Ruled surfaces as self-similar solutions to the harmonic mean curvature flow in Minkowski 3-space

In this paper, we fully classify the ruled surfaces in Minkowski 3-space E13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}<^>3_1$$\end{document} and Euclidean 3-space E3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {E}}<^>3$$\end{document} which are self-similar solutions to the harmonic mean curvature flow (HMCF) under the translation and homothetic motions. In contrast to the Euclidean case, we find a special class of ruled surfaces in Minkowski 3-space which are homothetic self-similar solution to the HMCF.