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An Introduction to Riemann Surfaces

  • Terrence Napier
  • Mohan Ramachandran

Part of the Cornerstones book series (COR)

Table of contents

  1. Front Matter
    Pages I-XVII
  2. Analysis on Riemann Surfaces

    1. Front Matter
      Pages 1-1
    2. Terrence Napier, Mohan Ramachandran
      Pages 3-23
    3. Terrence Napier, Mohan Ramachandran
      Pages 101-154
  3. Further Topics

    1. Front Matter
      Pages 155-155
    2. Terrence Napier, Mohan Ramachandran
      Pages 157-189
    3. Terrence Napier, Mohan Ramachandran
      Pages 191-309
    4. Terrence Napier, Mohan Ramachandran
      Pages 311-371
  4. Background Material

    1. Front Matter
      Pages 373-373
    2. Terrence Napier, Mohan Ramachandran
      Pages 375-405
    3. Terrence Napier, Mohan Ramachandran
      Pages 407-414
    4. Terrence Napier, Mohan Ramachandran
      Pages 415-476
    5. Terrence Napier, Mohan Ramachandran
      Pages 477-530
    6. Terrence Napier, Mohan Ramachandran
      Pages 531-543
  5. Back Matter
    Pages 545-560

About this book

Introduction

This textbook presents a unified approach to compact and noncompact Riemann surfaces from the point of view of the L² -method, a powerful technique used in the theory of several complex variables.  The work features a simple construction of a strictly subharmonic exhaustion function and a related construction of a positive-curvature Hermitian metric in a holomorphic line bundle, topics which serve as starting points for proofs of standard results such as the Mittag-Leffler, Weierstrass, and Runge theorems; the Riemann−Roch theorem; the Serre duality and Hodge decomposition theorems; and the uniformization theorem. The book also contains treatments of other facts concerning the holomorphic, smooth, and topological structure of a Riemann surface, such as the biholomorphic classification of Riemann surfaces, the embedding theorems, the integrability of almost complex structures, the Schönflies theorem (and the Jordan curve theorem), and the existence of smooth structures on second countable surfaces. 

Although some previous experience with complex analysis, Hilbert space theory, and analysis on manifolds would be helpful, the only prerequisite for this book is a working knowledge of point-set topology and elementary measure theory. The work includes numerous exercises—many of which lead to further development of the theory—and  presents (with proofs) streamlined treatments of background topics from analysis and topology on manifolds in easily-accessible reference chapters, making it ideal for a one- or two-semester graduate course.

Keywords

DeRham-Hodge decomposition Morse theory complex manifolds

Authors and affiliations

  • Terrence Napier
    • 1
  • Mohan Ramachandran
    • 2
  1. 1., Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2., Deptartment of MathematicsState University New York at BuffaloBuffaloUSA

Bibliographic information