Parabolic differential equations and the geometry of manifolds

本研究涉及非线性抛物型偏微分方程组及其在流形几何和拓扑学中的应用研究中的几个问题。一个一般的集中领域将是利玛窦流动中奇点形成的领域，重点是古解和利玛窦孤子的研究。哈密尔顿、佩雷尔曼和其他人发展的理论的一个中心主题是，发展中奇点的局部几何具有相对严格的结构，在许多重要情况下，以这些特殊类型的解为模型。因此，需要尽可能详细地了解它们可以假定的形式的多样性。此外，合作者还建议研究类型II奇点的发展和Ricci流动方程的唯一延拓问题。另一个一般的集中领域将是超曲面的曲率流，重点是这些流的微分Harnack不等式的应用、解释和进一步发展。其中一个目标是在Minkowski空间中演化的类空超曲面的背景下提炼这种不等式，着眼于永恒解的研究和孤子的翻译。在平均曲率流的情况下，广义相对论的研究人员对后一类物体作为洛伦兹时空的自然叶状结构感兴趣。在另一个方向上，合作者建议寻求从这种环境到交叉曲率流动的潜在联系，交叉曲率流动是研究具有负曲率的三维流形的潜在用途的内在流动。在这个建议中考虑的Ricci流和其他几何演化方程是几何学中“热流”方法的代表，其技术和目标跨越了该领域与拓扑、分析和数学物理的生动接口。这种方法被证明在攻击数学中最基本的问题之一的某些情况下是有效的，即哪些流形承认常曲率或其他正则几何？这个问题不仅对宇宙的物理模型有影响，而且伴随着这种方法的工具的发展，有望继续为分析许多结构相似的非线性偏微分方程组带来红利，这些偏微分方程组作为整个物理科学中不同现象的模型出现。

The proposed research concerns several problems in the study of nonlinear parabolic PDE and their application to the geometry and topology of manifolds. One general area of concentration will be that of singularity formation in the Ricci flow, with an emphasis on the study of ancient solutions and Ricci solitons. A central theme in the theory developed by Hamilton, Perelman, and others, is that the local geometry of a developing singularity possesses a relatively rigid structure, modelled, in many important cases, upon these special types of solutions. It is thus desirable to have as detailed a knowledge as possible of the diversity of forms which they can assume.Additionally, the co-PI proposes to investigate Type-II singularity development and the question of unique continuation for the Ricci flow equation. Another general area of concentration will be curvature flows of hypersurfaces, with an emphasis on the application, interpretation, and further development of differential Harnack inequalities for these flows. One aim will be to refine such inequalities in the setting of evolving spacelike hypersurfaces in Minkowski space, with an eye toward the study of eternal solutions and translating solitons. In the case of the mean curvature flow, the latter objects are of interest to researchers in general relativity as natural foliations of Lorentzian spacetimes. In another direction, the co-PI proposes to pursue a potential connection from this setting to the cross-curvature flow, an intrinsic flow of potential use in the study of three-manifolds with negative curvature.The Ricci flow and other geometric evolution equations considered in this proposal are representatives of the "heat-flow" method in geometry, the techniques and objectives of which straddle the field's lively interface with topology, analysis, and mathematical physics.This method has proven effective in attacking certain cases of one of the most fundamental questions in mathematics, namely, which manifolds admit constant curvature or otherwise canonical geometries? Not only does this question have ramifications for physical models of the universe, but the development of tools attendant to the approach promise to pay continued dividends to the analysis of the many structurally similar nonlinear PDE which occur as models of diverse phenomena throughout the physical sciences.