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The moduli of compact continuations of an open Riemann surface of genus one. (English) Zbl 0626.30046
Let R be an open Riemann surface of genus one with a fixed canonical homology basis \(\{\) A,B\(\}\) modulo dividing cycles. We call the pair (R,\(\{\) A,B\(\})\) a marked open Riemann surface. By a marked realization of (R,\(\{\) A,B\(\})\) we mean a triple \((T,\{A_ T,B_ T\},i)\), where \((T,\{A_ T,B_ T\})\) is a marked torus and i is a conformal embedding of R into T with i(A) and i(B) homotopic respectively to \(A_ T\) and \(B_ T\). \((T,\{A_ T,B_ T\},i)\) and \((T',\{A_{T'},B_{T'}\},i')\) are said to be equivalent if \(i'\circ i^{-1}\) extends to a conformal mapping of T onto T’. 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 \(M=M(R,\{A,B\})\). For any t, \(-1<t\leq 1\), there exists a unique holomorphic differential \(\phi_ t\) on R such that \(Im[e^{-\pi it/2}\phi_ t]\) is distinguished and \(\int_{A}\phi_ t=1\). Furthermore, there exists a marked realization \((T_ t,\{A_ t,B_ t\},i_ t)\) of (R,\(\{\) A,B\(\})\) such that \(\phi_ t\circ i_ t^{-1}\) extends to the normal holomorphic differential \(\phi^{T_ t}\) on \(T_ t\) w.r.t. the basis \(\{A_ t,B_ t\}\). The compact continuation represented by \((T_ t,\{A_ t,B_ t\},i_ t)\) is called the hydrodynamic continuation of (R,\(\{\) A,B\(\})\) w.r.t. \(\phi_ t.\)
The author proves the following results: (i) M(R,\(\{\) A,B\(\})\) is a closed disk in the Teichmüller space of genus one (the upper half plane); (ii) To each boundary point of M(R,\(\{\) A,B\(\})\) there corresponds a unique hydrodynamic continuation of (R,\(\{\) A,B\(\})\), and vice versa; (iii) To an interior point of M(R,\(\{\) A,B\(\})\) there correspond in general more than one compact continuation of (R,\(\{\) A,B\(\})\); (iv) The radius \(\rho\) (R) of M(R,\(\{\) A,B\(\})\) vanishes if and only if \(R\in O_{AD}\). The quantity \(\rho\) (R) gives a generalization of Schiffer’s span for planar regions. In the proof of (i)-(iv), the results of the author’s preceding papers [M. Shiba, Hiroshima Math. J. 14, 371-399 (1984; Zbl 0567.30033); M. Shiba and K. Shibata, Complex Variables 8, 205-211 (1987; Zbl 0586.30041)] are used.
Reviewer: H.Mizumoto

30F99 Riemann surfaces
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14H15 Families, moduli of curves (analytic)
Full Text: DOI
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