Compactness of the 0-Neumann problem on domains with bounded intrinsic geometry

By considering intrinsic geometric conditions, we introduce a new class of domains in complex Euclidean space. This class is invariant under biholomorphism and includes strongly pseudoconvex domains, finite type domains in dimension two, convex domains, C -convex domains, and homogeneous domains. For this class of domains, we show that compactness of the a-Neumann operator on (0, q) -forms is equivalent to the boundary not containing any q-dimensional analytic varieties (assuming only that the boundary is a topological submanifold). We also prove, for this class of domains, that the Bergman metric is equivalent to the Kobayashi metric and that the pluricomplex Green function satisfies certain local estimates in terms of the Bergman metric. 0 2021 Elsevier Inc. All rights reserved.

Automated summary

By considering intrinsic geometric conditions, a new class of domains in complex Euclidean space has been introduced, which is invariant under biholomorphism and includes several types of domains. It has been shown that for this class of domains, compactness of the a-Neumann operator on certain forms is equivalent to the boundary not containing any q-dimensional analytic varieties. Additionally, it has been proven that the Bergman metric is equivalent to the Kobayashi metric for this class of domains.