Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamen...Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of Mn,respectively.Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for Mn immersed in ~S^n+p _p(c) with parallel normalized mean curvature vector field is proved.When n≥3, the pinching constant is the best.Thus,the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math,1998,95:499-505) is corrected.Moreover,the reduction of the codimension when Mn is a complete submanifold in S^n+p _p(c) with parallel normalized mean curvature vector field is investigated.展开更多
Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau's self-adjoint differential operator, here we obtain some rigidity results for compac...Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau's self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.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 f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then un...Let f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.展开更多
In this paper,we study Ricci flow on compact manifolds with a continuous initial metric.It was known from Simon(2002)that the Ricci flow exists for a short time.We prove that the scalar curvature lower bound is preser...In this paper,we study Ricci flow on compact manifolds with a continuous initial metric.It was known from Simon(2002)that the Ricci flow exists for a short time.We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W^(1,p) for some n<p∞.As an application,we use this result to study the relation between the Yamabe invariant and Ricci flat metrics.We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense,then the manifold is isometric to a Ricci flat manifold.展开更多
On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of...On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the K?hler metric, the balanced metric, the locally conformal K?hler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier(2020) and a conjecture by Angella et al.(2018).展开更多
We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature.As a consequence,Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.For(α,β)-metrics on ma...We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature.As a consequence,Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.For(α,β)-metrics on manifold of dimension greater than 2,if the mean Landsberg curvature and the Berwald scalar curvature both vanish,then the Berwald curvature also vanishes.展开更多
In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of ...In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.展开更多
We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifold M^n(n≥3) with strongly negative curvature. By employing the super- subsolution method and a careful constructi...We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifold M^n(n≥3) with strongly negative curvature. By employing the super- subsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation.展开更多
In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.
A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K = 3c + δ where δ and c are scalar functions on M. In this paper, we establish the intrinsic relation bet...A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K = 3c + δ where δ and c are scalar functions on M. In this paper, we establish the intrinsic relation between scalar functions c(x) and a(x). More general, by invoking the Ricci identities for a one form, we investigate Finsler metric of weakly isotropic flag curvature K = 3θ/F + δ and show that F has constant flag curvature if θ is horizontally parallel.展开更多
In this paper,we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature,noncollasping volume and bounded diameter.Under such conditions,we can show that the manifold has finitely many di...In this paper,we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature,noncollasping volume and bounded diameter.Under such conditions,we can show that the manifold has finitely many diffeomorphism types,which generalizes Cheeger–Naber’s result to bounded scalar curvature along Ricci flow.In particular,this implies the manifold has uniform L^(2) Riemann curvature bound.As an application,we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound,lower volume bound and upper diameter bound.展开更多
Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H(n+1)(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that ...Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H(n+1)(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk(r)×H(n-k)(-1/(r2 + ρ2)), where r 〉 0 and 1 〈 k 〈 n - 1;(2)if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S(n-1)(r) × H1(-1/(r2 +ρ2)) or S1(r) × H(n-1)(-1/(r2 +ρ2)),r 〉 0, if one of the following conditions is satisfied (i) S≤(n-1)t22+c2t(-2)2 on Mn or (ii)S≥ (n-1)t21+c2t(-2)1 on Mn or(iii)(n-1)t22+c2t(-2)2≤ S≤(n-1)t21+c2t(-2)1 on Mn, where t_1 and t_2 are the positive real roots of (1.5).展开更多
This paper discusses the existence problem in the study of some partial differential equations. The author gets some bifurcation on the prescribed mean curvature problem on the unit ball, the scalar curvature problem ...This paper discusses the existence problem in the study of some partial differential equations. The author gets some bifurcation on the prescribed mean curvature problem on the unit ball, the scalar curvature problem on the n-sphere, and some field equations. The author gives some natural conditions such that the standard bifurcation or Thom-Mather theory can be used.展开更多
In this paper,we prove that for some completions of certain fiber bundles there is a Maxwell-Einstein metric conformally related to any given Kahler class.
