常曲率空间中单形的中面公式及其应用

本文给出了n维常曲率空间中单形的中面面积公式，利用它我们还得到了一类几何不等式．     

A Geometric Problem and the Hopf Lemma. Ⅱ

Abstract A classical result of A. D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in ℝⁿ⁺¹, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane X_n+1 =constant in case M satisfies: for any two points (X′,X_n+1), on M, with , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0401118.     

类拟的Euler公式

经典的欧拉公式描述了两曲面交线的曲率和曲面的法曲率以及曲面夹角之间的关系。本文给出了R3中曲面的法曲率的欧拉公式的简单证明。并得到一个关于曲面交线曲率及曲面测地曲率的类似的欧拉公式。     

Finsler几何的回顾与进展

本文回顾了近年来Finsler几何的进展．特别，我们描述了Finsler流形上几何不变量所满足的基本方程及其应用，并提出了相关的未解决的问题。     

三维欧氏空间中的一类Weingarten螺旋面

通过求解相关的非线性常微分方程,构造了三维欧氏空间中主曲率之差为常数的螺旋面,并证明这类曲面的广泛存在性.     

有关法曲率的若干性质

以[2]中经典微分几何问题为切入点,运用复数与三角工具广泛深入地探讨了“过曲面上一点有n条切线,若相邻两条切线的交角为2nπ,曲面法线与切线所定平面截得曲线的曲率半径为ρ1,ρ2,ρ3,…,ρn时,∑ni=11ρim,∑ni=1ρi,∏ni=1ρi的结果”,得到了法曲率与相关的三个有趣定理.     

中曲率大于零的非凸曲面

本文研究了曲面的曲率问题.利用积分几何方法和旋转曲面性质,构造出一个欧氏空间R3中紧致光滑的中曲率H大于零的非凸曲面,并得到了关于紧致光滑曲面曲率的几个不等式.     

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.     

关于曲率的深入研究

本文就曲线上一点处曲率的存在条件做了讨论,提出了曲率存在的一个定理,给出了单侧曲率的概念,从而说明了曲率并非在光滑曲线上的每一点处都存在.     

具有平行平均曲率向量场的子流形

本文改进了Ｓ．Ｔ．Ｙａｕ（文［１］）中关于单位球面中具有平行平均曲率向量场的子流形的一个结果。然而从截面曲率这一角度出发，给出了空间形式Ｒ＾ｎ＋ｐ（ｃ）（ｎ＞１，ｐ＞１）中具有平行平均曲率向量场的可定向闭子流形Ｍ＾ｎ的有关结果和积分不等式。     

DEFORMING CONVEX HYPERSURFACES BY THE LINEAR COMBINATION OF THE MEAN CURVATURE AND THE N-TH ROOT OF THE GAUSS-KRONECKER CURVATURE

1IntroductionTherehavebeensomeinterestingresultsinstudyingtheflowofconvexhypersurfacesintheEuclideanspacebyfunctionsOftheirprincipalcurvatures.BeingviewedasanextensionOfthetheoremOfGageandHedton[3],Huiskellprovedin16]thatdeformingconvexhypersurforesbytheirmeancurysturefunctiollsconvergetoaroundsphereinasense.FollowingthemethodsOfHuiskell[6]andTso[IOI,Chowshowedin[1]thatthesame.statementasin.[6]reconstrueifthemeancurvatureisreplacedbythen-throotoftheGauss-Kroneckercurvature.FOrgenerality…     

Almost nonnegative curvature operator and cohomology rings

We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author.     

Relative mean curvature configurations for surfaces in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>, <Emphasis Type="Italic">n</Emphasis> ≥ 5

We define the relative mean curvature directions on surfaces immersed in ℝⁿ, n ≥ 4, generalizing the concept of mean curvature directions for surfaces in 4-space studied by Mello. We obtain their differential equations and study their corresponding generic configurations. *Work partially supported by DGCYT grant no. MTM2004-03244 and Unimontes-BR. †Work partially supported by DGCYT grant no. MTM2004-03244. ‡Work partially supported by DGCYT grant no. BFM2003-0203.     

Expression of curvature tensors and some applications

We get an explicit expression of curvature operators in terms of at most eight terms of sectional curvatures. Some applications of this result are also given, particularly we improve a result of Chen-Tian related to the first Chern class of admissible surfaces in pinched manifolds. We also characterize in a simple way all functionsk(x, y) which can be sectional curvatures of some curvature operatorR.Supported by CNPq, Brazil and NNSFC.     

曲面上的曲率新讲

阐述一种讲解曲面上的各种曲率形式的新教学模式.我们利用平面向量分析,矩阵,求导运算等工具推导出所有的结论和公式.这种新观点和方法避免了微分几何传统讲法中使用Weingarten映射及其特征值的思路,而较之更加直观明了.     

单位球面中极小子流形的曲率拼挤

本文证明了单位球面中极小子流形的一些拼挤定理,特别注意到单位球面中的极小超曲面、给出了截曲率的拼挤常数,我们也改进了由Ｎ．Ｅｊｉｒｉ得到的Ｒｉｃｃｉ曲率拼挤常数。     

A Schur type lemma for the mean Berwald curvature in Finsler geometry

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.     

On Complete Hypersurfaces with Constant Mean Curvature and Finite L^P-norm Curvature in R^n＋1

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.