FRG: Collaborative Research: Geometric Flows and Applications

FRG：协作研究：几何流和应用

• 批准号: 0354639
• 负责人: Richard Hamilton
• 金额: --
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0354639&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Geometric; Flows

Proposals DMS-0354639/0354621/0354737Title: 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.

提案 DMS-0354639/0354621/0354737 标题：FRG-几何流和应用 PI：R.Hamilton、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。它可用于理解几何结构如何演化为更规范的结构或规范结构的并集。在大多数情况下，存在一个控制演化的张力场。最值得注意的情况是调和图流、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对几何和拓扑的深刻理解。即使存在的时间很短，也会产生平滑结构的直接后果。例如，具有有界曲率的完全流形的Ricci流的短时存在性，为通过曲率有界协变导数的度量来逼近度量提供了平滑效应。所有这些几何流都有许多共同的特征，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点有了深入的理解，并得出良好的估计：就像在奇点形成中发挥重要作用的李-丘-汉密尔顿估计一样。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获得洞察力，反之亦然。惠斯肯和辛斯特拉里的作品对于这一目的非常重要。 Huisken-Ilmanenon 的逆平均曲率流工作也是如此。佩雷尔曼最近的突破当然将成为整个项目讨论的核心。我们不仅希望确定三个流形的几何化的整个程序，而且还希望加强该技术并将其应用于各种重要的几何情况：具有正Chern类的紧凯勒流形的Ricci流以及四维流形。请注意，曹辰朱最近的著作已经指出了佩雷尔曼在凯勒案中的论点的重要性。佩雷尔曼最近在凯勒案中的工作取得了进一步的进展。我们希望将其纳入凯勒几何的更大图景中。当人们研究凯勒几何时，理解卡拉比-丘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。 M.-T. Wang 利用拉格朗日平均曲率流来追求这一点。这种子流形的存在和规律性将在未来的几何学中发挥重要作用。如上所述，逆平均曲率流对于我们的讨论也很重要，因为它是由 Huisken-Ilmanen 在解决黎曼彭罗斯猜想时所证明的。在广义相对论方面，Bray, Huisken, M.-T.王和丘将大量参与对出现的各种流量的分析。 （Huisken-Yau用平均曲率流研究重心，Bray研究彭罗斯猜想）总的来说，这个联合项目将会有密切的合作，并且会培养很多学生。 我们也期待进行共同磋商。还将就多孔介质流、油扩散、成像锐化等问题咨询应用数学家。达斯卡洛普洛斯一直活跃于多孔介质流、高斯曲率流和相关问题。