Mean curvature flow of Lagrangian submanifolds with isolated conical singularities

2011/07/24 by Behrndt, Tapio
#35K55 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1107.4803

Abstract

In this paper we study the short time existence problem for the (generalized) Lagrangian mean curvature flow in (almost) Calabi--Yau manifolds when the initial Lagrangian submanifold has isolated conical singularities modelled on stable special Lagrangian cones. Given a Lagrangian submanifold F₀:L→ M in an almost Calabi--Yau manifold M with isolated conical singularities at x₁,...,x_n∈ M modelled on stable special Lagrangian cones C₁,...,C_n in ℂ^m, we show that for a short time there exist one-parameter families of points x₁(t),... x_n(t)∈ M and a one parameter family of Lagrangian submanifolds F(t,⋅):L→ M with isolated conical singularities at x₁(t),...,x_n(t)∈ M modelled on C₁,...,C_n, which evolves by (generalized) Lagrangian mean curvature flow with initial condition F₀:L→ M.

Related

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• Singularities of the Lagrangian Mean Curvature Flow
• On short time existence of Lagrangian mean curvature flow
• Singularity formations in Lagrangian mean curvature flow
• Singularities of Lagrangian mean curvature flow: monotone case