Analysis of singularities of the Lagrangian mean curvature flow with pseudo-holomorphic curves

这个项目的目的是详细研究拉格朗日平均曲率流的性质，这是一个关于拉格朗日子流形到凯勒-爱因斯坦流形的黎曼浸入的抛物方程的拟线性系统。在这种热流中特别感兴趣的是对奇点形成的精确理解。一个主要的问题是收敛到自相似的解决方案的流动单调拉格朗日环面。一个主要的动机，这种流动是至关重要的存在问题的体积最小化拉格朗日子流形，特别是特殊的拉格朗日环面研究镜像对称猜想。该项目的新方面之一是结合了抛物系统的分析与辛拓扑技术，即伪全纯曲线的方法。拉格朗日子流形上具有边界的伪全纯圆盘的存在性结果和填充技术的研究提供了关于平均曲率流爆破行为的重要信息。

It is the purpose of this project to study in details the properties of the Lagrangian mean curvature flow, which is a quasilinear system of parabolic equations for Riemannian immersions of Lagrangian submanifolds into a Kähler-Einstein manifold. Of particular interest in this heat flow is the precise understanding of the formation of singularities. A main question is the convergence towards self-similar solutions for flows of monotone Lagrangian tori. A prime motivation for this flow is the crucial existence problem of volume minimizing Lagrangian submanifolds, in particular special Lagrangian tori as studied for the Mirror Symmetry Conjecture. One of the new aspects of this project is the combination of the analysis of the parabolic system with techniques from symplectic topology namely the method of pseudo-holomorphic curves. The study of existence results and filling techniques for pseudo-holomorphic disks with boundary on the Lagrangian submanifolds provides crucial information about the blow-up behaviour of the mean curvature flow.