Spacelike solitons of the r-th mean curvature flow in pp-wave spacetimes

Abstract Our aim in this paper is to investigate geometric conditions which guarantee the nonexistence of spacelike solitons of the r -th mean curvature flow (shortly, r -MCF) in a pp-wave spacetime, namely, a $$(n+1)$$ ( n + 1 ) -dimensional connected Lorentzian manifold admitting a parallel lightlike vector field, where $$n\ge 2$$ n ≥ 2 and $$r\in \{1,\ldots ,n\}$$ r ∈ { 1 , … , n } . Our approach is based on three main analytical frameworks: a Liouville type result, a maximum principle for complete noncompact Riemannian manifolds with polynomial volume growth or satisfying a certain convergence at infinity, and a suitable version of the Omori-Yau maximum principle. Furthermore, we also exhibit nice nontrivial examples of spacelike solitons of the r -MCF in a pp-wave spacetime, showing that the main hypotheses that we assume in our nonexistence results are, indeed, necessary.