Riemann Surfaces

This graduate text on Riemann surface theory proves the fundamental analytical results on the existence of meromorphic functions and the Uniformisation Theorem. The approach taken emphasises PDE methods, applicable more generally in global analysis. The connection with geometric topology, and in particular the role of the mapping class group, is also explained. To this end, some more sophisticated topics have been included, compared with traditional texts at this level. While the treatment is novel, the roots of the subject in traditional calculus and complex analysis are kept well in mind.

I Preliminaries 1: Holomorphic Functions 2: Surface Topology II Basic Theory 3: Basic Definitions 4: Maps between Riemann Surfaces 5: Calculus on Surfaces 6: Elliptic functions and integrals 7: Applications of the Euler characteristic III Deeper Theory 8: Meromorphic Functions and the Main Theorem for Compact Riemann Surfaces 9: Proof of the Main Theorem 10: The Uniformisation Theorem IV Further Developments 11: Contrasts in Riemann Surface Theory 12: Divisors, Line Bundles and Jacobians 13: Moduli and Deformations 14: Mappings and Moduli 15: Ordinary Differential Equations Bibliography Index

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