Hyperbolic span and pseudoconvexity

Abstract

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 ${\sigma }_H\left(R\right)$. We establish the variation formulas of ${\sigma }_H\left(t\right):={\sigma }_H\left(R\right(t\left)\right)$ for the deforming open Riemann surface $R\left(t\right)$ of genus one with complex parameter $t$ in a disk $\Delta$ of center $0$, and we show that if the total space $\mathcal{R}={\bigcup }_{t\in \Delta }(t,R(t\left)\right)$ is a two-dimensional Stein manifold, then ${\sigma }_H\left(t\right)$ is subharmonic on $\Delta$. In particular, ${\sigma }_H\left(t\right)$ is harmonic on $\Delta$ if and only if $\mathcal{R}$ is biholomorphic to the product $\Delta \times R\left(0\right)$.