Behavior of the Ricci Flow and Related Curature Flows

点击翻译按钮获取中文摘要

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.