Research on Lagrangian Mean Curvature Flow and Yamabe Invariants

这个项目的目的是研究几何流动和曲率与拓扑之间的关系。关于几何流，主要研究者计划研究更高的余维平均曲率流，这是面积泛函的梯度流。 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，首席研究员表明，有限时间奇点是不可避免的，即， 它们发生在专家们希望它们不会发生的许多情况下，然后他证明了关于奇点无穷小行为的最佳结果。我们计划研究其他的设置，我们可以理解的奇异性，并了解在第一个奇点的时候奇异集的大小。最后一个问题非常具有挑战性，一个令人满意的答案将是该领域的一个突破。关于曲率与拓扑的关系，我们研究了一个三维流形允许哪些常数量曲率度量。更具体地说，我们打算延长调查员以前的工作，并希望推出一大类的流形$L $的$M $和$M\#L $有相同的"类型"的常数正标量曲率度量。几何分析中许多问题的基本原理是，给定一个几何对象，能够找到另一个携带相同类型信息但具有更好属性的几何对象。例如，如果一个人正在研究从A到B的路径，最好的可能路径将是具有最短长度的路径。本研究项目所涉及的两个问题都遵循这一指导原则。在第一种方法中，我们尝试使用热方程流方法将某些类型的拉格朗日子流形变形为仍然是拉格朗日子流形但具有最小面积的子流形。预计这在数学物理学中有很好的应用（更确切地说是在SYZ猜想中）。在第二个问题中，我们试图了解几何信息的常数标量曲率度量进行。已知对于曲面，这些度量决定其拓扑类型。对于3-流形，已知常Ricci曲率决定流形。这是一个重要的开放性问题，了解，为3维流形，可以提取的信息，从常数标量曲率度量。

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.