COLLABORATIVE RESEARCH: FRG: Geometric Flows and Applications

提案DMS-0354603/0354621/0354737题目：联邦德国-几何流和应用P.I.s：R.汉密尔顿，P.Daskalopoulos（哥伦比亚大学）/H-D Cao（利哈伊大学）/S-T Yau（哈佛大学）摘要几何流产生非线性抛物型偏微分方程。它可以用来理解一个几何结构如何演化为一个更规范的结构或规范结构的并集。在大多数情况下，有一个张力场控制着进化。最著名的例子是调和映射流、Ricci流、平均曲率流、高斯曲率流和逆平均曲率流。该几何结构的长期存在性和渐近性态揭示了对几何和拓扑的深刻理解。即使是短暂的存在也会立即使结构变得平滑。例如，有界曲率完备流形的Ricci流的短时存在性提供了光滑效应，使得度量可以用曲率的有界协变导数来逼近，所有这些几何流都有许多共同的特征，最显著的是流的孤立解的基本作用。它给出了对非线性系统奇异性的深刻理解，并得到了很好的估计：如Li-Yau-汉密尔顿估计，它对奇异性的形成起着重要作用。在研究Ricci流的同时，也不断地研究平均曲率流和其他几何流，反之亦然。Huisken和Sinestrari的作品将是重要的这一目的。Huisken-Ilmanen关于反平均曲率流的工作也是如此。佩雷尔曼的最新突破当然将成为整个项目讨论的中心。我们不仅希望确定三维流形的几何化的整个程序，而且希望加强和应用该技术到各种重要的几何情况：具有正Chern类的紧致Kaehler流形的Ricci流，以及四维流形。请注意，Cao-Chen-Zhu最近的工作已经指出了Perelman在Kaehler案例中的论点的重要性。佩雷尔曼最近在凯勒案中的工作取得了进一步的进展。我们希望把它纳入一个更大的图片ofKaehler几何。在研究Kaehler几何时，对特殊Lagrange子流形的研究是理解Calabi-Yau流形的镜像几何的一个重要组成部分。这是M所追求的。T.这类子流形的存在性和正则性将在未来的几何中发挥重要作用。如上所述，反平均曲率流对我们的讨论也很重要，因为Huisken-Ilmanen在解决RiemannianPenrose猜想中的工作证明了这一点。根据广义相对论，Bray，Huisken，M. T. Wang和Yau将非常积极地参与分析出现的各种流动。（Huisken-Yau用平均曲率流研究重心，Bray研究Penrose猜想）总的来说，将有密切的合作，许多学生将在这个联合项目下接受培训。 我们还期待进行联合磋商。应用数学家也将咨询问题，如多孔介质流，石油扩散，成像锐化等Daskalopoulos一直活跃在多孔介质流，高斯曲率流和相关问题。

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.