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

• Applicant: Professor Dr. Knut Smoczyk
• Subject Area: Mathematics
• Term: from 2003 to 2008
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 5406743

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.

• DFG Programme: Priority Programmes
• Subproject of: SPP 1154: Global Differential Geometry