  Single partially confined collapsed polymers are studied in two dimensions.
They are described by self-avoiding random walks with nearest-neighbour
attractions below the $\Theta$-point, on the surface of an infinitely long
cylinder. For the simulations we employ the pruned-enriched-Rosenbluth method
(PERM). The same model had previously been studied for free polymers (infinite
lattice, no boundaries) and for polymers on finite lattices with periodic
boundary conditions. We verify the previous estimates of bulk densities, bulk
free energies, and surface tensions. We find that the free energy of a polymer
with fixed length $N$ has, for $N\to \infty$, a minimum at a finite cylinder
radius $R^*$ which diverges as $T\to T_\theta$. Furthermore, the surface
tension vanishes roughly as $(T_\theta-T)^\alpha$ for $T\to T_\theta$ with
$\alpha\approx 1.7$. The density in the interior of a globule scales as
$(T_\theta-T)^\beta$ with $\beta \approx 0.32$.
