  The paper studies a scalar auxiliary variable (SAV) method to solve the
Cahn-Hilliard equation with degenerate mobility posed on a smooth closed
surface {\Gamma}. The SAV formulation is combined with adaptive time stepping
and a geometrically unfitted trace finite element method (TraceFEM), which
embeds {\Gamma} in R3. The stability is proven to hold in an appropriate sense
for both first- and second-order in time variants of the method. The
performance of our SAV method is illustrated through a series of numerical
experiments, which include systematic comparison with a stabilized
semi-explicit method.
