Lowener Theory on Analytic Universal Covering Maps

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 $r={\lim }_{t\nearrow \sup I}{f}_t^{\prime }(0)$, and $\{g_t\}$ is a classical Loewner chain of univalent functions. Under a mild regularity assumption on $t\mapsto f_t^{\prime }(0)$, 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 $\{\hat{I}{\text{c◯}}_t\}$: continuity and monotonicity of $\{f_t\}$ are equivalent to kernel continuity and monotonicity of $\{\hat{I}{\text{c◯}}_t\}$. We further show that the connectivity $C(\hat{I}{\text{c◯}}_t)=\#(\hat{\mathbb{C}}\setminus \hat{I}{\text{c◯}}_t)$ 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.

65 pages