The $\overline{\partial}$-Neumann operator and the Kobayashi metric

Abstract

We introduce a condition, called Property -K, which encodes information about the holomorphic structure of fat subdomains. We obtain an equivalence between this condition and the compactness of the $\overline{\partial}$-Neumann operator in any convex domain. We also exhibit a local property of the Kobayashi metric under which the domain is locally a product space.