ANALYTIC AND GEOMETRIC ASPECTS OF RICCI FLOW

项目负责人：Bennett chow项目的目的是研究黎曼度量的Hamilton Ricci流的解析和几何方面，以及相关主题。该流已成功地对满足正曲率条件的黎曼流形进行了拓扑分类。Hamilton在三维流形上的Ricciflow程序是理解thurston的几何化猜想的一种方法，该猜想包含了poincare猜想。该项目考虑的主要主题包括寻找新的Ricci流的Harnack不等式，进一步了解现有的harnack不等式，以及研究Ricci流的坍缩解的行为。哈纳克不等式是偏微分方程领域的基础，由李、丘和汉密尔顿的工作开创了微分几何中的抛物方程。汉密尔顿矩阵-哈纳克不等式在分析里奇流下出现的奇点时尤为重要。理解这些奇点对研究其他演化方程（如平均曲率流）和依赖工具（如与注入半径估计相关的紧性定理）有重大影响。我们将在没有这种估计的情况下考虑利玛窦流的各个方面，在这些方面解会崩溃。许多现象都是用演化方程来模拟的，比如热的传递和金属的熔融态和固态之间界面的演化。演化方程几何学的研究在过去几年中有了极大的发展。在许多情况下，几何演化方程使初始几何结构变形为改进的几何结构，但在其他情况下，结构发展为奇点。对这些奇异点进行分析是非常重要的。这种研究在很大程度上是针对里奇流（一种使流形上的几何结构变形的演化方程）、爱因斯坦相对论中出现的局部欧几里得空间以及数学和理论物理的大多数主要分支进行的。例如，时空是一个四维流形，而我们生活的宇宙是一个三维流形。数学家们普遍认为，任何三维流形都可以分解成具有标准几何结构的块。由于里奇流使几何结构变形，而所有流形都承认几何结构，因此，如果能证明里奇流使几何结构变形为典型几何结构，就可以作为解决上述问题的一种方法。

AbstractAward: DMS-9971891Principal Investigator: Bennett ChowThe objective of the project is to investigate analytic andgeometric aspects of Hamilton's Ricci flow of Riemannian metrics,and related topics. This flow has been successful intopologically classifying Riemannian manifolds satisfyingpositive curvature conditions. Hamilton's program for the Ricciflow on 3-dimensional manifolds is an approach to understandingThurston's Geometrization Conjecture, which subsumes the PoincareConjecture. The main topics considered in the project includethe search for new Harnack inequalities for the Ricci flow,furthering the geometric understanding of the existing Harnackinequalities, and investigations in the behavior of the Ricciflow for collapsing solutions. Harnack inequalities arefundamental to the area of partial differential equations andthey have been pioneered for parabolic equations in differentialgeometry by the work of Li, Yau, and Hamilton. Hamilton's matrixHarnack inequality is especially important in the analysis ofsingularities that arise under the Ricci flow. Understandingthese singularities has had a major impact on the study of otherevolution equations such as the mean curvature flow, and dependsupon tools such as a compactness theorem related to injectivityradius estimates. We shall consider aspects of the Ricci flow inthe absence of such estimates, where solutions collapse.Many phenomena are modeled by evolutionary equations, such as thetransfer of heat and the evolution of interfaces between moltenand solid forms of metal. The study of evolution equations ingeometry has grown tremendously in the last several years. Inmany cases geometric evolution equations deform an initialgeometric structure to an improved one, but in other cases thestructures develop singularities. It is of fundamental importanceto analyze these singularities. Such a study has been carried outto a large extent for the Ricci flow, which is an evolutionequation deforming geometric structures on manifolds, the locallyEuclidean spaces arising in Einstein's Theory of Relativity andmost major branches of mathematics and theoretical physics. Forexample, space-time is a 4-dimensional manifold and the universewe live in is a 3-dimensional manifold. It is widely believed bymathematicians that any 3-dimensional manifold may be decomposedinto pieces which admit canonical geometric structures. Sincethe Ricci flow deforms geometric structures, and all manifoldsadmit geometric structures, it may be used as an approach to theabove question if it can be shown that the flow deforms geometricstructures to canonical geometric structures.