Helicoidal surfaces rotating/translating under the mean curvature flow

TL;DR

This paper classifies and constructs helicoidal surfaces in Euclidean space that exhibit self-similar motions under mean curvature flow, including a new two-parameter family of such surfaces and their limiting behaviors.

Contribution

It introduces a new two-parameter family of helicoidal surfaces rotating or translating under mean curvature flow and analyzes their limiting behaviors and classifications.

Findings

New two-parameter family of helicoidal surfaces

Limiting behavior as pitch approaches zero

Classification of immersed cylinders in constant mean curvature helicoids

Abstract

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.