Inventiones mathematicae

, Volume 168, Issue 3, pp 449–484

Singularities of Lagrangian mean curvature flow: zero-Maslov class case

Authors

    • Instituto Superior Técnico
    • Mathematics DepartmentStanford University
Article

DOI: 10.1007/s00222-007-0036-3

Cite this article as:
Neves, A. Invent. math. (2007) 168: 449. doi:10.1007/s00222-007-0036-3

Abstract

We study singularities of Lagrangian mean curvature flow in ℂn when the initial condition is a zero-Maslov class Lagrangian. We start by showing that, in this setting, singularities are unavoidable. More precisely, we construct Lagrangians with arbitrarily small Lagrangian angle and Lagrangians which are Hamiltonian isotopic to a plane that, nevertheless, develop finite time singularities under mean curvature flow.

We then prove two theorems regarding the tangent flow at a singularity when the initial condition is a zero-Maslov class Lagrangian. The first one (Theorem A) states that that the rescaled flow at a singularity converges weakly to a finite union of area-minimizing Lagrangian cones. The second theorem (Theorem B) states that, under the additional assumptions that the initial condition is an almost-calibrated and rational Lagrangian, connected components of the rescaled flow converges to a single area-minimizing Lagrangian cone, as opposed to a possible non-area-minimizing union of area-minimizing Lagrangian cones. The latter condition is dense for Lagrangians with finitely generated H1(L,ℤ).

Copyright information

© Springer-Verlag 2007