Lowener Theory on Analytic Universal Covering Maps
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arXiv
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| Format: | Preprint |
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2019
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| author | Yanagihara, Hiroshi |
| author_facet | Yanagihara, Hiroshi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1907_11987 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Lowener Theory on Analytic Universal Covering Maps Yanagihara, Hiroshi Complex Variables 30C35, 30F35 (Primary) 30C45(Secondary) 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. |
| title | Lowener Theory on Analytic Universal Covering Maps |
| topic | Complex Variables 30C35, 30F35 (Primary) 30C45(Secondary) |
| url | https://arxiv.org/abs/1907.11987 |