ANALYTIC AND GEOMETRIC ASPECTS OF RICCI FLOW

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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.