Balanced metrics on Hartogs domains

TL;DR

This paper proves that for a certain class of complex domains with a natural Kähler metric, the balanced condition implies the domain is essentially a part of complex hyperbolic space, revealing a rigidity property.

Contribution

It establishes a rigidity result linking balanced metrics on Hartogs domains to their isometry with complex hyperbolic space, extending understanding of geometric structures on these domains.

Findings

Balanced metric condition implies domain is hyperbolic space

m_0 must be greater than the complex dimension n

Hartogs domain with balanced metric is isometric to hyperbolic space

Abstract

An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped with a natural Kaehler metric g_F. In this paper we prove that if m_0g_F is balanced for a given positive integer m_0 then m_0>n and (D_F, g_F) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space.