Selfsimilar solutions to the mean curvature flow in $\mathbb{R}^{3}$
Abstract
In this paper we make an analysis of selfsimilar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\mathbb{R}^{3}$. We prove that selfsimilar solutions of the MCF by noncylindrival surfaces and conical surfaces in $\mathbb{R}^{3}$ are trivial. Moreover, we characterize the selfsimilar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in $\mathbb{R}^{3}$ in terms of the curvature of the generating curve. Finally, we characterize the selfsimilar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in $\mathbb{R}^3$. Explicit families of exact solutions for the MCF by cylindrical surfaces in $\mathbb{R}^{3}$ are also given.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 arXiv:
 arXiv:2005.10688
 Bibcode:
 2020arXiv200510688L
 Keywords:

 Mathematics  Differential Geometry