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Proceedings of the American Mathematical Society

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The Schottky problem on pants

Author: R. C. Penner
Journal: Proc. Amer. Math. Soc. 104 (1988), 253-256
MSC: Primary 30C20; Secondary 30F20, 32G15, 57N05
MathSciNet review: 958077
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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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Article copyright: © Copyright 1988 American Mathematical Society