A Hermitian metric on hyperbolic complex manifolds

TL;DR

This paper introduces a new Hermitian metric for Kobayashi hyperbolic manifolds that is distance decreasing under holomorphic maps and is expected to have better regularity than classical metrics.

Contribution

The paper presents a novel method for defining a Hermitian metric on hyperbolic complex manifolds that differs from Wu's classical approach and offers improved regularity.

Findings

The new metric is distance decreasing under holomorphic mappings.

It is distinct from Wu's classical construction.

The metric is expected to have superior regularity properties.

Abstract

We describe a method of defining a Hermitian metric on Kobayashi hyperbolic manifolds. The metric is distance decreasing under holomorphic mappings, up to a multiplicative constant. This method is distinct from the classical construction of Wu, and yields a metric which is expected to have superior regularity properties.