FRG: Collaborative Research: Geometric Flows and Applications

建议DMS-0354603/0354621/0354737标题：FRG-几何流动和应用P.I.S.哈密尔顿，P.Daskalopoulos(哥伦比亚大学)/H-D CaO(利哈伊大学)/S-T Yau(哈佛大学)抽象几何流动产生非线性抛物型偏微分方程组。它可以用来理解几何结构如何演变为更规范的结构或规范结构的并集。在大多数情况下，有一个紧张场支配着进化。最显著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。几何结构的长期存在和渐近行为揭示了人们对几何和拓扑学的深刻理解。即使是短时间的存在，也会立即产生使结构变得平滑的后果。例如，具有有界曲率的完备流形的Ricci流的短时间存在性提供了用曲率的有界协变导数的度量来逼近度量的光滑化效果，所有这些几何流都有许多共同的特征，其中最值得注意的是流的孤立解的基本作用。它使人们对非线性系统的奇异性有了很好的理解，并得到了很好的估计：就像Li-Yau-Hamilton估计一样，它对奇点的形成起着重要的作用。在研究Ricci流时，通过研究平均曲率流和其他几何流，可以得到持续的洞察力，反之亦然。惠斯肯和Sinestrari的作品对于这一目的将是重要的。Huisken-Ilmanenon逆平均曲率流的工作也是如此。佩雷尔曼的最新突破当然将是整个项目讨论的中心部分。我们不仅希望确定三个流形的整个几何化程序，而且还希望加强和应用这一技术来处理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，曹振柱最近的工作已经指出了佩雷尔曼在凯勒案件中的论点的重要性。佩雷尔曼最近在凯勒案中的工作取得了进一步的进展。我们希望把它融入到更大的Kaehler几何图景中。在研究Kaehler几何时，要理解Calabi-Yau流形的镜像几何，一个非常重要的因素就是研究特殊的拉格朗日子流形。M-T。Wang利用拉格朗日平均曲率流来追求这一点。此子流形的存在和正则性将在未来的几何学中发挥重要作用。如上所述，逆平均曲率流对于我们的讨论也将是重要的，正如Huisken-Ilmanen在求解黎曼彭罗斯猜想时所做的工作所证明的那样。从广义相对论的角度来看，布雷、惠斯肯、王文通和邱文通将会非常积极地参与到对出现的各种流动的分析中。(Huisken-Yau使用平均曲率流研究重心，Bray研究彭罗斯猜想)作为一个整体，在这个联合计划下，将会有密切的合作和许多学生的背叛。我们也期待进行共同磋商。Daskalopoulos还将在多孔介质流动、油的扩散、成像锐化等问题上咨询应用数学家。Daskalopoulos一直活跃在多孔介质流动、高斯曲率流动和相关问题上。

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.