Directional Moduli and Pseudoconvexity

Sachiko Hamano ()
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Sachiko Hamano: Kyoto Sangyo University, Department of Mathematics, Faculty of Science

Chapter Chapter 5 in Essays on Geometry, 2025, pp 65-79 from Springer

Abstract: Abstract This is a survey chapter on our works related to the rigidity of pseudoconvex domains fibered by open Riemann surfaces according to directional moduli. For a marked open Riemann surface R of finite genus g and a real g-vector a = ( a 1 , … , a g ) ≠ 0 $$\mathbf {a}=(a_1, \ldots , a_g)\ne \mathbf {0}$$ , we introduce the a $$\mathbf {a}$$ -span ρ a $$\rho _{\mathbf {a}}$$ , and establish a new relation between ρ a $$\rho _{\mathbf {a}}$$ and the set of period matrices of all closings of R. From the viewpoint of several complex variables, a variational formula of ρ a ( t ) $$\rho _{\mathbf {a}}(t)$$ is obtained for a smooth family ℛ $$\mathcal R$$ of open Riemann surfaces R ( t ) $$R(t)$$ with a complex parameter t in a disk Δ $$\Delta $$ . As an application, we state the subharmonicity of the diameter of the a $$\mathbf {a}$$ -directional moduli disk for higher genera when ℛ $$\mathcal R$$ is a two-dimensional pseudoconvex domain fibered by open Riemann surfaces of the same topological type.

Date: 2025
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DOI: 10.1007/978-3-031-76257-4_5

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