黎曼流形上 Ricci 曲率的几何

项目名称： 黎曼流形上 Ricci 曲率的几何

项目编号： No.11501285

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 许奕彦

作者单位： 南京大学

项目金额： 18万元

中文摘要： 本项目主要研究 Riemannian 流形上 Ricci 曲率的几何。特别地，我们将研究 Ricci 曲率假设下 Riemannian 度量的形变理论及 Ricci 流的奇点理论。在第一部分中，我们将重点探索 Ricci 曲率与流形几何拓扑结构之间的关系，Ricci 曲率有下界 Riemannian 流形的收敛理论，以及 Ricci 曲率一致有界 Riemannian 度量 Gromov-Hausdorff 极限的正则性。这三个研究课题密切相关，对 Ricci 曲率几何的新观点将解决很多与 Ricci 曲率有关的几何问题。在第二部分，我们的主要目标是分类 Ricci 孤立子。特别的，我们希望能够刻画具有非负 Ricci 曲率的 Ricci 收缩子的几何结构。如果获得成功，这将帮助理解 Ricci 流的奇点理论，进而在研究流形的几何拓扑问题上获得新进展。

中文关键词： Riemannian；流形；Ricci；曲率；Gromov-Hausdorff；收敛；Ricci；孤立子

英文摘要： The proposed research is concerned with geometric aspects of Ricci curvature on Riemannian Manifolds. In particular, we plan to study the degeneration theory of Riemannian metrics under Ricci curvature bound, as well as the singularity formations of Ricci flow. In the first part, we will focuses on studying the connections between Ricci curvature and the geometric, topological structure of Riemannain manifolds, the convergence theory of Riemannian metric with lower Ricci curvature, and the regularity theory of Gromov-Hausdorff limits of Riemannian metircs with bounded Ricci curvature. These three projects are intimately connected, the further understanding in geometric aspects of Ricci curvature should solve many open geometric problems involving Ricci curvature. In the second part, the main goal is to classify gradient Ricci soliton. Specifically, we hope to study the geometric structure of Ricci shrinkers with nonnegative Ricci curvature. The success will benefit the understanding of the singularity formations of the Ricci flow, and has profound applications in geometry and topology of manifolds.

英文关键词： Riemannian Manifold;Ricci Curvature;Gromov-Hausdorff Convergence;Ricci Soliton