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Hyperbolic span and pseudoconvexity. (English) Zbl 1371.32008
Summary: A planar open Riemann surface $$R$$ admits the Schiffer span $$s(R,\zeta)$$ to a point $$\zeta\in R$$. M. Shiba showed that an open Riemann surface $$R$$ of genus one admits the hyperbolic span $$\sigma_{H}(R)$$. We establish the variation formulas of $$\sigma_{H}(t):=\sigma_{H}(R(t))$$ for the deforming open Riemann surface $$R(t)$$ of genus one with complex parameter $$t$$ in a disk $$\Delta$$ of center $$0$$, and we show that if the total space $$\mathcal{R}=\bigcup_{t\in\Delta}(t,R(t))$$ is a two-dimensional Stein manifold, then $$\sigma_{H}(t)$$ is subharmonic on $$\Delta$$. In particular, $$\sigma_{H}(t)$$ is harmonic on $$\Delta$$ if and only if $$\mathcal{R}$$ is biholomorphic to the product $$\Delta\times R(0)$$.
##### MSC:
 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 32E10 Stein spaces, Stein manifolds
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