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

拉格朗日平均曲率流是拉格朗日子流形在Kähler-Einstein流形中黎曼浸入的拟线性抛物方程系统，本课题的目的是详细研究拉格朗日平均曲率流的性质。对这种热流特别感兴趣的是对奇点形成的精确理解。一个主要问题是单调拉格朗日环面流的自相似解的收敛性。这种流动的一个主要动机是拉格朗日子流形的体积最小化的关键存在性问题，特别是对镜面对称猜想所研究的特殊拉格朗日环面。该项目的一个新方面是将抛物系统的分析与辛拓扑的技术即伪全纯曲线的方法相结合。拉格朗日子流形上带边界的伪全纯盘的存在性结果和填充技术的研究，为研究平均曲率流的爆破行为提供了重要的信息。

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.