离散度量，离散曲率及离散曲率流方法

This project aims at studying the geometric, topological and combinatorial structure of triangulated manifolds by discrete curvature flow methods. Concretely, we plan to study the following topics..1. Define a new discrete Gauss curvature and scalar curvature. Introduce discrete Ricci flow, discrete Calabi flow, discrete Yamabe flow to study the discrete Yamabe problem, i.e. finding discrete metrics with constant discrete scalar curvature..2. Define discrete Einstein metric, and introduce discrete curvature methods to study the properties of discrete Einstein metric..3. Find the relationship between the constant discrete scalar curvature metrics, discrete Einstein metrics and the geometric, topological, combinatorial structures of manifolds. .4. Study the relation between discrete geometric objects and the corresponding smooth geometric objects. Find out the conditions on manifolds under which the discrete geometric objects, such as discrete metric, discrete curvature, and discrete curvature flows, converge to the corresponding smooth geometric objects..5. Apply the discrete curvature methods in engineering fields, especially computer graphics, computer vision, and medical imaging..The applicant has already published several papers in top journals regarding this research area, so there is no reason to doubt the applicants' ability and experience in this research topic. This project is a natural extension of the research the applicant has done. The applicant is sure that he is competent to do this project well.

本项目旨在引入离散曲率流方法，研究剖分流形的几何拓扑及组合结构。具体而言，本项目拟研究如下内容：.1. 定义新的离散 Gauss 曲率与离散数量曲率，并引入离散 Ricci 流、Calabi 流、Yamabe 流研究离散 Yamabe 问题——寻找离散常曲率度量；.2. 定义离散 Einstein 度量，引入离散曲率流研究该度量的性质；.3. 研究离散常曲率度量、离散 Einstein 度量与流形的几何拓扑及剖分的组合结构之间的关系；.4. 考察离散几何对象与相关光滑几何对象的关系，在适当条件下或者对于某些规则流形，实现离散度量、离散曲率及离散曲率流对相应光滑量的近似与逼近；.5. 把离散曲率流方法应用到曲面参数化、几何建模、医学成像、计算机视觉等领域。.申请人在该领域已有很多积累。本项目是申请人已有工作的自然延续，申请人有信心、有能力优质完成本项目。

本项目旨在引入离散曲率流方法，研究剖分流形的几何拓扑及组合结构。具体而言，本项目需要合理定义离散 Gauss 曲率与离散数量曲率，并引入离散 Ricci 流、Calabi 流、Yamabe 流研究离散 Yamabe 问题——寻找离散常曲率度量；本项目需要定义离散 Einstein 度量，引入离散曲率流研究该度量的性质，研究离散常曲率度量、离散 Einstein 度量与流形的几何拓扑及剖分的组合结构之间的关系；另外，本项目还关注离散曲率流方法在曲面参数化、几何建模等领域的应用。..经过本项目的培育，目前已经在GAFA，Adv. Math.，Math Ann，J. Funct. Analy.，Trans. AMS.，Int. Math.Res.Not.，Cal. Var. PDE. 等期刊发表SCI论文共20篇。其中有19篇是第一作者或者通讯作者，而且这19篇论文中，都是第一标注本项目基金。另外，项目执行人成功申请到国家自科基金面上项目。