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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Complex structures on Riemann surfaces
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by Garo Kiremidjian PDF
Trans. Amer. Math. Soc. 169 (1972), 317-336 Request permission


Let $X$ be a Riemann surface (compact or noncompact) with the property that the length of every closed geodesic is bounded away from zero. Then we show that sufficiently small complex structures on $X$ can be described without making use of Schwarzian derivatives or the theory of quasiconformal mappings. Instead, we use methods developed in Kuranishi’s work on the existence of locally complete families of deformations of compact complex manifolds. We introduce norms $|\quad {|_k}$ ($k$ a positive integer) on the space of ${C^\infty }(0,p)$-forms with values in the tangent bundle on $X$, which are similar to the usual Sobolev $||\quad |{|_k}$-norms. (In the compact case $|\quad {|_k}$ is equivalent to $||\quad |{|_k}$.) Then we prove that certain properties of $||\quad |{|_k}$, crucial for Kuranishi’s approach, are also satisfied by $|\quad {|_k}$.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 169 (1972), 317-336
  • MSC: Primary 32G15; Secondary 30A46
  • DOI:
  • MathSciNet review: 0310296