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Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings. Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem.

Geometric function theory. --- Functions of complex variables. --- Complex variables --- Elliptic functions --- Functions of real variables --- Function theory, Geometric --- Functions of complex variables

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Differential topology --- Algebraic topology --- Conformal mapping. --- Geometry, Hyperbolic. --- Measure theory. --- Differential topology. --- Complex manifolds. --- Hyperbolic spaces. --- Kleinian groups. --- Complex manifolds --- Conformal mapping --- Geometry, Hyperbolic --- Hyperbolic spaces --- Kleinian groups --- Measure theory --- Lebesgue measure --- Measurable sets --- Measure of a set --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Groups, Kleinian --- Discontinuous groups --- Hyperbolic complex manifolds --- Manifolds, Hyperbolic complex --- Spaces, Hyperbolic --- Geometry, Non-Euclidean --- Hyperbolic geometry --- Lobachevski geometry --- Lobatschevski geometry --- Geometry, Differential --- Topology --- Conformal representation of surfaces --- Mapping, Conformal --- Transformation, Conformal --- Geometric function theory --- Mappings (Mathematics) --- Surfaces, Representation of --- Transformations (Mathematics) --- Analytic spaces --- Manifolds (Mathematics)