Infinitesimal generators and the Loewner equation on complete hyperbolic manifolds

TL;DR

This paper characterizes infinitesimal generators on complete hyperbolic complex manifolds without regularity assumptions, establishing a general Loewner type equation and exploring related open problems.

Contribution

It provides a new characterization of infinitesimal generators on hyperbolic manifolds and develops a general Loewner equation applicable to various regularity levels.

Findings

Characterization of infinitesimal generators without Kobayashi distance regularity

Development of a general Loewner type equation for different regularity orders

Identification of open problems in the theory of hyperbolic complex manifolds

Abstract

We characterize infinitesimal generators on complete hyperbolic complex manifolds without any regularity assumption on the Kobayashi distance. This allows to prove a general Loewner type equation with regularity of any order $d\in [1,+\infty ]$. Finally, based on these results, we focus on some open problems naturally arising.