We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate t...We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.展开更多
The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method...The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].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
Let (M, g) be an n-dimensional Riemannian manifold and T*M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki ...Let (M, g) be an n-dimensional Riemannian manifold and T*M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T*M. Second, we construct locally decomposable Golden Riemannian structures on T*M . Finally we investigate curvature properties of T*M .展开更多
文摘Let Mn be a closed spacelike submanifold isometrically immersed in de Sitter space S^n+p _p(c).Denote by R,H and S the normalized scalar curvature,the mean curvature and the square of the length of the second fundamental form of Mn,respectively.Suppose R is constant and R≤c. The pinching problem on S is studied and a rigidity theorem for Mn immersed in ~S^n+p _p(c) with parallel normalized mean curvature vector field is proved.When n≥3, the pinching constant is the best.Thus,the mistake of the paper “Space-like hypersurfaces in de Sitter space with constant scalar curvature”(see Manus Math,1998,95:499-505) is corrected.Moreover,the reduction of the codimension when Mn is a complete submanifold in S^n+p _p(c) with parallel normalized mean curvature vector field is investigated.
文摘Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau's self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.
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.
文摘Let f: Mn(?)Sn+1 C Rn+2 be an n-dimensional complete oriented Rieman-nian manifold minimally immersed in an (n+1)-dimensional unit sphere Sn+1. Denote by S_+~n+1 the upper closed hemisphere. If f(Mn)(?)_+~n+1, then under some curvature conditions the authors can get that the isometric immersion is a totally embedding. They also generalize a theorem of Li Hai Zhong on hypersurface of space form with costant scalar curvature.
基金supported by National Natural Science Foundation of China (Grant Nos.12125105 and 12071425)the Fundamental Research Funds for the Central Universities+1 种基金supported by National Natural Science Foundation of China (Grant Nos.11971424 and 12031017)supported by National Natural Science Foundation of China (Grant No.11971424)。
文摘In this paper,we study Ricci flow on compact manifolds with a continuous initial metric.It was known from Simon(2002)that the Ricci flow exists for a short time.We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in the distributional sense provided that the initial metric is W^(1,p) for some n<p∞.As an application,we use this result to study the relation between the Yamabe invariant and Ricci flat metrics.We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in the distributional sense,then the manifold is isometric to a Ricci flat manifold.
基金supported by National Natural Science Foundation of China(Grant Nos.10831008,11025103 and 11501505)。
文摘On an almost Hermitian manifold, there are two Hermitian scalar curvatures associated with a canonical Hermitian connection. In this paper, two explicit formulas on these two scalar curvatures are obtained in terms of the Riemannian scalar curvature, norms of the components of the covariant derivative of the fundamental 2-form with respect to the Levi-Civita connection, and the codifferential of the Lee form. Then we use them to get characterization results of the K?hler metric, the balanced metric, the locally conformal K?hler metric or the k-Gauduchon metric. As corollaries, we show partial results related to a problem given by Lejmi and Upmeier(2020) and a conjecture by Angella et al.(2018).
基金supported in part by the National Natural Science Foundation of China(Grant Nos.11871126,11501067,11571184).
文摘We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature.As a consequence,Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.For(α,β)-metrics on manifold of dimension greater than 2,if the mean Landsberg curvature and the Berwald scalar curvature both vanish,then the Berwald curvature also vanishes.
文摘In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.
基金Project partially supported by the NNSF of China
文摘We study the conformal deformation for prescribing scalar curvature function on Cartan-Hadamard manifold M^n(n≥3) with strongly negative curvature. By employing the super- subsolution method and a careful construction for the supersolution, we obtain the best possible asymptotic behavior for near infinity so that the problem of complete conformal deformation is solvable. In more general cases, we prove an asymptotic estimation on the solutions of the conformal scalar curvature equation.
