A global pinching theorem for compact surfaces inS 3 with constant mean curvature

LetM be a compact minimal surface inS ³. Y. J. Hsu^[5] proved that if S₂22, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·₂ denotes theL ²-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS ³ with constant mean curvature.Supported by NSFH.     

曲面调和映照的广义Frenet公式及其应用

本文建立了从曲面到复Grassmamn流形调和映照的广义Frenet公式。作为应用，我们得到了调和映照为强共形的一个等价条件。我们也讨论了等距调和映照的曲率pinching性质。从而改进了有关伪全纯曲线的相应结果。     

子流形对外围空间的一个拼挤定理

首先得到一个推广的Simons型积分不等式，然后用它给出共形平坦黎曼流形中紧致极小子流形对外围空间的一个拼挤定理．     

Almost Extrinsically Homogeneous Submanifolds of Euclidean Space

Consider a closed manifold M immersed in R^m. Suppose that the trivial bundle M × R^m = T M ⊗ ν M is equipped with an almost metric connection ~ ∇ which almost preserves the decomposition of M × R^m into the tangent and the normal bundle. Assume moreover that the difference Γ = ∂~∇ with the usual derivative ∂ in R^m is almost ~∇-parallel. Then M admits an extrinsically homogeneous immersion into R^m. Mathematics Subject Classifications (2000): 53C20, 53C24, 53C30, 53C42, 53C40.     

复射影空间的拟共形平坦Kaehler完备子流形

研究复射影空间的拟共形平坦Kaehler完备子流形得到局部结构与关于数量曲率的拼挤常数．     

Collapsed 5-manifolds with pinched positive sectional curvature

Let M be a closed 5-manifold of pinched curvature 0<δ?secM?1. We prove that M is homeomorphic to a spherical space form if one of the following conditions holds:

(i)

The center of the fundamental group has index ?w(δ), a constant depending on δ;

(ii)

and the fundamental group is a non-cyclic group of order ?C, a constant;

(iii)

The volume is less than ?(δ) and the fundamental group is either isomorphic to a spherical 5-space group or has an odd order, and it has a center of index ?w, a constant.

     

Long time existence of Ricci-harmonic flow

We give a survey about recent results on Ricci-harmonic flow.     

Eigenvalue pinching theorems on compact symmetric spaces

We prove two first eigenvalue pinching theorems for Riemannian symmetric spaces (Theorems 1 and 2). As their application, we answer negatively a question raised by Elworthy and Rosenberg, who proposed to show that for every compact simple Lie group with a bi-invariant Riemannian metric on with respect to , being the Killing form of the Lie algebra , the first eigenvalue would satisfy

for all orthonormal bases of tangent spaces of (cf. Corollary 3). This problem arose in an attempt to give a spectral geometric proof that for a Lie group .

     

Normal scalar curvature and a pinching theorem in S~m×R and H~m×R

We obtain an inequality in Sm×R and Hm×R which is similar to DDVV conjecture.As an application,we show that a minimal submanifold in H m×R with nonnegative scalar curvature must be a surface of the type γ×R,where γ is a geodesic in H m.In addition,we get a pinching theorem in Sm×R.     

稳定拟调和映照的不存在性

本文讨论了稳定拟调和映照的不存在性，得到了一个拼挤定理。