Solitons of the mean curvature flow in $s^2\times \mathbb{R}$

Abstract

A soliton of the mean curvature flow in the product space $s^2\times \mathbb{R}$ as a surface whose mean curvature $H$ satisfies the equation $H=⟨N,X⟩$, where $N$ is the unit normal of the surface and $X$ is a Killing vector field. In this paper we consider the vector field tangent to the fibers and the vector field associated to a rotations about an axis of $s^2$, respectively. We give a classification of the solitons with respect to these vector fields assuming that the surface is invariant under a one-parameter group of vertical translations or under a group of rotations of $s^2$.

14 pages, 5 figures