Mean curvature flow of Lagrangian submanifolds with isolated conical singularities

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 $F_0:L\to M$ 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 $F(t,\cdot ):L\to M$ 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 $F_0:L\to M$.

40 pages, 1 figure