## Kyoto Journal of Mathematics

### 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 $\sigma_{H}(R)$. 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 $\Delta\times R(0)$.

#### Article information

Source
Kyoto J. Math., Volume 57, Number 1 (2017), 165-183.

Dates
Received: 4 December 2015
Revised: 12 February 2016
Accepted: 15 February 2016
First available in Project Euclid: 11 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1489201235

Digital Object Identifier
doi:10.1215/21562261-3759558

Mathematical Reviews number (MathSciNet)
MR3621784

Zentralblatt MATH identifier
1371.32008

#### Citation

Hamano, Sachiko; Shiba, Masakazu; Yamaguchi, Hiroshi. Hyperbolic span and pseudoconvexity. Kyoto J. Math. 57 (2017), no. 1, 165--183. doi:10.1215/21562261-3759558. https://projecteuclid.org/euclid.kjm/1489201235

#### References

• [1] L. V. Ahlfors and L. Sario, Riemann Surfaces, Princeton Math. Ser. 26, Princeton Univ. Press, Princeton, 1960.
• [2] R. C. Gunning and R. Narasimhan, Immersion of open Riemann surfaces, Math. Ann. 174 (1967), 103–108.
• [3] S. Hamano, Variation formulas for $L_{1}$-principal functions and the application to the simultaneous uniformization problem, Michigan Math. J. 60 (2011), 271–288.
• [4] S. Hamano, Uniformity of holomorphic families of non-homeomorphic planar Riemann surfaces, Ann. Polon. Math. 111 (2014), 165–181.
• [5] S. Hamano, Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans, Math. Z. 284 (2016), no. 1-2, 491–505.
• [6] S. Hamano, F. Maitani, and H. Yamaguchi, Variation formulas for principal functions, II: Applications to variation for harmonic spans, Nagoya Math. J. 204 (2011), 19–56.
• [7] Y. Kusunoki, Theory of Abelian integrals and its applications to conformal mappings, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 32 (1959), 235–258.
• [8] N. Levenberg and H. Yamaguchi, The metric induced by the Robin function, Mem. Amer. Math. Soc. 92 (1991), no. 448.
• [9] M. Schiffer, The span of multiply connected domains, Duke Math. J. 10 (1943), 209–216.
• [10] M. Shiba, The moduli of compact continuations of an open Riemann surface of genus one, Trans. Amer. Math. Soc. 301, (1987), no. 1 299–311.
• [11] M. Shiba, The Euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems, Kodai Math. J. 16 (1993), 118–137.