非负曲率流形上的丛

本文主要证明了射影空间乃至齐性空间上各种向量丛的全空间可以赋予截面曲率非负的完备度量.     

曲率渐近非负流形

本文主要研究了在曲率渐近非负流形上的一些性质：其上不存在非常数的正调和函数；若其调和函数是对一固定指数d的多项式增长,那么由这些调和函数所组成的空间的维数≤Cd~(CD),而且此流形的体积为无穷大等．     

曲率渐近非负流形

本文主要研究了在曲率渐近非负流形上的一些性质其上不存在非常数的正调和函数;若其调和函数是对一固定指数d的多项式增长,那么由这些调和函数所组成的空间的维数≤CdCD,而且此流形的体积为无穷大等.     

非负Ricci曲率开流形的拓扑

我们证明了对于具有非负Rieei曲率,大体积增长且内半径下有界的完备n维Riemann流形,只要存在常数C>0使得 则它微分同胚于欧式空间Rn.我们还证明了在某些pinching条件下具有非负射线曲率的完备n维Riemarm流形微分同胚与Rn,改进了已知的结果.     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

完备非紧具非负曲率流形之拓扑结构

本文给出完备非紧具非负曲率的Ｒｉｅｍａｎｎ流形具有限拓扑型的一个简单证明     

R^n空间上具有正曲率和负曲率的度量一些例子

From the class of conformally flat metric of gij=f(|x|^2)δij defined on R^n,where f:[0,∞]→（0,∞）is a C^2 function,we find one class of metrics with positive curvature and two classes of complete metrics with negative curvature.     

负曲率流形上给定数量曲率的共形形变

本文研究具强负曲率Cartan－Hadamard流形M~n（n≥3）上给定数量曲率函数S的共形形变问题．利用上下解方法,并通过精心构造上解,我们获得了当完备的共形形变度量存在时,函数S在无穷远附近的最佳渐近性态．在较一般情况下,我们还给出了共形数量曲率方程解的渐近估计．     

完备非紧具非负曲率流形之拓扑结构

本文给出完备非紧具非负曲率的Riemann流形具有限拓扑型的一个简单证明.     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

HERMITIAN-EINSTEIN METRICS FOR HIGGS BUNDLES OVER COMPLETE HERMITIAN MANIFOLDS

In this paper, we solve the Dirichlet problem for the Hermitian-Einstein equations on Higgs bundles over compact Hermitian manifolds. Then we prove the existence of the Hermitian-Einstein metrics on Higgs bundles over a class of complete Hermitian manifolds.     

BRILL－NOETHER MATRIX FOR RANK TWO VECTOR BUNDLES

Lei X be an arbitrary smooth irreducible complex projective curve, E (?) X a rank two vector bundle generated by its sections. The author first represents E as a triple {D1,D2,f}, where D1 , D2 are two effective divisors with d = deg(D1) + deg(D2), and f ∈ H0(X, [D1] |D2) is a collection of polynomials. E is the extension of [D2] by [D1] which is determined by f. By using f and the Brill-Noether matrix of D1 + D2, the author constructs a 2g X d matrix WE whose zero space gives Im{H0(X,[D1]) (?) H0(X, [D1] |D1)}(?)Im{H0(X, E) (?) H0(X,[D2]) (?) H0(X,[D2] |D2)}. From this and H0(X,E) = H0(X, [D1]) (?) Im{H0(X, E) (?) H0(X, [D2])}, it is got in particular that dimH0(X, E) = deg(E) - rank(WE) + 2.     

SOME EXTENSIONS OF THE MEAN CURVATURE FLOW IN RIEMANNIAN MANIFOLDS

Given a family of smooth immersions of closed hypersurfaces in a locally symmetric Riemannian manifold with bounded geometry,moving by mean curvature flow,we show that at the first finite singular time of mean curvature flow,certain subcritical quantities concerning the second fundamental form blow up.This result not only generalizes a result of Le-Sesum and Xu-Ye-Zhao,but also extends the latest work of Le in the Euclidean case.     

SOME RESULTS ON PRODUCT COMPLEX FINSLER MANIFOLDS

Let (M, F ) be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds (M 1 , F 1 ) and (M 2 , F 2 ). In this paper, we obtain the relationship between the Chern Finsler connection coefficients Γ i ; k associated to F and the Chern Finsler connection coefficients Γ a ; c , Γα ; γ associated to F 1 , F 2 , respectively. As applications we prove that, if both (M 1 , F 1 ) and (M 2 , F 2 ) are strongly Ka¨hler Finsler (complex Berwald, or locally complex Minkowski, respectively) manifolds, so does (M, F ). Furthermore, we prove that the holomorphic curvature K F = 0 if and only if K F1 = 0 and K F2 = 0.     

FUNDAMENTAL GROUPS OF CLOSED POSITIVELY CURVED MANIFOLDS WITH ALMOST DISCRETE ABELIAN GROUP ACTIONS

Let M be a closed n-manifold of positive sectional curvature.Assume that M admits an effiective isometrical T 1×Zkp-action with p prime.The main result of the article is that if k=1 for n=3 or k (n+1)/4 for n≥5,then there exists a positive constant p(n),depending only on n,such that π1(M) is cyclic if p≥p(n).     

A Simple Proof of Tyurin's Babylonian Tower Theorem

Using the method of Coand? and Trautmann [4 Coand? , I. , Trautmann , G. ( 2006 ). The splitting criterion of Kempf and the Babylonian tower theorem . Comm. Algebra 34 : 2485 – 2488 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we give a simple proof of a theorem due, in the smooth case, to Tyurin [9 Tyurin , A. N. ( 1976 ). Finite dimensional vector bundles over infinite varieties . Math. USSR Izv. 10 : 1187 – 1204 .[Crossref], [Web of Science ®] , [Google Scholar]]: if a vector bundle E on a c-codimensional locally Cohen–Macaulay closed subscheme X of ? ⁿ extends to a vector bundle F on a similar closed subscheme Y of ? ^N , for every N > n, then E is the restriction to X of a direct sum of line bundles on ? ⁿ . Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.     

透镜空间L~n(4)上的向量丛及相关的对合

本文研究了透镜空间Ln(4)的上同调群生成元的运算性质,利用这些生成元,并借助于KO-理论计算出了透镜空间Ln(4)上任意向量丛的全Stiefel—Whitney示性类.更进一步地,给出了不动点集为透镜空间Ln(4)的带对合的流形的协边分类.     

POSITIVELY CURVED COMPLETE NONCOMPACT KAHLER MANIFOLDS

In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. Among other things, we prove that a complete noncompact Kahler surface with positive and bounded sectional curvature and with finite analytic Chern number c1（M）^2 is biholomorphic to C2.