Research on Lagrangian Mean Curvature Flow and Yamabe Invariants

拉格朗日平均曲率流与Yamabe不变量的研究

• 批准号: 0604164
• 负责人: Andre Neves
• 金额: $ 11.14万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604164&HistoricalAwards=false
• 关键词: Research; Lagrangian; Mean; Curvature; Flow

The aim of this project is to study geometric flows and the relationship between curvature and topology. Regarding geometric flows, the principal investigator plans to study higher codimension mean curvature flow, which is the gradient flow for the area functional. More precisely, we plan to study mean curvature flow deformation of Lagrangian submanifolds. In his thesis, the principal investigator showed that finite time singularities are unavoidable, i.e., they occur in many cases where experts were hoping they would not occur, and then he proved the optimal result about the infinitesimal behavior of singularities. We plan to investigate other settings on which we can understand singularities and also to understand the size of the singular set at the time of the first singularity. This last question is very challenging and a satisfactory answer would be a breakthrough in the field. Regarding the relationship between curvature and topology we plane to investigate which constant scalar curvature metrics does a 3-manifold admit. More specifically, we intent to extend the investigator's previous work and hope to unveil a large class of manifolds $L$ for which $M$ and $M\#L$ have the same ``type'' of constant positive scalar curvature metrics. The underlying philosophy of many problems in geometric analysis is to given a geometric object being able to find another geometric object carrying the same type of information but having better properties. For instance, if one is studying the paths that go from A to B, the best possible path would be the one with shortest lenght. Both problems addressed in this research project obey this guiding principle. In the first one we try to use a heat -equation flow method to deform certain kinds of Lagrangian submanifolds into those that are still Lagrangian but have the least area possible. This is expected to have very nice applications in mathematical physics (more precisely in the SYZ conjecture). In the second problem we try to understand which geometric information does a constant scalar curvature metric carry. It is known that for surfaces these metrics determine its topological type. For 3-manifolds it is know that constant Ricci curvature determines the manifold. It is an important open problem to understand, for 3-dimensional manifolds, the information that can be extracted from constant scalar curvature metrics.

这个项目的目的是研究几何流动以及曲率和拓扑之间的关系。在几何流方面，本课题拟研究高协维平均曲率流，即面积泛函的梯度流。更确切地说，我们计划研究拉格朗日子流形的平均曲率流变形。在他的论文中，首席研究员证明了有限时间奇点是不可避免的，即在许多专家希望奇点不发生的情况下，它们会发生，然后他证明了奇点的无穷小行为的最优结果。我们计划研究其他可以理解奇点的设置，并了解第一个奇点时奇异集的大小。最后一个问题是非常具有挑战性的，一个令人满意的答案将是该领域的一个突破。针对曲率与拓扑的关系，探讨了三流形中哪些常数标量曲率度量是允许的。更具体地说，我们打算扩展研究者之前的工作，并希望揭示一类大型流形$L$，其中$M$和$M\#L$具有相同的常数正标量曲率度量的“类型”。几何分析中许多问题的基本原理是，给定一个几何对象，能够找到另一个几何对象，该几何对象携带相同类型的信息，但具有更好的性质。例如，如果要研究从A到B的路径，最佳路径就是长度最短的路径。本课题研究的两个问题都遵循这一指导原则。在第一个例子中，我们尝试用热方程流动法将某些类型的拉格朗日子流形变形为那些仍然是拉格朗日的但面积尽可能小的流形。这有望在数学物理（更准确地说，在SYZ猜想中）中有很好的应用。在第二个问题中，我们试着理解常数标量曲率度量所携带的几何信息。众所周知，对于曲面，这些指标决定了它的拓扑类型。对于3流形，已知恒定的里奇曲率决定了流形。对于三维流形，如何从常数标量曲率度量中提取信息是一个重要的开放性问题。