Kyoto Journal of Mathematics

Hyperbolic span and pseudoconvexity

Sachiko Hamano, Masakazu Shiba, and Hiroshi Yamaguchi

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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

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

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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

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