子流形几何与曲率流

In this project, we will invite some famous experts and outstanding young scholarsin the field of the geometry of submanifolds and curvature flows to give lectures,teaching courses and discuss questions in our seminars. We will focus on several frontier topics, including the geometry and topology of foliations, index theorems on manifolds with boundary, curvature and topology of submanifolds, isoparametric hypersurfaces and focal submanifolds, isoparametric functions and the geometry of exotic spheres, the Möbius differential geometry, the generalized Willmore conjecture for higher dimensional submanifolds; the mean curvature flow, the Ricci flow of submanifolds, curvature flows of hypersurfaces, the Schrödinger flow, etc., and the applications of curvature flows in geometry and topology; the Yau conjecture on the first eigenvalue of minimal hypersurfaces in spheres, isoperimetric inequality and related problems, the eigenvalue pinching problem of submanifolds, the estimates of eigenvalues and the evolution of eigenvalues along curvature flows, the existence of bounded harmonic functions, the heat kernel and heat equations on submanifolds; the Chern conjecture on the scalar curvature of minimal hypersurfaces in spheres, the rigidity and geometric inequalities of minimal submanifolds and self-similar submanifolds, etc. By carrying out the project, a number of internationally influential, original and innovational results will be obtained and a group of outstanding young talents in mathematics will be trained, and the research of differential geometry in China will be greatly promoted.

邀请本领域著名专家和优秀青年学者围绕子流形几何与曲率流的前沿专题开展讲学和研讨活动，内容包括：叶层结构的几何与拓扑，带边流形的指标定理，子流形的曲率与拓扑，等参超曲面与焦流形，等参函数与怪球面的几何，Möbius微分几何，高维Willmore猜想；子流形的平均曲率流、Ricci流，超曲面的曲率流，Schrödinger流等及其在几何与拓扑和物理学中的应用；球面中极小超曲面第一特征值的丘成桐猜想，等周不等式及相关问题，子流形的特征值拼挤问题，特征值估计及沿曲率流的演化，有界调和函数的存在性问题，子流形的热核与热方程；球面中极小超曲面数量曲率的陈省身猜想，极小子流形、自相似子流形等几类子流形的刚性与几何不等式。通过本项目的实施，产生一批具有原始创性的新成果，并培养出一批年轻的几何学人才，提升我国微分几何的研究水平。

本项目执行期限为2022.01.01-2022.12.31。我们在全国抗疫的形势下，克服执行期间新冠疫情管控带来的困难，围绕子流形几何与曲率流及其相关课题，举办了两次线下会议、两次线上会议与一个系列专题讲座等。同时还邀请国内外20余位著名专家和优秀青年学者做线上报告和交流活动。这些活动吸引了国内子流形几何和曲率流研究方向的近百名青年教师与研究生参加，取得很好的效果。通过这些活动，加深了国内微分几何专家学者之间的高层次的学术交流与探讨，这对于促进培养一批年轻的几何学才俊起到了推动作用。在高级研讨班项目的有力推动下，项目组成员及其合作者在子流形几何与曲率流及相关课题的研究中获得了若干研究成果，在国际著名数学期刊发表高水平学术论文6篇，完成论文4篇。有力促进了对优秀研究生、博士后以及青年教师的培养工作，其中指导培养毕业博士生两名，指导博士后三名，指导在读博士生十名。