Behavior of the Ricci Flow and Related Curature Flows

摘要 DMS - 0202796。 PI：Dan Knopf 我的研究重点是几何演化方程，特别是里奇流和相关的曲率流。我打算研究七个领域，在这些领域我已经取得了先前的成果，并且继续工作可能会产生新的有用的数学。 [1] 当流收敛时，研究其极限稳定性很有价值，以提高我们对流动力学的整体理解。 [2] 如果a流无法收敛但表现出非奇异方式，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崩溃的动力学。 [3] 在大多数情况下，流确实会变得单一；因此，开发更好的奇点分类至关重要（特别是关于解决瑟斯顿几何化猜想的汉密尔顿计划）。 [4] 研究奇点的基本方法是构造一系列抛物线膨胀（爆炸）。为了限制这些解决方案，必须通过各种方式获得（部分）注入半径估计。 [5] 获得这种射射半径估计的最有力（但也许是最困难）的方法是研究和扩展由 Li 和 Yau 首创并由 Hamilton 进一步发展的现有 Harnack 估计。 [6] 研究某些模型奇点处抛物线膨胀的渐近行为和稳定性也很有用（这种方法在研究平均曲率流方面非常富有成效）。 [7] 关于奇点的更多信息可以通过构建和研究孤子来获得：自相似解通常作为爆炸的极限而出现。此外，凯勒·里奇孤子与复杂几何和代数几何有着有趣的联系。几何演化研究物体形状变化的方式。在某些情况下，例如平均曲率流和多孔介质流，其动机是模拟某些物理现象，例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的油流。在其他情况下，目标是改善对象的形状，要么找到最佳（最有效）的形状，要么帮助数学家识别和分类几何对象。我自己的研究是一个大型计划的一部分，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 3 维形状的愿望。但无论其动机来自材料科学还是纯数学，所有几何进化问题都有很多共同点；使田地受益于丰富的异花施肥。特别是，针对任何这些高度非线性问题开发的想法和技术通常可以快速适应相关应用。

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.