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By considering intrinsic geometric conditions, we introduce a new class ofdomains in complex Euclidean space. This class is invariant underbiholomorphism and includes strongly pseudoconvex domains, finite type domainsin dimension two, convex domains, $\mathbb{C}$-convex domains, and homogeneousdomains. For this class of domains, we show that compactness of the$\bar{\partial}$-Neumann operator on $(0,q)$-forms is equivalent to theboundary not containing any $q$-dimensional analytic varieties (assuming onlythat the boundary is a topological submanifold). We also prove, for this classof domains, that the Bergman metric is equivalent to the Kobayashi metric andthat the pluricomplex Green function satisfies certain local estimates in termsof the Bergman metric.

Compactness of the $\bar{\partial}$-Neumann problem on domains with bounded intrinsic geometry