Bergman kernels, TYZ expansions and Hankel operators on the Kepler manifold

TL;DR

This paper investigates Bergman kernels, TYZ expansions, and Hankel operators on the Kepler manifold, revealing their properties, asymptotic behaviors, and the non-existence or non-uniqueness of balanced metrics.

Contribution

It provides explicit descriptions of reproducing kernels, analyzes their asymptotic expansions, and studies Hankel operators on the Kepler manifold, extending understanding of complex geometric analysis.

Findings

Reproducing kernels are explicitly described for certain invariant measures.

Asymptotic TYZ expansions are established for these kernels.

Kepler manifold does not admit balanced metrics or such metrics are not unique.

Abstract

For a class of $O(n+1,R)$ invariant measures on the Kepler manifold possessing finite moments of all orders, we describe the reproducing kernels of the associated Bergman spaces, discuss the corresponding asymptotic expansions of Tian-Yau-Zelditch, and study the relevant Hankel operators with conjugate holomorphic symbols. Related reproducing kernels on the minimal ball are also discussed. Finally, we observe that the Kepler manifold either does not admit balanced metrics, or such metrics are not unique.