基金This work was supported by the National Natural Science Foundation of China(Nos.11811530291,11831006,11771092)。
文摘In this paper, the authors give a new proof of Block and Weinberger’s Bochner vanishing theorem built on direct computations in the K-theory of the localization algebra.
基金Supported by the National Natural Science Foundation of China(11071005)Research Fund for the Doctoral Program of Higher Education of China 20110001110069
文摘A Finsler metric on a manifold M with its flag curvature K is said to be almost isotropic flag curvature if K = 3c + δ where δ and c are scalar functions on M. In this paper, we establish the intrinsic relation between scalar functions c(x) and a(x). More general, by invoking the Ricci identities for a one form, we investigate Finsler metric of weakly isotropic flag curvature K = 3θ/F + δ and show that F has constant flag curvature if θ is horizontally parallel.
基金Supported by NSFC (Grant No. 11701507)the Fundamental Research Funds for the Central Universities 2019QNA3001 and DECRA 190101471。
文摘In this paper,we consider Ricci flow on four dimensional closed manifold with bounded scalar curvature,noncollasping volume and bounded diameter.Under such conditions,we can show that the manifold has finitely many diffeomorphism types,which generalizes Cheeger–Naber’s result to bounded scalar curvature along Ricci flow.In particular,this implies the manifold has uniform L^(2) Riemann curvature bound.As an application,we point out that four dimensional Ricci flow would not have uniform scalar curvature upper bound if the initial metric only satisfying lower Ricci curvature bound,lower volume bound and upper diameter bound.
基金supported by NSF of Shaanxi Province (SJ08A31)NSF of Shaanxi Educational Committee (2008JK484+1 种基金2010JK642)Talent Fund of Xi'an University of Architecture and Technology
文摘Let Mn be an n-dimensional complete connected and oriented hypersurface in a hyperbolic space H(n+1)(c) with non-zero constant mean curvature H and two distinct principal curvatures. In this paper, we show that (1) if the multiplicities of the two distinct principal curvatures are greater than 1,then Mn is isometric to the Riemannian product Sk(r)×H(n-k)(-1/(r2 + ρ2)), where r 〉 0 and 1 〈 k 〈 n - 1;(2)if H2 〉 -c and one of the two distinct principal curvatures is simple, then Mn is isometric to the Riemannian product S(n-1)(r) × H1(-1/(r2 +ρ2)) or S1(r) × H(n-1)(-1/(r2 +ρ2)),r 〉 0, if one of the following conditions is satisfied (i) S≤(n-1)t22+c2t(-2)2 on Mn or (ii)S≥ (n-1)t21+c2t(-2)1 on Mn or(iii)(n-1)t22+c2t(-2)2≤ S≤(n-1)t21+c2t(-2)1 on Mn, where t_1 and t_2 are the positive real roots of (1.5).
基金This work is partially supported by NNSF of ChinaSRF for Returned Overseas Chinese Scholars+1 种基金State Education Commission a scientific grant of Tsinghua University at Beijing
文摘This paper discusses the existence problem in the study of some partial differential equations. The author gets some bifurcation on the prescribed mean curvature problem on the unit ball, the scalar curvature problem on the n-sphere, and some field equations. The author gives some natural conditions such that the standard bifurcation or Thom-Mather theory can be used.
文摘In this paper,we prove that for some completions of certain fiber bundles there is a Maxwell-Einstein metric conformally related to any given Kahler class.
文摘We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.
文摘The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].
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.
文摘Let (M, g) be an n-dimensional Riemannian manifold and T*M be its cotan-gent bundle equipped with the rescaled Sasaki type metric. In this paper, we firstly study the paraholomorphy property of the rescaled Sasaki type metric by using some compati-ble paracomplex structures on T*M. Second, we construct locally decomposable Golden Riemannian structures on T*M . Finally we investigate curvature properties of T*M .