Schwarz–Ahlfors–Pick theorem (original) (raw)
Das Lemma von Schwarz-Pick (nach Hermann Schwarz und Georg Alexander Pick) ist eine Aussage aus der Funktionentheorie über holomorphe Endomorphismen des Einheitskreises, die das Schwarzsche Lemma verallgemeinert.Im Rahmen der hyperbolischen Geometrie bedeutet es, dass holomorphe Endomorphismen Kontraktionen sind.
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| dbo:abstract | Das Lemma von Schwarz-Pick (nach Hermann Schwarz und Georg Alexander Pick) ist eine Aussage aus der Funktionentheorie über holomorphe Endomorphismen des Einheitskreises, die das Schwarzsche Lemma verallgemeinert.Im Rahmen der hyperbolischen Geometrie bedeutet es, dass holomorphe Endomorphismen Kontraktionen sind. (de) In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to itself, will not increase the Poincaré distance between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces: Theorem (Schwarz–Ahlfors–Pick). Let U be the unit disk with Poincaré metric ; let S be a Riemann surface endowed with a Hermitian metric whose Gaussian curvature is ≤ −1; let be a holomorphic function. Then for all A generalization of this theorem was proved by Shing-Tung Yau in 1973. (en) Теорема Пика, или теорема Шварца — Пика — инвариантная формулировка и обобщение леммы Шварца. (ru) Теорема Піка, або теорема Шварца — Піка — інваріантне формулювання та узагальнення леми Шварца. (uk) |
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| rdfs:comment | Das Lemma von Schwarz-Pick (nach Hermann Schwarz und Georg Alexander Pick) ist eine Aussage aus der Funktionentheorie über holomorphe Endomorphismen des Einheitskreises, die das Schwarzsche Lemma verallgemeinert.Im Rahmen der hyperbolischen Geometrie bedeutet es, dass holomorphe Endomorphismen Kontraktionen sind. (de) Теорема Пика, или теорема Шварца — Пика — инвариантная формулировка и обобщение леммы Шварца. (ru) Теорема Піка, або теорема Шварца — Піка — інваріантне формулювання та узагальнення леми Шварца. (uk) In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to itself, will not increase the Poincaré distance between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces: for all (en) |
| rdfs:label | Lemma von Schwarz-Pick (de) Schwarz–Ahlfors–Pick theorem (en) Теорема Пика (комплексный анализ) (ru) Теорема Піка (комплексний аналіз) (uk) |
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