$K^{\alpha }$-translators of offset 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 $w$.

13 pages, 4 figures