A planar open Riemann surface admits the Schiffer span to a point . M. Shiba showed that an open Riemann surface of genus one admits the hyperbolic span . We establish the variation formulas of for the deforming open Riemann surface of genus one with complex parameter in a disk of center , and we show that if the total space is a two-dimensional Stein manifold, then is subharmonic on . In particular, is harmonic on if and only if is biholomorphic to the product .
"Hyperbolic span and pseudoconvexity." Kyoto J. Math. 57 (1) 165 - 183, April 2017. https://doi.org/10.1215/21562261-3759558