Classification Theory of Riemann Surfaces

1. Front Matter

   Pages I-XX

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2. Introduction

   • L. Sario, M. Nakai

   Pages 1-9

3. Dirichlet Finite Analytic Functions

   • L. Sario, M. Nakai

   Pages 10-78

4. Other Classes of Analytic Functions

   • L. Sario, M. Nakai

   Pages 79-144

5. Dirichlet Finite Harmonic Functions

   • L. Sario, M. Nakai

   Pages 145-221

6. Other Classes of Harmonic Functions

   • L. Sario, M. Nakai

   Pages 222-285

7. Functions with Logarithmic Singularities

   • L. Sario, M. Nakai

   Pages 286-363

8. Functions with Iversen’s Property

   • L. Sario, M. Nakai

   Pages 364-390

9. Back Matter

   Pages 391-449

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The purpose of the present monograph is to systematically develop a classification theory of Riemann surfaces. Some first steps will also be taken toward a classification of Riemannian spaces. Four phases can be distinguished in the chronological background: the type problem; general classification; compactifications; and extension to higher dimensions. The type problem evolved in the following somewhat overlapping steps: the Riemann mapping theorem, the classical type problem, and the existence of Green's functions. The Riemann mapping theorem laid the foundation to classification theory: there are only two conformal equivalence classes of (noncompact) simply connected regions. Over half a century of efforts by leading mathematicians went into giving a rigorous proof of the theorem: RIEMANN, WEIERSTRASS, SCHWARZ, NEUMANN, POINCARE, HILBERT, WEYL, COURANT, OSGOOD, KOEBE, CARATHEODORY, MONTEL. The classical type problem was to determine whether a given simply connected covering surface of the plane is conformally equivalent to the plane or the disko The problem was in the center of interest in the thirties and early forties, with AHLFORS, KAKUTANI, KOBAYASHI, P. MYRBERG, NEVANLINNA, SPEISER, TEICHMÜLLER and others obtaining incisive specific results. The main problem of finding necessary and sufficient conditions remains, however, unsolved.

• Riemannsche Fläche
• Surfaces
• function
• proof
• theorem

• Unversity of California, California, USA

  L. Sario

• Nagoya Unversity, Germany

  M. Nakai

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