Geometric flows give rise to nonlinear parabolic partial differential equations. It can be used to understand how a geometric structure evolves to a more canonical one or the union of canonical structures. In most cases, there is a tension field which governs the evolution. The most notable cases are the harmonic map flow, the Ricci flow, the mean curvature flow, the Gaussian curvature flow, and the inverse mean curvature flow. The long time existence and asymptotic behavior of the geometric structure has revealed deep understanding of geometry and topology. Even short time existence have immediate consequence of smoothing out the structure. For example, the short time existence of the Ricci flow for complete manifolds with bounded curvature provides smoothing effect to approximate the metric by metrics with bound covariant derivatives of curvature. All these geometric flows have many common features, most notable is the fundamental role of solitary solutions of the flow. It gives strong understanding of singularity of the nonlinear system and lead to good estimates: like the Li-Yau-Hamilton estimate which play important roles on singularity formations. While working on the Ricci flows, there are constant insight by working on the mean curvature flow and other geometric flows, and vice versa. The works of Huisken and Sinestrari will be important for this purpose. And so is the work of Huisken-Ilmanen on the inverse mean curvature flow. The most recent breakthrough of Perelman will of course be the central piece of discussion for the whole project. Not only that we like to make sure the whole program of geometrization for three manifolds, but also we like to strengthen and apply the technique to various important geometric situation: the Ricci flow for compact Kaehler manifolds with positive Chern class, and to four dimensional manifolds. Note that the recent work of Cao-Chen-Zhu has already pointed to the importance of the argument of Perelman in the Kaehler case. Perelman's most recent work in the Kaehler case made further progress. We hope to incorporate it in a bigger picture of Kaehler geometry. When one studies the Kaehler geometry, a very important ingredient to understand Mirror 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. As was mentioned above, the inverse mean curvature flow will also be important for our discussions as it was demonstrated by the work of Huisken-Ilmanen in solving the Riemannian Penrose conjecture. In terms of general relativity, Bray, Huisken, M.-T. Wang and Yau will be very much involved in the analysis of various flows that appeared. (Huisken-Yau used the mean curvature flow to study center of gravity, Bray studied the Penrose conjecture) As a whole, there will be close cooperation and many students will be trained under this joint program. We also expect to have joint consultations. Applied mathematicians will also 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.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Christopher W. Stark
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Columbia University
New York
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