4 research outputs found
Self-similar solutions to the mean curvature flow in the Minkowski plane
We introduce the mean curvature flow of curves in the Minkowski plane
and give a classification of all the self-similar solutions.
In addition, we describe five other exact solutions to the flow.Comment: 31 pages, 38 figures. Two exact solutions added from previous versio
Helicoidal surfaces rotating/translating under the mean curvature flow
We describe all possible self-similar motions of immersed hypersurfaces in
Euclidean space under the mean curvature flow and derive the corresponding
hypersurface equations. Then we present a new two-parameter family of immersed
helicoidal surfaces that rotate/translate with constant velocity under the
flow. We look at their limiting behaviour as the pitch of the helicoidal motion
goes to 0 and compare it with the limiting behaviour of the classical
helicoidal minimal surfaces. Finally, we give a classification of the immersed
cylinders in the family of constant mean curvature helicoidal surfaces.Comment: 21 pages, 22 figures, final versio