- In a first part, we show that compactness of the -Neumann operator is independent of the metric, among metrics smooth on the closure of the domain. This is analogous to subellipticity, but in contrast to global regularity. Our methods also give a new proof of the independence of subellipticity. In a second part, we look at the ideal of compactness multipliers. The zero set of this ideal may be viewed as the obstruction to compactness. We determine this zero set in the case of convex domains and in the case of certain Hartogs domains in 2.