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Riemann Surfaces

In: An Introduction to Compactness Results in Symplectic Field Theory

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  • Casim Abbas

    (Michigan State University, Department of Mathematics)

Abstract

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.

Suggested Citation

  • Casim Abbas, 2014. "Riemann Surfaces," Springer Books, in: An Introduction to Compactness Results in Symplectic Field Theory, edition 127, chapter 0, pages 1-99, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-31543-5_1
    DOI: 10.1007/978-3-642-31543-5_1
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