  We consider the Cahn-Hilliard equation with constant mobility and logarithmic
potential on a two-dimensional evolving closed surface embedded in $\mathbb
R^3$, as well as a related weighted model. The well-posedness of weak solutions
for the corresponding initial value problems on a given time interval $[0,T]$
have already been established by the first two authors. Here we first prove
some regularisation properties of weak solutions in finite time. Then, we show
the validity of the strict separation property for both the problems. This
means that the solutions stay uniformly away from the pure phases $\pm1$ from
any positive time on. This property plays an essential role to achieve
higher-order regularity for the solutions. Also, it is a rigorous validation of
the standard double-well approximation. The present results are a twofold
extension of the well-known ones for the classical equation in planar domains.
