The Schwarz LemmaThe Schwarz lemma is among the simplest results in complex analysis that capture the rigidity of holomorphic functions. This self-contained volume provides a thorough overview of the subject; it assumes no knowledge of intrinsic metrics and aims for the main results, introducing notation, secondary concepts, and techniques as necessary. Suitable for advanced undergraduates and graduate students of mathematics, the two-part treatment covers basic theory and applications. Starting with an exploration of the subject in terms of holomorphic and subharmonic functions, the treatment proves a Schwarz lemma for plurisubharmonic functions and discusses the basic properties of the Poincaré distance and the Schwarz-Pick systems of pseudodistances. Additional topics include hyperbolic manifolds, special domains, pseudometrics defined using the (complex) Green function, holomorphic curvature, and the algebraic metric of Harris. The second part explores fixed point theorems and the analytic Radon-Nikodym property. |
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admissible curve algebra arbitrary Aut(D Aut(T Banach manifold Banach space biholomorphically equivalent bounded domain bounded symmetric domains Carathéodory pseudodistance Cartan chapter compact subset complex Banach manifold complex manifold converges convex domains Corollary defined definition denote exists extreme point fe H(D finite dimensional Finsler metric fixed point geodesic Green function Hahn–Banach theorem Hence Hermitian Hilbert space holomorphic curvature holomorphic functions holomorphic mappings holomorphic retract implies inequality infinite infinitesimal Finsler pseudometric intrinsic metrics irreducible isometry lim sup Math Möbius transformations neighbourhood norm open unit ball plurisubharmonic functions positive integer properties proposition proved pseudoconvex pseudometric real numbers result Riemann Schwarz lemma Schwarz–Pick lemma Schwarz–Pick system sequence shows ſº subharmonic subspace suppose tanh tanhº theory topology triple system unit ball unit disc upper semicontinuous vector Vesentini Vigué