ANALYTIC AND GEOMETRIC ASPECTS OF RICCI FLOW

摘要奖：DMS-9971891首席研究员：Bennett Chow该项目的目标是研究黎曼度量的哈密尔顿利奇流的解析和几何方面，以及相关主题。这个流在理论上成功地对满足正曲率条件的黎曼流形进行了分类。哈密尔顿关于三维流形上的Ricciflow的程序是理解瑟斯顿几何化猜想的一种方法，该猜想包含Poincare猜想。项目中考虑的主要主题包括寻找Ricci流的新的Harnack不等式，进一步了解现有的Harnackin等式，以及研究Ricci流的崩溃解的行为。Harnack不等式是偏微分方程域中的基本问题，由Li，Yau和Hamilton的工作开创了微分几何中抛物型方程的先河。哈密尔顿矩阵的Harnack不等式在分析Ricci流下出现的奇点时特别重要。对这些奇点的理解对其他演化方程的研究产生了重大影响，例如平均曲率流，以及与注入量半径估计相关的紧致性定理等依赖工具。在没有这样的估计的情况下，我们将考虑Ricci流动的各个方面，其中解决方案崩溃。许多现象由演化方程模拟，例如热的传递和金属熔体和固体形式之间的界面的演化。在过去的几年里，演化方程计量学的研究有了很大的发展。在许多情况下，几何演化方程将初始的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。对这些奇点进行分析是非常重要的。这样的研究在很大程度上是对Ricci流进行的，它是一个演化方程，它在流形上变形几何结构，在爱因斯坦相对论中产生的局部欧氏空间，以及数学和理论物理的大多数主要分支。例如，时空是一个4维流形，而我们生活的宇宙是一个3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规范几何结构的碎片。由于Ricci流使几何结构变形，而所有流形都承认几何结构，如果能证明该流将几何结构变形为正则几何结构，则它可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.