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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, Teichm├╝ller theory (complex-analytic aspects in several variables)
32E10 Stein spaces, Stein manifolds
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