Weakly holomorphic homogeneous regular manifolds
Andrew Zimmer
TL;DR
The paper introduces weakly holomorphic homogeneous regular (wHHR) manifolds, extending the HHR framework to include a broader class of complex manifolds and domains. It develops pointwise and global metric control via the weak squeezing function, pluricomplex Green function, and Bergman kernel, proving that on wHHR Stein manifolds the Bergman metric is complete with bounded geometry and that the Bergman and Kobayashi metrics are uniformly biLipschitz. A key result is a local-to-global comparison: under suitable embeddings and Levi-form bounds, the Kobayashi metric is comparable to the Bergman metric, yielding Cauchy completeness of the Kobayashi distance on wHHR manifolds. The paper also shows a necessary condition for the compactness of the $\bar{\partial}$-Neumann operator on bounded wHHR domains with $C^0$ boundary, namely that boundary analytic varieties obstruct compactness. Throughout, it provides examples, rigidity results, and several open questions about Steinness, optimality of the wsq bound, and domain classifications within the wHHR framework.
Abstract
We introduce a class of complex manifolds which we call weakly holomorphic homogeneous regular manifolds (wHHR) manifolds. As the name suggests, this class contains the so-called holomorphic homogeneous regular manifolds but also other classes of complex manifolds such as two dimensional finite type domains and simply connected Kähler manifolds with pinched negative sectional curvature. For wHHR Stein manifolds we prove that the Bergman and Kobayashi metrics are biLipschitz equivalent.