On short time existence of Lagrangian mean curvature flow

Kim Moore,Tom Begley

doi:10.1007/s00208-016-1420-3

Abstract

We consider a short time existence problem motivated by a conjecture of Joyce (Conjectures on Bridgeland stability for Fukaya categories of Calabi–Yau manifolds, special Lagrangians, and Lagrangian mean curvature flow. arXiv:1401.4949, 2014). Specifically we prove that given any compact Lagrangian Lsubset mathbb {C}^n with a finite number of singularities, each asymptotic to a pair of non-area-minimising, transversally intersecting Lagrangian planes, there is a smooth Lagrangian mean curvature flow existing for some positive time, that attains L as t searrow 0 as varifolds, and smoothly locally away from the singularities.

Highlights

A long-standing open problem in the study of Calabi-Yau manifolds is whether given a Lagrangian submanifold, one can find a special Lagrangian in its homology or Hamiltonian isotopy class
The second component is a monotonicity formula for the self-expander equation, which allows us to show that the approximating family of initial conditions that we construct in the proof, which are self-expanders in a ball, remain weakly close to the self-expander for a short time. The combination of these results tells us that the evolution of the approximating flows is close to the evolution of the self-expander near the singularity
A mean curvature flow is a one parameter family of immersions F : M × [0, T ) → Rn+k such that the normal velocity at any point is given by the mean curvature vector, that is dF = H. dt

Summary

Introduction

A long-standing open problem in the study of Calabi-Yau manifolds is whether given a Lagrangian submanifold, one can find a special Lagrangian in its homology or Hamiltonian isotopy class. Special Lagrangians are always area minimising, so one way to approach the existence problem is to try to minimise area among all Lagrangians in a given class. This minimisation problem turns out to be very subtle and fraught with difficulties. Schoen and Wolfson [14] showed that when the real dimension is 4, given a particular class one can find a Lagrangian minimising area among Lagrangians in that class, but that the minimiser need not be a spe-.

B Tom Begley

Mean curvature flow

Lagrangian submanifolds and Lagrangian mean curvature flow

Evolution equations and monotonicity formulas

Stability of self-expanders

Main theorem

Short-time existence

Construction of approximating family