A boundary version of Ahlfors` lemma, locally complete by Kraus D. PDF

By Kraus D.

A boundary model of Ahlfors' Lemma is validated and used to teach that the classical Schwarz-Carathéodory mirrored image precept for holomorphic services has a in simple terms conformal geometric formula when it comes to Riemannian metrics. This conformally invariant mirrored image precept generalizes obviously to analytic maps among Riemann surfaces and comprises between different effects a characterization of finite Blaschke items because of M. Heins.

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Additional resources for A boundary version of Ahlfors` lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps

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Let Ω ⊆ be a domain and let Γ be a smooth subset of ∂Ω. Further, let λ(z) |dz| be a regular conformal metric on Ω with κλ ≥ −cλ , and let µ(z) |dz| be a regular conformal pseudo-metric on Ω with κµ ≤ −Cµ for some positive constants cλ and Cµ . If λ(z) |dz| is locally complete near Γ, then Cµ λ(z) lim inf ≥ z→ξ µ(z) cλ for every ξ ∈ Γ. 5. 4 is just a very special case of Bland’s boundary Schwarz Lemma (which in its original form applies to higher dimensional situations). 1, λ(z) |dz| is a regular conformal pseudo-metric and the regularity of µ(z) |dz| is of no importance.

1, the function fˆ has an analytic extension across I . Thus f = ϕˆα ◦ fˆ has an analytic continuation across Γ and consequently to a £ whole neighborhood of ξ0 . 5. 6. Let S and R be simply connected Riemann surfaces with analytic boundaries ∂S and ∂R, respectively; let Γ be an open and connected subset of ∂S ; and let R carry a complete regular conformal metric λ(w) |dw| with curvature bounded below and above by negative constants −cλ and −Cλ , respectively. Further, let f : S → R be an analytic map.

3) z→ξ λ(f (z)) |f ′ (z)| ≥ µ(z) Cµ cλ for every ξ in Γ. 3) is the quotient of two conformal pseudometrics on S . Since µ(z) |dz| → +∞ as z → Γ, this quotient is therefore a well-defined function on the surface S at least near Γ. Proof. 3 and let I = πS (Γ) ⊆ ∂ . Further, define the analytic map g : → S by g = f ◦ πS and the holomorphic function h : → −1 by h = πR ◦ f ◦ πS . (i)⇒(ii): Suppose f has an analytic extension across Γ with f (Γ) ⊆ ∂R. 1 applied to the analytic map g yields lim λ(g(u)) |g ′ (u)| = +∞ u→η for every η ∈ I .

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A boundary version of Ahlfors` lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps by Kraus D.


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