流形上整体几何与几何分析的若干研究

By systematically applying modern methods in global geometry and geometric analysis, we aim to study the convergence of the Ricci flow under integral curvature pinching condition and discuss its application in the study of the curvature and topology; to study the convergence of the mean curvature flow of arbitrary codimension and the volume-preserving mean curvature flow under integral curvature pinching condition and discuss the application in the study of the curvature and topology, and general relativity; to study the convergence of the mean curvature flow of arbitrary codimension in space forms under optimal pointwise curvature pinching condition; to study the uniqueness of geometric, topological, differentiable structures and the finiteness of topological structures on Riemannian manifolds with special restriction on the curvature; to find the relationship between geometric, analytic and topological invariants of the manifolds, obtain better estimates for the upper and lower bounds on the eigenvalues of the Laplace-Beltrami operator and give an estimate for the fundamental gap of the Schr?dinger operator on manifolds; to investigate the influence of eigenvalues on the geometric structure and topological property of manifolds and discuss the uniqueness of topological structures on manifolds under the eigenvalue pinching condition; to promote the studies on Yau's conjecture about the first eigenvalue of the Laplace-Beltrami operator on closed and embedded minimal hypersurfaces in the unit sphere and on Chern's conjecture about the scalar curvature pinching problem on closed minimal hypersurfaces in the unit sphere. These are the leading topics in the study of modern mathematics, which will be of great importance and very useful for many other disciplines.

系统地运用整体几何与几何分析的现代方法，深入研究Ricci流在曲率积分拼挤条件下的收敛性及其在曲率与拓扑中的应用；研究任意余维平均曲率流和保体积平均曲率流在曲率积分拼挤条件下的收敛性及其在曲率与拓扑和广义相对论中的应用；研讨空间形式中任意余维平均曲率流在最佳逐点曲率拼挤条件下的收敛性；研究黎曼流形在曲率限制条件下几何结构、拓扑结构、微分结构的唯一性及拓扑有限性；建立流形的几何量、分析量与拓扑量之间的关系式，探寻Laplace-Beltrami算子特征值上、下界的优化估计和流形上Schr?dinger算子的基本间隙估计；研究特征值对流形的几何、拓扑性质的影响，在特征值拼挤条件下探讨流形拓扑结构的唯一性；推进球面中极小超曲面的关于Laplace-Beltrami算子第一特征值的丘成桐猜想和关于数量曲率拼挤区间的陈省身猜想的研究。本课题属国际前沿，在许多领域有重要应用。

研究黎曼流形的曲率与几何、分析、拓扑的内在联系是现代微分几何的重要课题。本项目紧跟当今微分几何研究的前沿与热点，主要研究流形上的整体几何与几何分析，探讨了黎曼流形及子流形的曲率与拓扑、几何热流的收敛性及其应用、几何刚性与特征值拼挤等问题。我们引入了子流形上的一个新外蕴不变量τ并证明了关于τ的拼挤条件下子流形上Ricci流的光滑收敛定理与子流形的微分球面定理。证得了截面曲率拼挤条件下子流形上Ricci流的光滑收敛定理与子流形的微分球面定理。证明了曲率拼挤条件下正数量曲率黎曼流形的微分球面定理与分类定理。获得了双曲空间中任意余维平均曲率流在最佳曲率拼挤条件下的光滑收敛定理及其拓扑应用。获得了球面中任意余维平均曲率流在曲率积分拼挤条件下的光滑收敛定理及其拓扑应用。得到了空间形式中保体积平均曲率流的光滑收敛定理。证明了平均曲率流的光滑延拓定理。证明了一类黎曼流形上Ricci流的曲率估计与光滑收敛定理。得到了一类共形平坦黎曼流形上Yamabe流的光滑收敛定理。证明了球面中平行平均曲率子流形关于不变量τ的外蕴刚性定理。得到了欧氏空间中完备极小子流形关于曲率衰减间隙的刚性定理。获得了平均曲率流自收缩解的几何刚性定理。证明了λ-超曲面的整体刚性定理并获得其高余维推广。证明了一类黎曼流形的积分不等式与最佳整体刚性定理。证明了欧氏空间中闭超曲面关于Laplace算子第一特征值的拼挤定理等。本项目共完成学术论文19篇，其中8篇论文发表在《J. Math. Pure. Appl.》、《Comm. Anal. Geom.》、《Ann. Glob. Anal. Geom.》等著名SCI国际期刊上。