The perpendicular Neumann problem for mean curvature flow with a timelike cone boundary condition

TL;DR

This paper proves long-term existence and describes the asymptotic behavior of spacelike mean curvature flow in Minkowski space with a convex cone boundary, showing convergence to a hyperbolic solution.

Contribution

It establishes the existence for all time of mean curvature flow with a cone boundary in Minkowski space and characterizes its asymptotic convergence.

Findings

Flow exists for all time under given conditions.

Flow converges to a hyperbolic solution after renormalization.

The boundary condition is a convex cone with perpendicular Neumann boundary.

Abstract

This paper demonstrates existence for all time of mean curvature flow in Minkowski space with a perpendicular Neumann boundary condition, where the boundary manifold is a convex cone and the flowing manifold is initially spacelike. Using a blowdown argument, we show that under renormalisation this flow converges towards a homothetically expanding hyperbolic solution.