子流形几何、谱理论与拓扑不变量

In this project, we primarily studied Geometry of Submanifolds, Spectrum Theory and Topological Invariants. Through various seminars, looking.up and studying a lot of corresponding papers, participating various civil and.foreign academic activities, visiting other countries and cooperating, we have made some great progress. In the last three years, we have published 24 papers on some important civil and foreign journals, four of which were on SCI journals. I have published a monogragh and fosterd five high-qualified doctors and six.high-qualified masters. My outstanding doctor, Jiaqiang Mei, has been a postdoctor of famous mathematician, Gang Tian and also my outstanding masters, Huadong Pang and Fangyun Yang, have been accepted as his doctors. In the following aspects, we have got some results top in China and of inernational level: 1. Estimate of the first eigenvalue, isospectrum problem and the properties of higher eigenvalues of Laplace operator; 2. Rigidity problem in Geometry of Submanifolds; 3. The relations between curvatures and topological invariants(Betti numbers, Homotopy groups, etc.); 4. ∞ C compactness for minimal submanifolds in the unit.sphere; 5. Small excess and the topology of open manifolds; 6. A class of Jiang spaces.

主要研究开流形在何种渐近曲率条件下它等距或微分同胚于Euclid空间，在何种曲率条件下该流形拓扑型有限；闭流形方面，正曲率对拓扑性质的影响，以及正Ricci曲率度量的存在裕唤糁铝餍紊螸aplace算子的谱理论和等谱问题；Yamabe问题，还应用Gromov的整体观点研究极小子流形。已积累了大量资料并发表了许多论文。期望作出具有国际水平的成果。