Homothetic $$\alpha $$ -Self-Similar Solutions to the Mean Curvature Flow in Minkowski 3-Space

In this paper, we classify the non-degenerate ruled surfaces which are homothetic \(\alpha \)-self-similar solutions to the mean curvature flow (MCF) in Minkowski 3-space \(\mathbb {E}^3_1\). Other than cylindrical surfaces as in \(\mathbb {E}^3\), we find a class of non-cylindrical ruled surface with null rulings, in particular, the null scrolls. Further, we investigate the non-degenerate surfaces of revolutions generated by non-null curve as homothetic \(\alpha \)-self-similar solutions to the MCF according to the causality of their rotation axes as spacelike, timelike and lightlike.