ANALYTIC AND GEOMETRIC ASPECTS OF RICCI FLOW

AbstractAward：DMS-9971891首席研究员：班尼特周该项目的目标是调查分析和几何方面的汉密尔顿的里奇流的黎曼度量，和相关的主题。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。三维流形上Ricciflow的汉密尔顿程序是理解Thurston几何化猜想的一个途径，它包含了Poincare猜想。 在该项目中考虑的主要议题包括寻找新的Harnack不等式的Ricci流，进一步的几何理解现有的Harnack不等式，并在行为的Ricciflow崩溃的解决方案的调查。Harnack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抛物方程的研究。 汉密尔顿矩阵Harnack不等式在分析Ricci流下的奇异性时特别重要。 理解这些奇异性对研究其他的卷积方程，如平均曲率流，以及依赖于工具，如与注入半径估计相关的紧致性定理，产生了重大影响。 我们将考虑在没有这种估计的情况下的Ricci流的各个方面，其中解决方案collaps.Many现象是由演化方程模拟的，如热的传递和熔融和固体形式的金属之间的界面的演化。发展方程的研究在过去的几年里有了很大的发展。 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，结构发展为奇异性。对这些奇异性的分析是十分重要的。这种研究在很大程度上已经对Ricci流进行了，Ricci流是流形上变形几何结构的演化方程，爱因斯坦相对论中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，时空是一个四维流形，而我们所处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被分解成具有规范几何结构的块。 由于Ricci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Ricci流使几何结构变形为正则几何结构，就可以作为解决上述问题的一种途径。

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.