Mean curvature flow of Lagrangian submanifolds with isolated conical singularities

TL;DR

This paper proves short-time existence of the generalized Lagrangian mean curvature flow starting from Lagrangian submanifolds with isolated conical singularities modeled on stable special Lagrangian cones in (almost) Calabi--Yau manifolds.

Contribution

It establishes the short-time existence of the flow for Lagrangian submanifolds with conical singularities, extending previous results to singular initial conditions.

Findings

Existence of short-time solutions to the flow.

Flow preserves the conical singularity structure.

Deformation of singularities modeled on stable cones.

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 $F_0:L\rightarrow M$ in an almost Calabi--Yau manifold $M$ with isolated conical singularities at $x_1,...,x_n\in M$ modelled on stable special Lagrangian cones $C_1,...,C_n$ in $\mathbb{C}^m$, we show that for a short time there exist one-parameter families of points $x_1(t),... x_n(t)\in M$ and a one parameter family of Lagrangian submanifolds $F(t,\cdot):L\rightarrow M$ with isolated conical singularities at $x_1(t),...,x_n(t)\in M$ modelled on $C_1,...,C_n$, which evolves by (generalized) Lagrangian mean curvature flow with initial condition $F_0:L\rightarrow M$.