Balls in complex hyperbolic manifolds

TL;DR

This paper establishes an explicit lower bound for the size of Bergman balls within fundamental polyhedra of certain complex hyperbolic manifolds, leading to volume bounds for these manifolds.

Contribution

It provides a new explicit lower bound for Bergman ball radii in complex hyperbolic manifolds, which was previously unknown.

Findings

Derived an explicit lower bound for Bergman ball radii.

Established a volume lower bound for complex hyperbolic manifolds.

Connected geometric bounds to manifold volume estimates.

Abstract

In this paper we get an explicit lower bound for the radius of a Bergman ball contained in the Dirichlet fundamental polyhedron of a torsion-free discrete group $G\subset PU(n,1)$ acting on complex hyperbolic space. Consequently the volume of all complex hyperbolic n-manifolds is bounded below by the volume of this ball.