CAREER:Singularities and singularity models in curvature flows

本项目的主要目的是研究微分几何问题中的非线性抛物方程，如欧几里得空间的超曲面由其主曲率函数、Ricci流和Yamabe流的演化。更准确地说，PI侧重于对这些演化方程中可能出现的有限时间奇点进行奇点分析。这些方程出现在量子场论、等离子体物理、薄膜动力学中。更准确地说，PI想要研究的是非线性几何流的规律性，比如找到最小几何条件，保证Ricci流和平均曲率流的解的光滑存在。PI想要了解的另一件事是非线性几何流的古老解及其分类。众所周知，在有限时间奇点处，古代解是作为奇点模型（爆破极限）出现的。它们的分类对于更好地理解在有限时间内可能出现的奇点是至关重要的。二维Ricci流的古老解描述了某些渐近自由局域量子场论的重整化群方程在紫外状态下的轨迹。一类特殊的古代解是利玛奇孤子。PI想要研究这些，其最终目标是对一般里奇流的一般奇点进行分类。PI提出的项目将许多不同的数学活跃领域联系起来，如非线性分析、微分几何和拓扑学。本文提出的关于非线性抛物型几何演化方程的奇异性分析和正则性的研究活动可能会在几何和拓扑学中产生有趣的应用。在物理学中也可能有潜在的应用。众所周知，里奇流理论导致了拓扑学上庞加莱猜想的一个解。希望几何流可以帮助解决其他重要的拓扑问题，如高维流形的分类。用流动理论来解决这样一个难题的主要障碍之一是理解奇点的形成和奇点的分类，因为人们不能指望流动永远存在。它很可能在有限时间内发展出奇点。PI提出理解流中奇点的形成，如Ricci流、平均曲率流、Yamabe流，因此有助于找到解决上面提到的大问题的方法。

The main objective of this project is the study of nonlinear parabolic equations which come from differential geometry problems, such as the evolution of a hypersurface of Euclidean space by functions of its principal curvatures, the Ricci flow and the Yamabe flow. More precisely, the PI focuses on singularity analysis of possible finite time singularities occurring in those evolution equations. Such equations appear in quantum field theory, plasma physics, thin liquid film dynamics. More precisely, one of the things the PI wants to study is the regularity of nonlinear geometric flows, such as finding the minimal geometric conditions that will guarantee the smooth existence of a solution to the Ricci flow and the mean curvature flow. The other thing the PI would like to understand are the ancient solutions to nonlinear geometric flows and their classification. It is well known that ancient solutions arise as singularity models (blown up limits) at finite time singularities. Their classification is crucial for better understanding the singularities that may occur in finite time. Ancient solutions to the two dimensional Ricci flow describe trajectories of the renormalization group equations of certain asymptotically free local quantum field theories in the ultra-violet regime. One special class of ancient solutions are Ricci solitons. The PI would like to study those, having an ultimate goal of classifying generic singularities of a generic Ricci flow. The project the PI proposes links many different active fields of mathematics, such as nonlinear analysis, differential geometry and topology. The proposed research activity on singularity analysis and regularity of nonlinear parabolic geometric evolution equations may result in interesting applications in geometry and topology. There may be potential application in physics as well. It is well known that the Ricci flow theory has lead to a solution of the Poincare conjecture in topology. The hope is that geometric flows may help solving other important topological question such as the classification of manifolds in higher dimensions. One of the main obstacles in order to even approach such a difficult question like that by using the flow theory is understanding the singularity formation and the classification of singularities, since one can not hope the flow will exist forever. Most likely it will develop singularities in finite time. The PI proposes to understand the formation of singularities in the flows such as the Ricci flow, mean curvature flow, the Yamabe flow and therefore contribute to finding a way to approach the big mentioned problem above.