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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The moduli of compact continuations of an open Riemann surface of genus one


Author: M. Shiba
Journal: Trans. Amer. Math. Soc. 301 (1987), 299-311
MSC: Primary 30F30; Secondary 14H15, 30F25, 32G15
MathSciNet review: 879575
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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^{\prime},\,\{ {A_T^{\prime}},\,{B_T^{\prime}}\} ,i^{\prime})$ are equivalent if $ i^{\prime} \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

DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0879575-2
PII: S 0002-9947(1987)0879575-2
Keywords: Continuation (prolongation, extension) of Riemann surfaces, modulus (of a torus), Teichmüller space, extremal slit torus, span, $ {O_{AD}}$, hydrodynamic continuation, complex velocity potential, Strömungsfunktion, distinguished harmonic differential, principal function
Article copyright: © Copyright 1987 American Mathematical Society