Behavior of the Ricci Flow and Related Curature Flows

摘要DMS - 0202796。PI: Dan knopf我的研究集中在几何演化方程上，特别是里奇流和相关的曲率流。我计划研究七个领域，在这些领域我已经取得了先前的成果，并在继续的工作中可能产生新的和有用的数学。当气流收敛时，研究其极限的稳定性对于提高我们对气流动力学的整体认识是有价值的。如果流不能收敛，但表现为非奇异方式，人们仍然可以通过对附近解的渐近行为进行分类来研究这种坍缩的动力学。[3]在大多数情况下，流确实是单一的；因此，发展一种更好的奇点分类是至关重要的（特别是关于汉密尔顿解决瑟斯顿几何化猜想的方案）。研究奇点的基本方法是构造一个抛物线膨胀（膨胀）序列。为了求这些解的极限，必须用各种方法求得（部分）注入半径估计。[5]获得这种注入半径估计的最有力（但可能是最困难）的方法是研究和扩展现有的由Li和Yau首创并由Hamilton进一步发展的哈纳克估计。研究抛物膨胀在某些模型奇点处的渐近行为和稳定性也是有用的（这种方法在研究平均曲率流方面非常有成果）。关于奇点的进一步信息可以通过构造和研究孤子获得：自相似解通常作为爆炸的极限出现。此外，Kaehler Ricci孤子与复几何和代数几何有着有趣的联系。几何演化研究的是物体形状变化的方式。在某些情况下，如平均曲率流和多孔介质流，动机是模拟某些物理现象，如形成金属合金时界面的运动，高粘性油薄膜的形状，或页岩中油的流动。在其他情况下，目标是改进物体的形状，或者找到最优（最有效）的形状，或者帮助数学家识别和分类几何物体。我自己的研究是一个大型项目的一部分，该项目旨在解决数学中最引人注目的开放性问题之一：理解和分类所有可能的三维形状的愿望。但无论他们的动机是来自材料科学还是纯粹的数学，所有几何演化问题都有很多共同点；这样田地就能从丰富的异花受精中获益。特别是，为这些高度非线性问题开发的思想和技术通常很快适用于相关应用。

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.