$K^\alpha$-translators of offset surfaces

TL;DR

This paper investigates $K^{ ext{alpha}}$-translators on parallel and canal surfaces in 3D space, showing conditions for their existence, characterizing canal surface translators as surfaces of revolution, and providing explicit examples and non-existence results.

Contribution

It characterizes $K^{ ext{alpha}}$-translators on canal surfaces, proves they must be surfaces of revolution, and establishes non-existence of such translators on parallel surfaces of rotational surfaces.

Findings

Canal surface $K^{ ext{alpha}}$-translators are surfaces of revolution.

Explicit examples of $K$-flow and inverse $K$-flow surfaces are provided.

No $K^{ ext{alpha}}$-translators exist on parallel surfaces of certain rotational surfaces.

Abstract

In this paper, we study $K^{\alpha}$--translators on parallel surfaces and canal surfaces in 3-dimensional Euclidean space $\mathbb{E}^3$. First, we investigate the condition under which two parallel surfaces can become $K^{\alpha}$--translators moving with the same speed $w$. Then, we examine $K^{\alpha}$--translators on canal surfaces and we show that if a canal surface is $K^{\alpha}$--translator, then it must be a surface of revolution in $\mathbb{E}^3$. We also provide examples for moving a surface of revolution under $K$--flow (Gauss curvature flow) and $K^{-1/2}$--flow (inverse Gauss curvature flow) along a direction $w=(0,0,1)$ and we illustrate such surfaces using Wolfram Mathematica 10.4. Finally, we prove that no $K^{\alpha}$--translators exist on the parallel surface of a rotational surface obtained from a canal surface with the same speed $w$, while the such rotational surfaces itself is a $K^{\alpha}$--translator with speed $w$.