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
DOI:
https://doi.org/10.1090/S0002-9947-1987-0879575-2
MathSciNet review:
879575
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 $(Tâ, \{ {A_Tâ}, {B_Tâ}\} ,iâ)$ are equivalent if $iâ \circ {i^{ - 1}}$ 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 Schifferâs 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
Keywords:
Continuation (prolongation, extension) of Riemann surfaces,
modulus (of a torus),
Teichmüller space,
extremal slit torus,
span,
<IMG WIDTH="48" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${O_{AD}}$">,
hydrodynamic continuation,
complex velocity potential,
Strömungsfunktion,
distinguished harmonic differential,
principal function
Article copyright:
© Copyright 1987
American Mathematical Society