Holomorphic Structures on Topological Surfaces

Part of the Cornerstones book series (COR)


In this chapter, we address the natural problem of determining conditions for a topological surface to admit a holomorphic structure. One necessary condition is, of course, that the surface be orientable. According to Radó’s theorem (Theorem 2.11.1), another necessary condition is that the surface be second countable. It turns out that these two conditions are also sufficient.


Riemann Surface Dirichlet Problem Complex Surface Topological Surface Linear Differential Operator 
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  1. [AhS]
    L. Ahlfors, L. Sario, Riemann Surfaces, Princeton University Press, Princeton, 1960. MATHGoogle Scholar
  2. [BerG]
    C. Berenstein, R. Gay, Complex Variables. An Introduction, Graduate Texts in Mathematics, 125, Springer, New York, 1991. MATHCrossRefGoogle Scholar
  3. [De3]
    J.-P. Demailly, Complex Analytic and Differential Geometry, online book. Google Scholar
  4. [For]
    O. Forster, Lectures on Riemann Surfaces, Graduate Texts in Mathematics, 81, Springer, Berlin, 1981. MATHCrossRefGoogle Scholar
  5. [Hö]
    L. Hörmander, An Introduction to Complex Analysis in Several Variables, third edition, North-Holland, Amsterdam, 1990. MATHGoogle Scholar
  6. [KnR]
    H. Kneser, T. Radó, Aufgaben und Lösungen, Jahresber. Dtsch. Math.-Ver., 35 (1926), issue 1/4, 49, 123–124. Google Scholar
  7. [Mi]
    J. W. Milnor, Topology from the Differentiable Viewpoint, based on notes by David W. Weaver, revised reprint of the 1965 original, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, 1997. MATHGoogle Scholar
  8. [Mu]
    J. R. Munkres, Topology: A First Course, Prentice-Hall, Englewood Cliffs, 1975. MATHGoogle Scholar
  9. [T]
    C. Thomassen, The Jordan–Schönflies theorem and the classification of surfaces, Am. Math. Mon. 99 (1992), no. 2, 116–130. MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsLehigh UniversityBethlehemUSA
  2. 2.Department of MathematicsSUNY at BuffaloBuffaloUSA

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