In this chapter we define the notion of convergence of a sequence of Riemann surfaces, and we prove that any sequence of smooth stable Riemann surfaces has a subsequence which converges to a noded Riemann surface. This is a special case of the celebrated result by P. Deligne and D. Mumford concerning the compactification of the moduli space of algebraic curves. We follow an approach by W. Thurston which is more geometric in nature, viewing Riemann surfaces as surfaces equipped with a hyperbolic metric. The exposition has been made self-contained because the details are scattered throughout the existing literature. In particular, we explain all the necessary background material from hyperbolic geometry.


Riemann Surface Marked Point Boundary Component Conformal Structure Hyperbolic Plane 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Casim Abbas
    • 1
  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA

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