Invariant translators of the solvable group

Abstract

We classify the translators to the mean curvature flow in the three-dimensional solvable group $$Sol_3$$ that are invariant under the action of a one-parameter group of isometries of the ambient space. In particular, we show that $$Sol_3$$ admits graphical translators defined on a half plane, in contrast with a rigidity result of Shahriyari (Geom Dedicata 175:57–64, 2015) for translators in the Euclidean space. Moreover, we exhibit some nonexistence results.