子流形与曲率流

Developing the modern methods and techniques in global differential geometry and geometric analysis, we will investigate the longtime existence and convergence theorems for the mean curvature flow of higher codimensions, the Ricci flow on submanifolds and the Willmore flow, etc., and their applications in the study of curvature and topology on Riemannian submanifolds as well. Using various curvature flows, we will prove differentiable sphere theorems, topological classification theorems and geometric inequalities for Riemannian submanifolds. We will prove the longtime existence and convergence theorems for the mean curvature flow of arbitrary codimension in space forms under optimal curvature pinching conditions and obtain optimal differentiable sphere theorems for submanifolds. Based on the investigation of the Willmore flow, we will discuss a new proof of the Willmore conjecture for closed surfaces and the generalized Willmore conjecture for submanifolds of high dimensions. Around the Chern conjecture and the generalized Bernstein problem, we will investigate the geometric gap phenomena of minimal submanifolds, the geometric rigidity of stable minimal submanifolds and their generalizations as well. We will study the rigidity and classification of self-similar solutions for the mean curvature flow, and the rigidity of Willmore submanifolds. We will also investigate the eigenvalue pinching problem on Riemannian submanifolds, the formula of first eigenvalue for embedded hypersurfaces and the Yau conjecture, the estimates of heat kernels and higher eigenvalues on submanifolds, and the existence of bounded nontrivial harmonic functions on nonpositively curved manifolds. This project belongs to frontier research fields in the core of mathematics, and will have many important applications.

发展整体微分几何、几何分析的研究方法与技巧，研究子流形上高余维平均曲率流、Ricci流和Willmore流等若干几何曲率流解的存在性定理、收敛性定理及其在几何与拓扑中的应用；运用曲率流等工具，证明子流形的球面定理、拓扑分类定理和几何不等式；在最优曲率拼挤条件下证明高余维平均曲率流解的存在性与收敛性定理，获得子流形的最佳微分球面定理；基于Willmore流的研究，研讨闭曲面的Willmore猜想的新证明、高维Willmore猜想；围绕陈省身猜想和广义Bernstein问题，研究极小子流形的几何间隙现象、稳定极小子流形的几何刚性问题及其推广；研究平均曲率流的自相似子流形的刚性与分类，Willmore子流形的刚性问题；研究子流形的特征值拼挤问题，嵌入超曲面第一特征值公式与丘成桐猜想，子流形的热核与高阶特征值估计，非正曲率流形上有界调和函数存在性问题等。本课题属核心数学的前沿领域,有许多重要应用。

本项目对子流形与曲率流的若干重要问题进行了深入系统的研究。首次在具有正Ricci曲率的最佳逐点拼挤条件下证明了双曲空间中任意余维平均曲率流的光滑收敛性定理，获得了首个具有正Ricci曲率子流形的最佳微分球面定理；在优化逐点曲率拼挤条件下证明了球面中任意余维平均曲率流解的光滑收敛性定理和紧致子流形的微分球面定理；在全新的逐点曲率拼挤条件下证明了复空间形式中任意余维平均曲率流解的光滑收敛性定理和紧致子流形的微分球面定理；证明了曲率积分拼挤条件下欧氏空间中任意余维平均曲率流的光滑收敛性定理、可延拓性定理和紧致子流形的微分球面定理；获得了优化逐点曲率拼挤条件或曲率积分拼挤条件下空间形式中任意余维平均曲率流古代解的几何刚性定理；证明了单位球面中紧致极小超曲面数量曲率的第二拼挤区间长度至少为n/18，获得了关于加强版陈省身猜想的最新研究结果；证明了欧氏空间中体积元不大于3的极小子流形一定是仿射线性子空间，这是Lawson-Ossermann问题研究的最新进展；利用Schrödinger流、bi-Schrödinger流，把曲线的几何理论推广到高维的对称Lie代数之上；建立了一般黎曼空间在度量的共形形变下其子流形的几何理论，找到了共形群下子流形的完备共形不变量系统并建立其完全可积条件等。项目组在《Trans. Amer. Math. Soc.》(2篇)、《Amer. J. Math.》、《J. Funct. Anal.》、《Calc. Var. & PDEs》(2篇)、《J. Geom. Anal.》(2篇)、《Math. Z.》等重要学术刊物发表或录用论文38篇，完成论文预印本13篇，修订出版了学术专著《Minimal submanifolds and related topics》(World Scientific, Singapore)的第二版，获得了2017年度高等学校科学研究优秀成果奖自然科学二等奖，获得了3项世界华人数学家大会最佳论文奖(若琳奖)。指导研究生和博士后80余位，培养了一批极具潜力的优秀青年几何人才（含6位国家层面的优秀青年人才）。