  We prove an abstract structure theorem for weighted manifolds supporting a
weighted $f$-Poincar\'e inequality and whose ends satisfy a suitable
non-integrability condition. We then study how our arguments can be used to
obtain full topological control on two important classes of hypersurfaces of
the Euclidean space, namely translators and self-expanders for the mean
curvature flow, under either stability or curvature asumptions. As an important
intermediate step in order to get our results we get the validity of a
Poincar\'e inequality with respect to the natural weighted measure on any
translator and we prove that any end of a translator must have infinite
weighted volume. Similar tools can be obtained for properly immersed
self-expanders permitting to get topological rigidity under curvature
assumptions.
