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 \({\mathbb {E}}^3_1\) ${\mathbb{E}}_1^3$ and Euclidean 3-space \({\mathbb {E}}^3\) ${\mathbb{E}}^3$ 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.