Self-similar solutions to the mean curvature flow in Euclidean and Minkowski space

Abstract

In the first part of this thesis, we give a classification of all self-similar solutions to the curve shortening flow in the Euclidean plane R² and discuss basic properties of the curves. The problem of finding the curves is reduced to the study of a twodimensional system of ODEs with two parameters that determine the type of the self-similar motion. In the second part, 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. In the third part, we introduce the mean curvature flow of curves in the Minkowski plane R¹,¹ and give a classification of all the self-similar solutions. In addition, we demonstrate five non-self-similar exact solutions to the flow.