Lagrangian mean curvature flow with boundary

TL;DR

This paper introduces a new boundary condition for Lagrangian mean curvature flow in Calabi-Yau manifolds, demonstrating preservation of the Lagrangian condition and analyzing specific flows with long-term behavior.

Contribution

It defines a natural mixed Dirichlet-Neumann boundary condition for the flow and proves preservation of the Lagrangian condition under this flow.

Findings

Flow preserves Lagrangian condition with boundary.

Long-time existence and convergence for equivariant Lagrangian discs.

Convergence of rescaled flow for the Clifford torus.

Abstract

We introduce Lagrangian mean curvature flow with boundary in Calabi--Yau manifolds by defining a natural mixed Dirichlet-Neumann boundary condition, and prove that under this flow, the Lagrangian condition is preserved. We also study in detail the flow of equivariant Lagrangian discs with boundary on the Lawlor neck and the self-shrinking Clifford torus, and demonstrate long-time existence and convergence of the flow in the first instance and of the rescaled flow in the second.