The polydisk theorem for Hartogs domains over symmetric domains

TL;DR

This paper generalizes the polydisk theorem to Hartogs domains over arbitrary bounded symmetric domains and explores their geometric rigidity, showing dual domains cannot embed into compact Riemannian manifolds.

Contribution

It extends the polydisk theorem to more general Hartogs domains and establishes a duality result, broadening understanding of their geometric properties.

Findings

Generalization of the polydisk theorem to arbitrary bounded symmetric domains

Establishment of a dual counterpart of the theorem

Demonstration that dual Hartogs domains cannot be totally geodesically immersed into compact manifolds

Abstract

We extend the polydisk theorem of [21], originally established for classical Cartan-Hartogs domains, to Hartogs domains over arbitrary (possibly reducible and exceptional) bounded symmetric domains. We further establish a dual counterpart of this result. As an application, we show that the dual of a Hartogs domain over a bounded symmetric domain admits no totally geodesic immersion into any compact Riemannian manifold, thereby broadening the rigidity phenomena obtained in [13].