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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

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