CAREER:Singularities and singularity models in curvature flows

职业：曲率流中的奇点和奇点模型

• 批准号: 1056387
• 负责人: Natasa Sesum
• 金额: $ 48万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1056387&HistoricalAwards=false
• 关键词: CAREER; Singularities; singularity; models; curvature; CAREER; Singularities; singularity; models; curvature

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.

该项目的主要目的是研究来自差异几何问题的非线性抛物线方程，例如通过其主要曲率的功能，RICCI流量和Yamabe流的功能，欧几里得空间的超表面的演变。更确切地说，PI侧重于对这些演化方程中可能发生的有限时间奇异性的奇异性分析。这种方程出现在量子场理论，血浆物理学，薄液膜动力学中。更确切地说，PI想要研究的一件事是非线性几何流量的规律性，例如找到最小的几何条件，这些条件将确保将溶液平稳地存在到RICCI流动和平均曲率流量。 PI想了解的另一件事是对非线性几何流量及其分类的古老解决方案。众所周知，在有限的时间奇点，古代解决方案是作为奇异模型（炸毁限制）出现的。他们的分类对于更好地理解有限时间内可能发生的奇异性至关重要。对二维RICCI流的古老解决方案描述了超紫罗兰色策略中某些渐近不自由局部量子场理论的重新归一化组方程的轨迹。 Ricci Solitons是一类古老的解决方案。 PI希望研究这些人，其最终目标是对通用RICCI流的通用奇异性进行分类。 PI提出的项目链接了许多不同的数学活性领域，例如非线性分析，差异几何和拓扑。提出的关于奇异性分析和非线性抛物线几何进化方程的规则性的研究活动可能会导致有趣的几何和拓扑应用。物理也可能存在潜在的应用。众所周知，RICCI流程理论已导致拓扑中的Poincare猜想解决方案。希望的是，几何流量可能有助于解决其他重要的拓扑问题，例如在更高维度中的流形分类。 通过使用流理论，甚至通过使用流程理论来解决这样一个棘手的问题的主要障碍之一是理解奇异性的形成和奇异性的分类，因为人们不能希望流动将永远存在。它很可能会在有限的时间内发展出奇异性。 PI提议了解诸如RICCI流，平均曲率流，Yamabe流量等流中奇点的形成，因此有助于找到一种解决上述大问题的方法。