可定向的具非负曲率完备非紧黎曼流形

本文研究了具非负曲率完备非紧黎曼流形的一些几何性质,包括闭测地线,体积等.证明了核心的余维数为奇数的可定向具非负曲率完备非紧黎曼流形在其核心的任一法测地线均为射线的条件下可等距分裂为R×N,其中N为低一维的流形.     

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     

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.     

具非负Ricci曲率和次大体积增长的流形

设M是具非负Ricci曲率的n维黎曼流形,其截曲率有下界,对M中的任意的点p有vol[B(p,r)]/rn-1=αM+o(1/rn-1)且假设函数f(r)=vol[B(p,r)]/2In(r)rn-1是单调递减的,则M具有限拓扑型,其中In(r)是一有界函数.     

具非负曲率完备非紧曲面的几何性质

本文证明了单连通完备非紧具非负曲率之曲面的任一测地线γ：[0,+∞)→M均趋于∞处这一几何性质,指出了一般的高维流形不具有此性质.本文还证明了单连通完备非紧具非负曲率的曲面的割迹与第一共轭轭迹是一致的;并且讨论了一般高维流形的共轭点与测地线的关系.     

具有非负Ricci曲率和大体积增长的开流形

In this paper,we prove that a complete n-dimensional Riemannian manifold with n0nnegative kth-Ricci curvature,large volume growth has finite topological type provided that lim{((vol[B(p,r))]/(ω_nr~n)-αM)r(k(n-1))/(k 1)(1-α/2)}＜=εfor some constantε＞0.We also prove that a complete Riemannian manifold with nonnegative kth-Ricci curvature and under some pinching conditions is diffeomorphic to R~n.     

具非负Ricci曲率和强有界几何条件的流形

设M是具非负Ricci曲率的n维完备非紧黎曼流形,若M具次大体积增长vol{B(p,r)1≥βM*, p ∈M, r≥1和满足强有界几何条件,则M具有限拓扑型.     

拟常曲率流形的完备子流形

本文应用广义极大值原理,对完备子流形的情形,给出了它是紧致的以及基本群有限的充分条件。     

完备非紧黎曼流形上一类曲率流的单调体积公式

本文研究了黎曼流形上一类一般的曲率流问题.利用Perelman在Ricci流下导出体积单调性的方法,在初始流形完备非紧的情况下,获得了这类曲率流的一个单调性的体积公式,推广了Reto Müller在紧致情形的结果.     

只有一个B—函数的完备黎曼流形

讨论了只有一个Ｂｕｓｅｍａｎｎ函数的完备非紧黎曼流形的几何拓扑性质。     

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.     

一类完备Riemann流形上的有界调和函数

本文我们将对一类完备Ｒｉｅｍａｎｎ流形上的有界调和函数所组成的线性空间的维数的上界进行估计，同时给出了一个关于测地球体积的Ｂｉｓｈｏｐ－Ｇｒｏｍｏｖ型体积比较定理。     

Riemannian Submersions and Riemannian Manifolds with Einstein--Weyl Structures

The main results of this paper are as follows. (a) Let : M N be a non-trivial Riemannian submersion with totally geodesic fibers of dimension 1 over an Einstein manifold N. If M is compact and admits a standard Einstein--Weyl structure with constant Einstein--Weyl function, then N admits a Kähler structure andM a Sasakian structure. (b) Let be a Riemannian submersion with totally geodesic fibers and N an Einstein manifold of positive scalar curvature . If M admits a standard Sasakian structure, then M admits an Einstein--Weyl structure with constant Einstein--Weyl function.     

The Extension of the H~k Mean Curvature Flow in Riemannian Manifolds

In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.     

The Yamabe Problem and Applications on Noncompact Complete Riemannian Manifolds

We let (M,g) be a noncompact complete Riemannian manifold of dimension n 3 whose scalar curvature S(x) is positive for all x in M. With an assumption on the Ricci curvature and scalar curvature at infinity, we study the behavior of solutions of the Yamabe equation on –u+[(n–2)/(4(n–1))]Su=qu ^{(n+2)/(n–2)} on (M,g). This study finds restrictions on the existence of an injective conformal immersion of (M,g) into any compact Riemannian n -manifold. We also show the existence of a complete conformal metric with constant positive scalar curvature on (M,g) with some conditions at infinity.     

带边界Riemann流形上的Sobolev空间等价范数

对具光滑边界αＭ的Ｒｉｅｍａｎｎ流形（Ｍ，ｇ），本文建立了Ｓｏｂｏｌｅｖ空间Ｈ（Ｍ）的等价范数     

The Riemannian Manifolds with Boundary and Large Symmetry

Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most 1/2 dim M（dim M - 1）. Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.