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

By Kraus D.

A boundary model of Ahlfors' Lemma is confirmed 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 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 resources for A boundary version of Ahlfors` lemma, locally complete conformal metrics and conformally invariant reflection principles for analytic maps

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This implies lim λ(f (z)) |f ′ (z)| |dz| = +∞ z→ξ for every ξ ∈ Γ. (ii) ⇒ (iii): Let µ(z) |dz| be a conformal pseudo-metric on S with curvature κµ ≤ −Cµ < 0. , (πS∗ µ)(u) |du| = µ(πS (u)) |πS′ (u)| |du|, gives a pseudo-metric on with curvature κπS∗ µ ≤ −Cµ . 4 follows lim inf z→ξ λ(f (z)) |f ′ (z)| λ(g(u)) |g ′ (u)| = lim−1 inf ≥ µ(z) πS∗ µ(u) u→πS (ξ) for every ξ ∈ Γ. Cµ cλ ∗ ′ (iii)⇒(i): Define on the regular conformal metric πR λ(v) := λ(πR (v)) |πR (v)| ∗ ′ ′ with curvature κπR∗ λ ≥ −cλ . Note that πR λ(h(u)) |h (u)| = λ(g(u)) |g (u)| for u ∈ .

T. Yau, A general Schwarz Lemma for K¨ahler Manifolds, Amer. J. Math. 100 (1978), 197–203.

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| = +∞ . z→ξ 254 D. KRAUS, O. ROTH AND S. 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 .

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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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