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

TL;DR

This paper analyzes self-similar solutions to the mean curvature flow in three-dimensional space, characterizing solutions for surfaces of revolution and ruled surfaces, and providing explicit solutions for cylindrical cases.

Contribution

It characterizes self-similar solutions for various surface types under mean curvature flow, including explicit solutions for cylindrical surfaces in .

Findings

Self-similar solutions of non-cylindrical and conical surfaces are trivial.

Characterization of solutions for surfaces of revolution under homothetic helicoidal motion.

Explicit families of solutions for cylindrical surfaces under mean curvature flow.

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.