Self-similar solutions to the mean curvature flow in ${\mathbb{R}}^3$

Abstract

In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in ${\mathbb{R}}^3$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in ${\mathbb{R}}^3$ are trivial. Moreover, we characterize the self-similar 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 self-similar 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.

References in corpus (3)

• Self-similar solutions to the curve shortening flow
• Helicoidal surfaces rotating/translating under the mean curvature flow
• A family of MCF solutions for the Heisenberg Group