The perpendicular Neumann problem for mean curvature flow with a   timelike cone boundary condition
  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. 
Ben Lambert
math.DG