  A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving
surface with non-matched densities is derived using methods from rational
thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic)
potential short time well-posedness of strong solutions together with a
separation property is shown, under the assumption of a priori prescribed
surface evolution. The problem is reformulated with the help of a pullback to
the initial surface. Then a suitable linearization and a contraction mapping
argument for the pullback system are used. In order to deal with the linearized
system, it is necessary to show maximal $L^2$-regularity for the surface Stokes
operator in the case of variable viscosity and to obtain maximal
$L^p$-regularity for the linearized Cahn-Hilliard system.
