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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality 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 1.43.

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The moduli of compact continuations of an open Riemann surface of genus one
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by M. Shiba PDF
Trans. Amer. Math. Soc. 301 (1987), 299-311 Request permission

Abstract:

Let $(R, \{ A, B\} )$ be a marked open Riemann surface of genus one. Denote by $(T, \{ {A_T}, {B_T}\} ,i)$ a pair of a marked torus $(T, \{ {A_T}, {B_T}\} )$ and a conformal embedding $i$ of $R$ into $T$ with $i(A)$ and $i(B)$ homotopic respectively to ${A_T}$ and ${B_T}$. We say that $(T, \{ {A_T}, {B_T}\} ,i)$ and $(T’, \{ {A_T’}, {B_T’}\} ,i’)$ are equivalent if $i’ \circ {i^{ - 1}}$ extends to a conformal mapping of $T$ onto ${T^\prime }$. The equivalence classes are called compact continuations of $(R, \{ A, B\} )$ and the set of moduli of compact continuations of $(R, \{ A, B\} )$ is denoted by $M = M(R, \{ A, B\} )$. Then $M$ is a closed disk in the upper half plane. The radius of $M$ represents the size of the ideal boundary of $R$ and gives a generalization of Schiffer’s span for planar domains; in particular, it vanishes if and only if $R$ belongs to the class ${O_{AD}}$. On the other hand, any holomorphic differential on $R$ with distinguished imaginary part produces in a canonical manner a compact continuation of $(R, \{ A, B\} )$. Such a compact continuation is referred to as a hydrodynamic continuation of $(R, \{ A, B\} )$. The boundary of $M$ parametrizes in a natural way the space of hydrodynamic continuations; i.e., the hydrodynamic continuations have extremal properties.
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 301 (1987), 299-311
  • MSC: Primary 30F30; Secondary 14H15, 30F25, 32G15
  • DOI: https://doi.org/10.1090/S0002-9947-1987-0879575-2
  • MathSciNet review: 879575