2006/08/15 by André Neves, Andre' Neves · 51 citations
Mathematics · #Curvature #Flow (mathematics) #Geometric Analysis and Curvature Flows #Gravitational singularity #Hamiltonian (control theory) #Lagrangian #Mean curvature flow #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Singularity #Tangent #math.DG #math.SG #msc:53C44
paper · pdf · doi:10.1007/s00222-007-0036-3
published in Inventiones mathematicae 168(3), 449-484 (Springer Science+Business Media) · 34 pages. 3 figures. To appear in Inventiones
arxiv created 2006/08/15 · openalex publication_date 2007/03/13 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/06
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).