  We prove two results which are relevant for constructing marginally outer
trapped tubes (MOTTs) in de Sitter spacetime. The first one holds more
generally, namely for spacetimes satisfying the null convergence condition and
containing a timelike conformal Killing vector with a "temporal function". We
show that all marginally outer trapped surfaces (MOTSs) in such a spacetime are
unstable. This prevents application of standard results on the propagation of
stable MOTSs to MOTTs. On the other hand, it was shown recently that, for every
sufficiently high genus, there exists a smooth, complete family of CMC surfaces
embedded in the round 3-sphere S3. This family connects a Lawson minimal
surface with a doubly covered geodesic 2-sphere. We show by a simple scaling
argument that this result translates to an existence proof for complete MOTTs
with CMC sections in de Sitter spacetime. Moreover, the area of these sections
increases strictly monotonically. We compare this result with an area law
obtained before for holographic screens.
