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 features has a in basic terms conformal geometric formula when it comes to Riemannian metrics. This conformally invariant mirrored image precept generalizes clearly to analytic maps among Riemann surfaces and includes between different effects a characterization of finite Blaschke items because of M. Heins.
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Additional info for A boundary version of Ahlfors` lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps
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. Then the following conditions are equivalent. (i) f has an analytic extension across Γ such that f (Γ) ⊂ ∂R. (ii) For every ξ ∈ Γ, lim λ(f (z)) |f ′ (z)| |dz| = +∞ .
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.
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