Solitons of the spacelike mean curvature flow in standard static spacetimes

We study complete noncompact spacelike mean curvature flow solitons (SMCFS) in a standard static spacetime obeying a suitable constraint on the sectional curvature. In this context, we prove a version of the Omori–Yau generalized maximum principle and apply it to deduce that such an SMCFS must be maximal in the sense that its mean curvature vanishes identically. Next, we use other maximum principles which deal with the notions of convergence to zero at infinity and polynomial volume growth to prove rigidity results for SMCFS. Furthermore, we apply our previous results to establish nonexistence results concerning entire Killing graphs constructed over the Riemannian base of a standard static spacetime. Finally, we also exhibit an example showing the relevance of key hypotheses in our results.