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 capabilities has a in basic terms conformal geometric formula by way of 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 as a result of M. Heins.
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Extra resources for A boundary version of Ahlfors` lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps
Notice that ∂R is a (not necessarily connected) real analytic manifold of (real) dimension 1. Now let f : → R be an analytic map. We say that f has an analytic extension across an open subarc Γ of ∂ with f (Γ) ⊆ ∂R, if there exists an analytic map F defined on a neighborhood U ⊆ of Γ, which maps into the Schottky double R = R ∪ ∂R ∪ R∗ such that F (Γ) ⊆ ∂R and F = f in ∩ U . Here R∗ denotes the mirror of R. 1. Let Γ be an open subarc of ∂ , let R be a simply connected Riemann surface with analytic boundary ∂R, let λ(w) |dw| be a complete regular conformal metric on R with curvature −cλ ≤ κλ ≤ −Cλ for some positive constants 250 D.
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A) Note that λ(f (z)) |f ′ (z)| is the density of the pullback of the metric λ(w) |dw| under the map f , so f ∗ λ(z) |dz| = λ(f (z)) |f ′ (z)| |dz| is a conformal pseudometric on and therefore a well-defined function on . 1, it suffices to assume that λ(w) |dw| is a complete regular conformal metric with curvature bounded below. However, these assumptions cannot be weakened further. For instance, take R = , λ(w) |dw| = |dw| and any function holomorphic in a neighborhood of a point of the unit circle.