Uniformization of Riemann Surfaces

Luis T. Magalhães
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Luis T. Magalhães: University of Lisbon, Instituto Superior Técnico

Chapter Chapter 12 in Complex Analysis and Dynamics in One Variable with Applications, 2025, pp 353-440 from Springer

Abstract: Abstract Coverings and fundamental group, deck mappings, Riemann surface defined by a covering of a Riemann surface, compact Riemann surface of an affine plane curve, Riemann–Hurwitz formula, Perron method in Riemann surfaces, hyperbolic, parabolic or elliptic Riemann surfaces, Riemann surfaces uniformization theorem, Radó theorem on Riemann surfaces satisfying the topological spaces 2nd countability axiom, hyperbolic, parabolic and elliptic simply connected Riemann surfaces geometry characterized by their groups of conformal automorphisms, Pick theorem for Riemann surfaces, proof that families of holomorphic functions between hyperbolic Riemann surfaces are normal, Poincaré metric near the boundary of an hyperbolic Riemann surface properly contained in another hyperbolic Riemann surface, Montel–Carathéodory theorem for functions defined in hyperbolic Riemann surfaces, Koebe one quarter theorem in Poincaré metric, meromorphic 1-forms in Riemann surfaces, existence in compact Riemann surfaces of dipole Green functions and of meromorphic functions separating points, Riemann existence theorem, Riemann–Hurwitz formula with meromorphic 1-Forms, Riemann inequality, Riemann–Roch theorem, proof that every compact Riemann surface is conformal to the compact Riemann surface of an irreducible affine plane curve, harmonic 1-forms, 1-homology space and periods of a compact Riemann surface, normal or standard form of compact Riemann surface, 1st de Rham Cohomology space of compact Riemann surface, Clifford theorem, proof that every compact Riemann surface can be holomorphically embedded in a projective space with dimension greater than the surface genus, elliptic and Abelian integrals, Abel theorem for Abelian integrals, Jacobian of compact Riemann surface and Abel–Jacobi mapping, Abel–Jacobi theorem. Exercises, including on hyperelliptic Riemann surfaces, the Bézout theorem, and the gap theorems of Weierstrass and of Noether.

Date: 2025
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DOI: 10.1007/978-3-031-64999-8_12

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