Mean Curvature Flow with a Neumann Boundary Condition in Flat Spaces

Abstract

In this thesis I study mean curvature flow in both Euclidean and Minkowski space with a Neumann boundary condition. In Minkowski space I show that for a convex timelike cone boundary condition, with compatible spacelike initial data, mean curvature flow with a perpendicular Neumann boundary condition exists for all time. Furthermore, by a blowdown argument I show convergence as t →∞ to a homothetically expanding hyperbolic hyperplane. I also study the case of graphs over convex domains in Minkowski space. I obtain long time existence for spacelike initial graphs which are taken by mean curvature flow with a Neumann boundary condition to a constant function as t →∞. In Euclidean space I consider boundary manifolds that are rotational tori where I write t for the unit vector field in the direction of the rotation. If the initial manifold M₀ is compatible with the boundary condition, and at no point has t as a tangent vector, then mean curvature flow with a perpendicular Neumann boundary condition exists for all time and converges to a flat cross-section of the boundary torus. I also discuss other constant angle boundary conditions.