On Einstein Manifolds of Positive Sectional Curvature

Let (M, g) be a compact oriented four-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M, g) is ₂, with its standard Fubini–Study metric.     

Finsler Manifolds with Positive Constant Flag Curvature

It is shown that a Finsler metric with positive constant flag curvature and vanishing mean tangent curvature must be Riemannian. As applications, we also discuss the case of Cheng's maximal diameter theorem and Green's maximal conjugate radius theorem in Finsler manifolds.     

Fundamental Groups of Closed Manifolds with Positive Curvature and Torus Actions

We determine the fundamental group of a closed n-manifold of positive sectional curvature on which a torus T^k (k large) acts effectively and isometrically. Our results are: (A) If k>(n − 3)/4 and n ≥ 17, then the fundamental group π₁(M) is isomorphic to the fundamental group of a spherical 3-space form. (B) If k ≥ (n/6)+1 and n≠ 11, 15, 23, then any abelian subgroup of π₁(M) is cyclic. Moreover, if the T^k-fixed point set is empty, then π₁(M) is isomorphic to the fundamental group of a spherical 3-space form.Mathematics Subject Classification (2000). 53-XX*Supported partially by NSF Grant DMS 0203164 and by a reach found from Beijing normal university.**Supported partially by NSFC 10371008.     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N ⁿ ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M ^m , the Yamabe invariantof M ^m × N ⁿ is no less than K times the invariant ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

The Scalar Curvature on Totally Geodesic Fiberings

We give a generalization of a theorem of Llarull concerning thebehaviour of the scalar curvature while perturbing the metric. In thispaper the following is shown: let Ñ N be a Riemannian submersion with totally geodesic fibre. IfÑ has the property that perturbingits metric towards a bigger one implies that there is a point onÑ where the perturbed scalarcurvature is less than the original one, then also the base manifoldN possesses this property. This result is applied to theprojective spaces.     

Rigidity Theorems for Compact Manifolds with Boundary and Positive Ricci Curvature

We prove some boundary rigidity results for the hemisphere under a lower bound for Ricci curvature. The main result can be viewed as the Ricci version of a conjecture of Min-Oo.      

截面曲率有界的Riemann流形中的闭子流形

证明了截面曲率有界的Ｒｉｅｍａｎｎ流形中闭子流形的一个广义Ｓｉｍｏｎｓ型积分不等式,其次建立了Ｓ,│Ｈ│,│△＾⊥Ｈ│与闭子流形特征的一个关系,结论推广了Ｃｈｅｒｎ＾「１」,Ｌｉ＾「２」和Ｘｕ＾「３」中相应的结果。     

Stability of the p-Curvature Positivity under Surgeries and Manifolds with Positive Einstein Tensor

We establish the stability of the class of manifolds with positive p-curvature under surgeries in codimension p + 3. As a consequence of this result, we first obtain the classification of compact 2-connected manifolds of dimension 7 with positive Einstein tensor; and secondly the existence of metrics with positive Einstein tensor on any compact, simply connected, non-spin manifold of dimension 7 whose second homotopy group is isomorphic to Z₂.     

曲率二次衰减的完备流形的基本群

本文研究曲率二次衰减的完备黎曼流形,证明了若它的直径增长满足小的线性增长条件,则其基本群是有限生成的.     

具有非负Ricci曲率的开流形的基本群

我们对某些类型的Riemannian流形,通过点到极小测地圈端点的距离建立了它到极小测地圈中点的距离的一致估计,然后利用这种一致估计证明了具有非负Ricci 曲率Riemannian流形的基本群有限生成的一个定理,对著名的Milnor猜测起到更强的支持作用．     

拟常曲率Riemann流形中的伪脐点子流形

研究了拟常曲率Riemann流形中具有平行平均曲率向量的伪脐点子流形,得到了一个Simons型公式．     

An Obstruction to Fundamental Groups of Positively Ricci Curved Manifolds

In this note we propose a conjecture concerning fundamental groups of Riemannian n-manifolds with positive Ricci curvature. We prove a partial result under an extra condition on a lower bound of sectional curvature. Our main tool is the theory of Hausdorff convergence. We also extend Fukaya and Yamaguchi's resolution of a conjecture of Gromov to limit spaces which may have singular points.     

Mean Curvature Flow with Dirichlet Boundary Conditions in Riemannian Manifolds with Symmetries

Consider a hypermanifold M ₀ of a Riemannian manifold N whose Riccicurvature is bounded from below. If M ₀ is transversal to a conformalvector field on N, then conditions are given, such that the meancurvature evolution of M ₀ with Dirichlet boundary conditions has asolution for all times.     

关于单位球面的子流形的一个Pinching定理

设Ｍ是单位球面的一个浸入子流形，ＵＭ＝∪ＵＭｘ是Ｍ的单位切丛.本文研究函数ｆ（ｘ）＝ｍａｘ－Ｂ（ｕ，ｕ）－Ｂ（ｖ，ｖ）２。其中Ｂ是Ｍ的第二基本形式．当Ｍ具平行平均曲率时，我们给出关于第二基本形式的一个Ｐｉｎｃｈｉｎｇ定理．对Ｍ是极小的情形，我们有相同的讨论．     

On the Structure of Cimpact Simply Connected Manifolds of Positive Sectional Curvature

We present a first structure theorem for compact simply connected positively curved manifolds with arbitrarily small pinching constants: For each nN and 0<1, there exists a positive number V = V(n,) such that if (M,g) is a compact simply connected n-dimensional Riemannian manifold with sectional curvature 0相似文献   

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.     

THEOREMS OF BARTH-LEFSCHETZ TYPE ON KAHLER MANIFOLDS WITH PARTIALLY POSITIVE CURVATURE

The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partially positive bisectional curvature without the assumption of nonnegative bisectional curvature. Some applications of the results to holomorphic mappings are given.     

Positively Curved Manifolds with Almost Maximal Symmetry Rank

The symmetry rank of a Riemannian manifold is the rank of the isometry group. We determine precisely which closed simply connected 5-manifolds admit positively curved metrics with (almost maximal) symmetry rank two. We also determine the precise Euler characteristic and the fundamental groups of all closed positively curved n-manifolds with almost maximal symmetry rank [(n–1)/2] (n 6, 7).