Helicoidal surfaces rotating/translating under the mean curvature flow

Details Helicoidal surfaces rotating/translating under the mean curvature flow Helicoidal surfaces rotating/translating under the mean curvature flow

2011-06-22

arxiv.org/abs/1106.4583v2

Mathematics

Differential Geometry

20 pages, 11 figures

Scientific paper

We describe all possible self-similar motions of immersed hypersurfaces in Euclidean space under the mean curvature flow and derive the corresponding surface equations. Then we present a new two-dimensional 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.

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