Ricci-Hessian 型黎曼流形的刚性及分类问题研究

The investigations on rigidity and classification of typical Riemannian manifolds are important concerns of differential geometric theory since its beginning. In recent years, along with the successful applications of methods in those subjects like geometric analysis, theory of critical points on geometric functional, a number of concepts of Ricci-Hessian type manifolds are constantly emerging. These Ricci-Hessian type manifolds include not only quasi Einstein manifolds like Ricci soliton closely related to curvature flow, but some special critical points of the important Riemannian functionals under the restrictive conditions like Miao-Tam metric and CPE metric. The Ricci-Hessian type tensor equations can always characterize manifold essences to a great extent, for instance, the characterization of the standard sphere metric by Obata. In this project, relying on the previous research foundation on quasi Einstein manifolds, we will concentrate on studying problems related to rigidity and classification of Ricci-Hessian type Riemannian manifolds under some assumptions, especially focus on rigidity and classification of generalized m-quasi-Einstein manifold, h-almost Ricci soliton, Miao-Tam critical metric, CPE metric with typical geometric or topological properties.

典型黎曼流形的刚性与分类问题研究历来是微分几何的重要关切。近年来，随着几何分析方法、泛函临界点研究技巧在此类课题研究上的成功运用，大量典型 Ricci-Hessian 型流形不断涌现，其中既包括与曲率流密切相关的 Ricci soliton、广义 m-quasi-Einstein 流形等拟爱因斯坦流形，也包括 Miao-Tam 度量、CPE 度量等重要黎曼泛函在限制条件下的临界点。流形上 Ricci-Hessian 型方程往往在很大程度上刻画流形的本质，如 Obata 关于球面度量的经典刻画。在前期拟爱因斯坦流形的研究基础上，本项目将集中研究 Ricci-Hessian 型流形在一定前提条件下的分类及刚性问题，特别专注于具有特殊几何与拓扑性质的广义 m-quasi-Einstein 流形、h-almost Ricci soliton 、Miao-Tam 度量、CPE 度量的刚性与分类研究。

Ricci-Hessian型流形是指度量与Ricci曲率、势函数的Hessian满足特定张量方程的流形。它与曲率流、黎曼泛函临界点的研究联系密切，同时在物理学领域有着重要应用，近年来受到几何学家的广泛关注。本项目在前期拟爱因斯坦流形研究的基础上，围绕Ricci-Hessian型流形的刚性及分类问题，按计划开展研究，取得了一些成果。一方面，在仿切触几何框架下，本项目分类了满足Miao-Tam临界点方程的仿切触度量(κ,μ)流形。另一方面，充分应用加权流形上的曲率估计及重要分析工具，在零迹黎曼曲率和零迹Ricci曲率构成的逐点曲率拼挤条件下，得到了两个完备收缩型梯度Ricci soliton的刚性结果。除此之外，受秩二复格拉斯曼流形（复双平面格拉斯曼流形与复双曲双平面格拉斯曼流形）中具有Ricci soliton结构的实超曲面研究的启发，作为项目研究内容的拓展与延伸，本项目得到了秩二复格拉斯曼流形中局部共形平坦的实超曲面的不存在性。