On the mean curvature flow solitons in Riemannian spaces endowed with a Killing vector field

We study the uniqueness and nonexistence of mean curvature flow solitons (MCFS) with respect to a nowhere zero Killing vector field K globally defined in a Riemannian space, via suitable Liouville type results. For this, we consider the ambient space as a warped product of the type M^n×_ρℝ , where the base M^n , with n⩾ 3 , is an arbitrarily fixed integral leaf of the distribution orthogonal to K and the warping function ρ∈ C^∞ (M) is given by ρ =|K| . In particular, assuming that M^n is closed (that is, compact without boundary), we conclude that the only closed MCFS with respect to K are the totally geodesic slices. Furthermore, we establish new Moser–Bernstein type results concerning entire Killing graphs constructed through the flow of K and which are complete MCFS with respect to it.