Symplectic geometry of Cartan-Hartogs domains

TL;DR

This paper explores the symplectic geometry of Cartan-Hartogs domains, introduces a duality concept, and provides explicit Darboux coordinates, revealing that symplectic duality occurs only in complex hyperbolic spaces.

Contribution

It constructs a duality framework for Cartan-Hartogs domains and derives explicit symplectic coordinates, advancing understanding of their geometric properties.

Findings

Explicit Darboux coordinates for Cartan-Hartogs domains and their duals

Symplectic capacity calculations for these domains

Symplectic duality only in complex hyperbolic spaces

Abstract

This paper studies the geometry of Cartan-Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan-Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan-Hartogs and their dual. Further, we compute their symplectic capacity and show that a Cartan-Hartogs admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.