# Lowener Theory on Analytic Universal Covering Maps

## TL;DR

This paper extends Loewner theory to chains of universal covering maps, providing a decomposition, PDE generalization, and geometric characterizations, linking classical theory with hyperbolic geometry and Fuchsian groups.

## Contribution

It introduces a new Loewner framework for universal covering maps, including a factorization, PDE, and evolution equations for deck transformations, unifying classical and geometric aspects.

## Key findings

- Decomposition of Loewner chains into analytic and univalent parts.
- Generalization of Loewner--Kufarev PDE for non-univalent chains.
- Characterization of universal covering map chains via domain monotonicity.

## Abstract

We study Loewner chains in $\mathcal{H}_0(\mathbb{D})$ without assuming univalence of each element. We prove a decomposition: every chain admits a factorization $f_t=F\circ g_t$, where $F$ is analytic on $\mathbb{D}(0,r)$ with $r=\lim_{t \nearrow \sup I} f_t'(0)$, and $\{g_t\}$ is a classical Loewner chain of univalent functions. Under a mild regularity assumption on $t \mapsto f_t'(0)$, we derive a partial differential equation that generalizes the Loewner--Kufarev equation. We then develop a Loewner theory for chains of universal covering maps. We characterize such chains in terms of domain families $\{\Omega_t\}$: continuity and monotonicity of $\{f_t\}$ are equivalent to kernel continuity and monotonicity of $\{\Omega_t\}$. We further show that the connectivity $C(\Omega_t)=\#(\hat{\mathbb{C}}\setminus \Omega_t)$ is a left-continuous nondecreasing function of $t$. Building on these results, we formulate a Loewner theory on Fuchsian groups and obtain evolution equations for deck transformations. As an application, we study hyperbolic metrics and establish a formula for the logarithmic derivative of the hyperbolic density along the chain. Our results provide a unified framework linking classical Loewner theory, covering maps, and the geometry of hyperbolic domains.