Miinzner,Q.M.Wang,et al.As an application,we finally get a geometrical filtration for the focal submanifolds of isoparametric functions on a complete Riemannian manifold.
基金The first author is partially supported by the National Natural Science Foundation of China (No.10271106)The second author is partially supported by the 973-Grant of Mathematics in China and the Huo Y.-D. fund.
文摘By using curvature estimates, we prove that a complete non-compact hypersurface M with constant mean curvature and finite L^n-norm curvature in R^1+1 must be minimal, so that M is a hyperplane if it is strongly stable. This is a generalization of the result on stable complete minimal hypersurfaces of R^n+1. Moreover, complete strongly stable hypersurfaces with constant mean curvature and finite L^1-norm curvature in R^1+1 are considered.
文摘Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖<sub>2</sub> denotes the L<sup>2</sup>-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S<sup>3</sup> with constant mean curvature.
基金Sponsored by the National Natural Science Foundation of China(10871218)
文摘For a Riemannian manifold(Mn,g) with curvature tensor R,the Jacobi operator J(X) is give.In this paper,the flat Riemannian manifolds are characterized in terms of special commutation properties of their Jacobi operators.
文摘In this paper we obtain some formulas for totally umbilical submanifolds in a localiy symmetric manifold, and dcrivc some local rcsults on the submanifolds from these formulas.
基金supported by National Natural Science Foundation of China (Grant Nos. 11831005, 12061131014 and 12171437)China Postdoctoral Science Foundation (Grant No. 2022M721871)
文摘In this paper,we study hypersurfaces of H^(2)×H^(2).We first classify the hypersurfaces with constant principal curvatures and constant product angle functions.Then we classify homogeneous hypersurfaces and isoparametric hypersurfaces,respectively.Finally,we classify the hypersurfaces with at most two distinct constant principal curvatures,as well as those with three distinct constant principal curvatures under some additional conditions.
文摘It is well known that a totally geodesic Lagrangian surface in a Lorentzian complex space form M12(4ε) of constant holomorphic sectional curvature 4s is of constant curvature 6. A natural question is "Besides totally geodesic ones how many Lagrangian surfaces of constant curvature εin M12(46) are there?" In an earlier paper an answer to this question was obtained for the case e = 0 by Chen and Fastenakels. In this paper we provide the answer to this question for the case ε≠0. Our main result states that there exist thirty-five families of Lagrangian surfaces of curvature ε in M12(4ε) with ε ≠ 0. Conversely, every Lagrangian surface of curvature ε≠0 in M12(4ε) is locally congruent to one of the Lagrangian surfaces given by the thirty-five families.