Principal Investigator: Bennett Chow
The objective of the project is to investigate analytic and geometric aspects of Hamilton's Ricci flow of Riemannian metrics, and related topics. This flow has been successful in topologically classifying Riemannian manifolds satisfying positive curvature conditions. Hamilton's program for the Ricci flow on 3-dimensional manifolds is an approach to understanding Thurston's Geometrization Conjecture, which subsumes the Poincare Conjecture. The main topics considered in the project include the search for new Harnack inequalities for the Ricci flow, furthering the geometric understanding of the existing Harnack inequalities, and investigations in the behavior of the Ricci flow for collapsing solutions. Harnack inequalities are fundamental to the area of partial differential equations and they have been pioneered for parabolic equations in differential geometry by the work of Li, Yau, and Hamilton. Hamilton's matrix Harnack inequality is especially important in the analysis of singularities that arise under the Ricci flow. Understanding these singularities has had a major impact on the study of other evolution equations such as the mean curvature flow, and depends upon tools such as a compactness theorem related to injectivity radius estimates. We shall consider aspects of the Ricci flow in the absence of such estimates, where solutions collapse.
Many phenomena are modeled by evolutionary equations, such as the transfer of heat and the evolution of interfaces between molten and solid forms of metal. The study of evolution equations in geometry has grown tremendously in the last several years. In many cases geometric evolution equations deform an initial geometric structure to an improved one, but in other cases the structures develop singularities. It is of fundamental importance to analyze these singularities. Such a study has been carried out to a large extent for the Ricci flow, which is an evolution equation deforming geometric structures on manifolds, the locally Euclidean spaces arising in Einstein's Theory of Relativity and most major branches of mathematics and theoretical physics. For example, space-time is a 4-dimensional manifold and the universe we live in is a 3-dimensional manifold. It is widely believed by mathematicians that any 3-dimensional manifold may be decomposed into pieces which admit canonical geometric structures. Since the Ricci flow deforms geometric structures, and all manifolds admit geometric structures, it may be used as an approach to the above question if it can be shown that the flow deforms geometric structures to canonical geometric structures.
