Conformal Regions

Conformal homeomorphisms and conformal regions, group of conformal automorphisms of a set, Schwarz lemma, characterization of the group of automorphisms of the unit disk centered at 0 by Möbius transformations, normal families of functions, Arzelà–Ascoli theorem, Montel theorem, Vitali theorem on “propagation of convergence,” Riemann mapping theorem, Carathéodory theorem of boundary correspondence, canonical regions of multiply connected regions, unification of some topological, analytic, and algebraic characterizations of simply connected regions, existence of solution of Dirichlet problem in simply connected regions properly contained in ℂ $$\mathbb {C}$$ with boundary values specified by a continuous function, prime ends of a simply connected open subset of the Riemann sphere, square length-area inequality, F. and M. Riesz theorem on bounded holomorphic functions in disks with radial limit in positive Lebesgue measure set of polar angles, Carathéodory compactification and criterion for continuous extension of conformal homeomorphisms in a disk to its boundary, extremal length of a set of rectifiable curves and its invariance under conformal homeomorphisms, extremal metric, characterization of prime ends by extremal length, general properties of univalent and schlicht functions, Löwner method for univalent functions, proofs of the Milin, Robertson and Bieberbach conjectures using Löwner chains as in the proof of L. de Branges in 1985, but with simplifications published in 1991 and 1994. Exercises, including Blaschke and Osgood convergence theorems, Schwarz–Christoffel formula, hyperbolic geometry, generalized Möbius transformations, reflection principle, factorization and zero prescription of bounded holomorphic functions in disks, solutions of Poisson equation, and elliptic functions.