CAREER:Singularities and singularity models in curvature flows

这个项目的主要目的是研究来自微分几何问题的非线性抛物型方程，例如欧几里德空间的超曲面随其主曲率、Ricci流和Yamabe流的函数的演化。更准确地说，PI侧重于那些演化方程中可能出现的有限时间奇点的奇性分析。这样的方程出现在量子场论、等离子体物理学、薄膜液膜动力学中。更确切地说，PI想要研究的事情之一是非线性几何流的正则性，例如找到保证Ricci流和平均曲率流的解的光滑存在的最小几何条件。PI想要了解的另一件事是非线性几何流动的古老解及其分类。众所周知，古老的解在有限时间奇点时以奇点模型(爆炸极限)的形式出现。它们的分类对于更好地理解可能在有限时间内发生的奇点至关重要。二维Ricci流的古老解描述了某些渐近自由的局域量子场论在紫外区的重整化群方程的轨迹。一类特殊的古代解是利玛窦孤子。PI想要研究这些，有一个最终目标是对一般Ricci流的一般奇点进行分类。PI提出的这个项目连接了许多不同的数学活跃领域，如非线性分析、微分几何和拓扑学。所提出的关于非线性抛物型几何发展方程的奇性分析和正则性的研究活动可能在几何和拓扑学中产生有趣的应用。这在物理学上也可能有潜在的应用。众所周知，Ricci流理论导致了Poincare猜想在拓扑学中的解。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。要想用流动理论来解决这样一个难题，主要的障碍之一是理解奇点的形成和分类，因为人们不能指望流动永远存在。最有可能的是，它将在有限的时间内形成奇点。PI建议理解流动中奇点的形成，如Ricci流、平均曲率流、Yamabe流，从而有助于找到解决上述大问题的方法。

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.