Canonical metrics on Cartan--Hartogs domains

TL;DR

This paper investigates the geometric properties of Cartan-Hartogs domains, specifically when their natural Kähler metrics are extremal and when certain expansion coefficients are constant, contributing to complex differential geometry.

Contribution

It provides criteria for when the Kähler metric on Cartan-Hartogs domains is extremal and when the Engliš expansion coefficient is constant, advancing understanding of their geometric structure.

Findings

Characterization of extremal metrics on Cartan-Hartogs domains

Conditions for constant coefficient in Engliš expansion

Insights into the geometric structure of these domains

Abstract

In this paper we address two problems concerning a family of domains $M_{\Omega}(\mu) \subset \C^n$, called Cartan-Hartogs domains, endowed with a natural Kaehler metric $g(\mu)$. The first one is determining when the metric $g(\mu)$ is extremal (in the sense of Calabi), while the second one studies when the coefficient $a_2$ in the Engli\v{s} expansion of Rawnsley $\epsilon$-function associated to $g(\mu)$ is constant.