Behavior of the Ricci Flow and Related Curature Flows

Ricci 流和相关 Curature 流的行为

• 批准号: 0328233
• 负责人: Dan Knopf
• 金额: $ 6.53万
• 依托单位: University of Iowa
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-12-10 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0328233&HistoricalAwards=false
• 关键词: Behavior; Ricci; Flow; Related; Curature

ABSTRACT DMS - 0202796. PI: Dan KnopfMy research centers on geometric evolutionequations, notably the Ricci flow and related curvature flows. I planto study seven areas in which I have obtained prior results, and wherecontinued work is likely to yield new and useful mathematics. [1] When a flow converges, it is valuable to study the stability of its limit, inorder to improve our global understanding of the dynamics of flows. [2] Ifa flow fails to converge but behaves in a nonsingular way, one can stillstudy the dynamics of this collapse by classifying the asymptotic behaviorof nearby solutions. [3] In most cases, a flow does become singular; so itis of paramount importance (particularly in regard to Hamilton's programto resolve Thurston's Geometrization Conjecture) to develop a betterclassification of singularities. [4] The basic method of studyingsingularities is the construction of a sequence of parabolic dilations(blow-ups). To take limits of these solutions, one must obtain (partial)injectivity radius estimates by various means. [5] The most powerful (butperhaps most difficult) way to obtain such injectivity radius estimateswould be to study and extend existing Harnack estimates of the typepioneered by Li and Yau and further developed by Hamilton. [6] It is also useful to study the asymptotic behavior and stability of parabolicdilations at certain model singularities (a method which has been veryfruitful in studying the mean curvature flow). [7] Further informationabout singularities can be obtained by constructing and studying solitons:self-similar solutions that often arise as limits of blow-ups. Moreover,Kaehler Ricci solitons have interesting connections with complex geometryand algebraic geometry.Geometric evolution studies the way anobject's shape changes. In some cases, such as the mean curvature flow andporous media flow, the motivation is to model certain physical phenomenasuch as the motion of an interface in forming metallic alloys, the shapeof a thin film of highly viscous oil, or the flow of oil in shale. Inother cases, the goal is to improve the shape of an object, either to findoptimal (most efficient) shapes, or else to help mathematicians recognizeand classify geometric objects. My own research is part of a large programto resolve one of the most compelling open questions in mathematics: thedesire to understand and classify all possible 3-dimensional shapes. Butregardless of whether their motivation comes from material science or puremathematics, all geometric evolution problems have much in common; so thatthe field benefits from rich cross-fertilization. In particular, ideas andtechniques that are developed for any of these highly nonlinear problemsare usually quickly adaptable to related applications.

摘要DMS -0202796。PI：Dan Knopfmy研究中心的几何进化等式，尤其是Ricci流量和相关曲率流。我的Planto研究了我获得了以前结果的七个领域，而WheContine的工作可能会产生新的和有用的数学。 [1]当流动收敛时，研究其极限的稳定性是很有价值的，即提高我们对流动动力学的全球理解。 [2] IFA流量无法收敛，但以非语言方式行为，可以通过对附近解决方案的渐近行为进行分类，可以仍然研究这种崩溃的动力学。 [3]在大多数情况下，流动确实变得单数；因此，重要的是至关重要（尤其是关于汉密尔顿解决瑟斯顿的几何化猜想的计划），以发展对奇点的更好分类。 [4]研究表现的基本方法是构造一系列抛物线扩张（爆炸）。为了限制这些解决方案，必须通过各种方式获得（部分）注射性半径估计。 [5]获得这种注射性半径估计的最强大的方法是研究和扩展Li和Yau键入的现有Harnack估计值，并由汉密尔顿进一步开发。 [6]研究某些模型奇异性（一种在研究平均曲率流非常有果实的方法）下，研究抛物线的渐近行为和稳定性也很有用。 [7]可以通过构建和研究孤子来获得进一步的信息，可以获得唯一的：自相似的解决方案通常会作为爆炸的限制。此外，Kaehler Ricci孤子与复杂的几何和代数几何形状具有有趣的联系。几何进化研究AnObject的形状变化的方式。在某些情况下，例如平均曲率流量和孔培养基流，动机是建模某些物理现象，例如界面在形成金属合金中的运动，高度粘性油的薄膜的形状，或者在页岩中的油流。在其他情况下，目标是改善对象的形状，以找到（最有效）形状，或者帮助数学家认识到几何对象。我自己的研究是一个大型计划的一部分，该程序可以解决数学中最引人注目的开放问题之一：TheDesire，以理解和对所有可能的三维形状进行分类。但没有他们的动机是来自材料科学还是丘脑的动力，所有几何进化问题都有很多共同点。因此，该领域受益于丰富的交叉施肥。特别是，为这些高度非线性问题中的任何一个开发的思想和技术通常会迅速适应相关的应用程序。