Short-Time Existence of Lagrangian MCF Past Conical Singularities

Abstract In his paper ‘Conjectures on Bridgeland Stability’ [15], Joyce asked if one can desingularise the transverse intersection point of an immersed Lagrangian using JLT expanders such that one gets a Lagrangian mean curvature flow via the desingularisations. Begley and Moore [4] answered this in the affirmative by constructing a family of desingularisations and showing that a certain limit along their flows satisfies LMCF along with convergence to the immersed Lagrangian in the sense of varifolds. We prove that there exists a solution with convergence in a stronger sense, using the notion of manifolds with corners and a-corners as introduced by Joyce [16]. Our methods are a direct P.D.E. based approach, along the lines of the proof of short-time existence for network flow by Lira, Mazzeo, Pluda and Saez [21].