常曲率空间中具平行平均曲率向量的子流形

本文利用第二基本形式的长度平方和平均曲率的关系研究常曲率空间中具平行平均曲率向量的子流形为全脐的ｐｉｎｃｈｉｎｇ问题，获得了一定条件下的最佳ｐｉｎｃｈｉｎｇ区间，并确定了ｐｈｉｎｃｎｉｎｇ区间端点处对应非全脐子流形的分类．     

拟常曲率流形中全脐点子流形的拼嵌定理

We study the global umbilic submanifolds with parallel mean curvature vector fields in a Riemannian manifold with quasi constant curvature and get a local pinching theorem about the length of the second fundamental form.     

局部对称空间中具有平行平均曲率向量的黎曼叶层的Pinching定理

本文利用Nakagawa和Takagi的计算散度的方法,求出局部对称空间中具有平行平均曲率向量的黎曼叶状结构${\cal F}$上向量场的散度,并证明了其上的整体Pinching定理.     

Volume-preserving flow by powers of the mth mean curvature

We consider the evolution of a closed convex hypersurface under a volume preserving curvature flow. The speed is given by a power of the mth mean curvature plus a volume preserving term, including the case of powers of the mean curvature or of the Gauss curvature. We prove that if the initial hypersurface satisfies a suitable pinching condition, the solution exists for all times and converges to a round sphere.     

球面的平行平均曲率子流形

本文讨论单位球面中具有平行单位平均曲率向量的子流形的第二基本形式长度拼挤问题,改进了已有的结论。     

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。     

Some local eigenvalue estimates involving curvatures

We first establish a local Faber–Krahn isoperimetric comparison in terms of scalar curvature pinching. Secondly we derive estimates of Cheeger constants related to the Dirichlet and Neumann problems via the (relative) isoperimetric profiles which allow us to obtain, in particular, lower bounds for first non-zero eigenvalues of the problem of Dirichlet and Neumann. These estimates involve scalar curvature and mean curvature respectively.     

球面中具有平行平均曲率向量的2—调和等距浸入

本文讨论了单位球面中具平行平均曲率向量的２－调和等距浸入，获得其成为极小浸入的充分条件和关于第二基本形式的Ｐｉｎｃｈｉｎｇ定理．     

不定复空间型中具有平行平均曲率向量的完备全实类空子流形

利用活动标架法和广义极大值原理研究了不定复射影空间中完备的具有平行平均曲率向量的全实类空子流形,得到这类子流形的Pinching定理.     

黎曼流形中紧子流形的拼挤定理

本利用几何不等式和曲率估计的方法，证明了黎曼流形N^n＋p,上的具有平行平均曲率的紧子流形M^n上的一个拼挤定理。若N上的截曲率KN满足- 1≤ KN≤δ≤0，且‖S- nH2‖n/2, ‖ S-nH^2‖n/n-s满足一些不等式，则δ= - 1。     

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.     

Minimal surfaces in a unit sphere pinched by intrinsic curvature and normal curvature

We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere Sn, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property K + K~N= 1 for the Gauss curvature K and the normal curvature K~N if the Gauss curvature is positive. Moreover, using this property we obtain the pinching on the intrinsic curvature and normal curvature, the pinching on the normal curvature, respectively.     

单位球面的三维紧致极小子流形

本文得到高维球面上三维紧致极小子流形的若干刚性定理．在一个整体拚挤条件下，数量曲率的拚挤常数被改进了，也对李奇曲率的拚挤问题进行了讨论．     

Curvature properties of compact spacelike hypersurfaces in de Sitter space

In this paper we establish a sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a lower bound for the square of its mean curvature. Our result will be a consequence of the maximum principle for the Laplacian operator. We also derive some other applications and consequences of our main result. In particular, we establish another sufficient condition for a compact spacelike hypersurface in de Sitter space to be spherical in terms of a pinching condition for its scalar curvature, as well as in terms of the Ricci curvature and in terms of the higher order mean curvatures     

局部对称流形中具有常平均曲率的完备超曲面

讨论了局部对称黎曼流形中具有常平均曲率的完备超曲面的性质,通过Laplace算子的计算,得到一个关于第二基本形式模长平方S的拼挤定理,推广了已有的结果．     

关于局部对称空间中的伪脐子流形

本文讨论了局部对称完备黎曼流形中的紧致伪脐子流形,且具有平行平均山率向量场。得到了这类子流形成为全脐子流形及其余维数减小的几个拼挤定理。     

EffectiveL p pinching for the concircular curvature

According to the work of Huisken, Margerin, and Nishikawa, we know that effective L_∞ pinching assumptions on the concircular curvature lead to the existence of a metric of constant positive sectional curvature. But what can we say under effectiveL _p pinching assumptions?     

李奇曲率平行的黎曼流形的曲率张量模长

李安民和赵国松［１］提出了下面的问题：找出李奇曲率平行的黎曼流形的曲率张量模长的最佳拼挤常数并确定达到该值的流形．本文确定了非爱因斯坦流形的最佳拼挤常数和达到该值的黎曼流形．在ｎ１２时，回答了［１］中提出的问题．     

Characterization of Totally Geodesic Totally Real 3-dimensional Submanifolds in the 6-sphere

In this paper, we study Lagrangian submanifolds of the nearly Kaehler 6-sphere. We derive a pinching result for the Ricci curvature of such submanifolds thus providing a characterisation of the totally geodesic submanifold. Our pinching result improves a previous result obtained by H. Li.