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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Schottky problem on pants
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by R. C. Penner
Proc. Amer. Math. Soc. 104 (1988), 253-256
DOI: https://doi.org/10.1090/S0002-9939-1988-0958077-5
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Abstract:

In this note, we consider the classical problem of Schottky of characterizing the set of period matrices which arise from all possible conformal structures on a fixed topological surface. Restricting to a planar surface with Euler characteristic $- 1$, we find that a real symmetric $3$-by-$3$ matrix arises as a period matrix if and only if the matrix has vanishing row sums, and the diagonal entries are positive and satisfy all three possible strict triangle inequalities. The technique of proof involves extremal and harmonic lengths of curve classes.
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Bibliographic Information
  • © Copyright 1988 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 104 (1988), 253-256
  • MSC: Primary 30C20; Secondary 30F20, 32G15, 57N05
  • DOI: https://doi.org/10.1090/S0002-9939-1988-0958077-5
  • MathSciNet review: 958077