FRG: Collaborative Research: Geometric Flows and Applications

题目：FRG-几何流动及其应用[j]: r.h ilton， P.Daskalopoulos（哥伦比亚大学）/H-D Cao （Lehigh大学）/ S-T Yau（哈佛大学）。它可以用来理解一个几何结构如何演变成一个更规范的结构或规范结构的结合。在大多数情况下，存在一个控制演化的张力场。最显著的例子是谐波流、里奇流、平均曲率流、高斯曲率流和逆平均曲率流。几何结构的长时间存在性和渐近性揭示了对几何和拓扑的深刻理解。即使是短暂的存在，也会对结构的平滑产生直接的影响。例如，曲率有界的完全流形的theRicci流的短时间存在性为用曲率有界协变导数的度量近似度规提供了平滑效果。所有这些几何流都有许多共同的特点，其中最显著的是流的孤立解的基本作用。它使我们对非线性系统的奇异性有了深刻的认识，并得到了很好的估计，如li - yu - hamilton估计，它对奇异性的形成起着重要的作用。在研究里奇流时，通过研究平均曲率流和其他几何流，可以不断地获得洞察力，反之亦然。huisken和Sinestrari的作品将在这方面发挥重要作用。huisken - ilmannon关于逆平均曲率流的研究也是如此。佩雷尔曼最近的突破当然会成为整个项目讨论的中心。我们不仅想要确定三维流形的几何化的整个程序，而且我们想要加强并应用该技术到各种重要的几何情况：具有正Chernclass的紧凑Kaehler流形的Ricci流，以及四维流形。请注意，曹晨-朱最近的工作已经指出佩雷尔曼的论点在凯勒案中的重要性。佩雷尔曼最近对凯勒一案的研究取得了进一步的进展。我们希望把它融入到更大的凯勒几何中。在研究Kaehler几何时，理解Calabi-Yau流形的镜像几何的一个非常重要的组成部分是对特殊拉格朗日子流形的研究。这已经被M.- T. Wang用拉格朗日平均曲率流进行了研究。这种子流形的存在性和规律性将在未来的几何学中发挥重要作用。如上所述，逆平均曲率流对我们的讨论也很重要，因为它在Huisken-Ilmanen解决黎曼彭罗斯猜想时的工作中得到了证明。在广义相对论方面，布雷，惠斯肯，m - t。Wang和Yau将深入分析出现的各种流动。（Huisken-Yau用平均曲率流来研究重心，bray研究彭罗斯猜想）作为一个整体，我们将密切合作，在这个联合项目下将培养很多学生。我们也期待着进行共同磋商。应用数学家也将咨询问题，如多孔介质流动，油的扩散，成像锐化等。Daskalopoulos一直活跃于多孔介质流动、高斯曲率流动和相关问题。

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.