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

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By Kraus D.

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

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

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Then the following are equivalent: (a) λ(z) |dz| is locally complete near Γ; (b) lim λ(z) = +∞ for every ξ ∈ Γ. z→ξ Proof. The domain Ω has at least two boundary points, since it has a smooth boundary set Γ. Thus Ω carries a complete regular conformal metric λΩ (z) |dz| with constant negative curvature. 1 applied to µ(z) |dz| := λΩ (z) |dz| yields implication “(b) ⇒ (a)”. 4 applied to µ(z) |dz| = λΩ (z) |dz|, using the well-known fact that λΩ (z) → +∞ as z → ξ for every ξ ∈ ∂Ω (see, for instance, [14]).

18] D. Minda, Regular analytic arcs and curves, Colloq. Math. 38 (1977), 73–82. [19] D. Minda, The strong form of Ahlfors’ lemma, Rocky Mountain J. Math. 17 (1987), 457–461. [20] D. Minda, A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables Theory Appl. 8 (1987), 129–144. [21] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992. [22] O. Roth, A general conformal geometric reflection principle, Trans. Amer.

The topology of the sphere ˆ also forces ∂D to be also connected. Therefore, ∂D is real analytic homeomorphic to the unit circle. This seems obvious, but is surprisingly difficult to prove (see, for instance, [18, Theorem 1]). Consequently, D is bounded by an analytic Jordan curve, and there is a conformal map Ψ defined on a neighborhood of which maps onto D and ∂ homeomorphically onto ∂D. Finally, π := πR ◦ Ψ is a conformal £ map defined on a neighborhood of such that π(∂ ) = ∂R. 1. 3. We pull the metric λ(w) |dw| back to the unit disk using w = π(u) and get a complete regular conformal metric ν(u) |du| := π ∗ λ(u) |du| = λ(π(u)) |π ′ (u)| |du| .

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