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 \(M^n\hspace{0.55542pt}{\times }_\rho \hspace{1.111pt}\mathbb {R}\) $M^n{\times }_{\rho }\mathbb{R}$ , where the base \(M^n\) $M^n$ , with \(n\geqslant 3\) $n⩾3$ , is an arbitrarily fixed integral leaf of the distribution orthogonal to K and the warping function \(\rho \in C^\infty (M)\) $\rho \in C^{\infty }(M)$ is given by \(\rho =|K|\) $\rho =|K|$ . In particular, assuming that \(M^n\) $M^n$ 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.