COLLABORATIVE RESEARCH: FRG: Geometric Flows and Applications

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Proposals DMS-0354603/0354621/0354737Title: FRG- Geometric flows and applicationsP.I.s: R.Hamilton, P.Daskalopoulos (Columbia University)/H-D Cao (Lehigh University)/ S-T Yau (Harvard University)ABSTRACT Geometric flows give rise to nonlinear parabolic partial differentialequations. It can be used to understand how a geometric structure evolvesto a more canonical one or the union of canonical structures. In most cases, thereis a tension field which governs the evolution. The most notable cases are the harmonicmap flow, the Ricci flow, the mean curvature flow, the Gaussian curvature flow, and theinverse mean curvature flow. The long time existence and asymptotic behavior of the geometricstructure has revealed deep understanding of geometry and topology. Even short time existencehave immediate consequence of smoothing out the structure. For example, the short time existence of theRicci flow for complete manifolds with bounded curvature provides smoothing effect toapproximate the metric by metrics with bound covariant derivatives of curvature.All these geometric flows have many common features, most notable is the fundamental roleof solitary solutions of the flow. It gives strong understanding of singularity of the nonlinearsystem and lead to good estimates: like the Li-Yau-Hamilton estimate which play importantroles on singularity formations. While working on the Ricci flows, there are constant insight byworking on the mean curvature flow and other geometric flows, and vice versa. The works ofHuisken and Sinestrari will be important for this purpose. And so is the work of Huisken-Ilmanenon the inverse mean curvature flow. The most recent breakthrough of Perelman will of course be the central pieceof discussion for the whole project. Not only that we like to make sure the whole program ofgeometrization for three manifolds, but also we like to strengthen and apply the technique to variousimportant geometric situation: the Ricci flow for compact Kaehler manifolds with positive Chernclass, and to four dimensional manifolds. Note that the recent work of Cao-Chen-Zhu hasalready pointed to the importance of the argument of Perelman in the Kaehler case. Perelman'smost recent work in the Kaehler case made further progress. We hope to incorporate it in a bigger picture ofKaehler geometry. When one studies the Kaehler geometry, a very important ingredient to understandMirror geometry for Calabi-Yau manifolds is the study of special Lagrangian submanifolds. This has been pursued by M.- T. Wang using the Lagrangian mean curvature flow .The existence and regularity of such submanifolds will play important roles in the future of geometry. Aswas mentioned above, the inverse mean curvature flow will also be important for our discussions as it wasdemonstrated by the work of Huisken-Ilmanen in solving the Riemannian Penrose conjecture. In termsof general relativity, Bray, Huisken, M.-T. Wang and Yau will be very much involved in the analysis ofvarious flows that appeared. (Huisken-Yau used the mean curvature flow to study center of gravity, Braystudied the Penrose conjecture) As a whole, there will be close cooperation and many students will betrained under this joint program. We also expect to have joint consultations. Applied mathematicians willalso be consulted on questions like porous media flow, diffusion of oil, imaging sharpening, etc. Daskalopoulos has been active on porous media flow, the Gaussian curvature flow and related questions.