Lowener Theory on Analytic Universal Covering Maps

TL;DR

The paper develops a unified Loewner theory extending classical univalent function theory to analytic universal coverings and covering maps, by proving a decomposition $f_t=F\circ g_t$ that reduces the general (possibly nonunivalent) chain to a univalent core plus an analytic precomposition. It derives generalized Loewner–Kufarev equations under a mild regularity on $t\mapsto f_t'(0)$ and establishes a kernel-convergence framework linking domain evolution $\{\Omega_t=f_t(\mathbb{D})\}$ to chain continuity and monotonicity, including left-continuity of connectivity and kernel-based continuity criteria. The work further extends the theory to universal coverings, yields a classification and extension principle for chains on Fuchsian groups, and provides applications to hyperbolic metrics and deck-transform evolution. Overall, it offers a comprehensive, structurally unified approach connecting classical Loewner theory, covering maps, kernel convergence, and hyperbolic geometry. These results pave the way for a Loewner calculus on more general hyperbolic domains and group actions, with potential implications for geometric function theory and related areas.

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 $\{Ω_t\}$: continuity and monotonicity of $\{f_t\}$ are equivalent to kernel continuity and monotonicity of $\{Ω_t\}$. We further show that the connectivity $C(Ω_t)=\#(\hat{\mathbb{C}}\setminus Ω_t)$ 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.

Figures (1)

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Theorems & Definitions (167)

• Definition 1.1
• Theorem 1.2: Decomposition Theorem
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Example 1.6
• Theorem 1.7
• Corollary 1.8
• Lemma 1.9
• Theorem 1.10