Kyoto Journal of Mathematics

Hyperbolic span and pseudoconvexity

Sachiko Hamano, Masakazu Shiba, and Hiroshi Yamaguchi

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Abstract

A planar open Riemann surface R admits the Schiffer span s(R,ζ) to a point ζR. M. Shiba showed that an open Riemann surface R of genus one admits the hyperbolic span σH(R). We establish the variation formulas of σH(t):=σH(R(t)) for the deforming open Riemann surface R(t) of genus one with complex parameter t in a disk Δ of center 0, and we show that if the total space R=tΔ(t,R(t)) is a two-dimensional Stein manifold, then σH(t) is subharmonic on Δ. In particular, σH(t) is harmonic on Δ if and only if R is biholomorphic to the product Δ×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

Subjects
Primary: 32Txx: Pseudoconvex domains 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]
Secondary: 32E10: Stein spaces, Stein manifolds 30Fxx: Riemann surfaces

Keywords
Closings of an open torus pseudoconvex family of open tori variational formula for the modulus (sub)harmonity of the hyperbolic span

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


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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.