Balanced metrics on Cartan and Cartan-Hartogs domains

TL;DR

This paper characterizes all balanced metrics on Cartan domains and shows that only the complex hyperbolic space among Cartan-Hartogs domains admits such metrics, also providing a new example related to Kähler-Einstein metrics.

Contribution

It completely describes balanced metrics on Cartan domains and identifies the unique Cartan-Hartogs domain with a balanced metric, also presenting a novel example of a Kähler-Einstein metric with specific properties.

Findings

All balanced metrics on Cartan domains are described.

Only the complex hyperbolic space admits a balanced metric among Cartan-Hartogs domains.

Provides the first example of a Kähler-Einstein metric that is not balanced after scaling.

Abstract

This paper consists of two results dealing with balanced metrics (in S. Donaldson terminology) on nonconpact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan-Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in [13] (Kaehler-Einstein submanifolds of the infinite dimensional projective space, to appear in Mathematische Annalen) we also provide the first example of complete, Kaehler-Einstein and projectively induced metric g such that $\alpha g$ is not balanced for all $\alpha >0$.