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

拟全纯曲线拉格朗日平均曲率流的奇异性分析

• 批准号: 5406743
• 负责人: Professor Dr. Knut Smoczyk
• 金额: --
• 依托单位: Institut für Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2003
• 资助国家: 德国
• 起止时间: 2002-12-31 至 2007-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/5406743?language=en
• 关键词: Analysis; singularities; Lagrangian; mean; curvature

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.

本项目的目的是详细研究拉格朗日平均曲率流的性质，拉格朗日平均曲率流是拉格朗日子流形的黎曼浸入Kähler-Einstein流形的拟线性抛物方程组。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流的自相似解的收敛问题。这种流动的一个主要动机是体积最小化拉格朗日子流形的关键存在问题，特别是对于镜像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型系统的分析与辛拓扑学的技巧相结合，即伪全纯曲线法。对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巧的研究提供了关于平均曲率流的爆破行为的重要信息。