Hamano, Sachiko On rigidity of pseudoconvex domains fibered by open Riemann surfaces according to directional moduli. (English) Zbl 1487.30037 Math. Z. 300, No. 1, 979-993 (2022). Summary: Schmieder and Shiba investigated conformal embeddings of a marked open Riemann surface \(R\) of finite genus \(g\) into closed Riemann surfaces of the same genus, and they showed that for each given \(j\) (\(1\le j \le g\)), the set \({\mathfrak{M}}_j\) of \(j\)-th diagonal elements of the period matrices of all closings of \(R\) is a closed disk. In this paper, for a marked open Riemann surface \(R\) of genus \(g\) (\(1 \le g < \infty\)) and a real \(g\)-vector \(\mathbf{a}=(a_1, \ldots, a_g)\ne \mathbf{0} \), we introduce the \(\mathbf{a} \)-span \(\rho_\mathbf{a}\) of \(R\), defined in terms of \(L_1\)- and \(L_0\)-canonical semi-exact differentials normalized by \(\mathbf{a} \), and establish a new relation between \(\rho_\mathbf{a}\) and \({\mathfrak{M}}_j\). From the viewpoint of several complex variables, a variational formula of \(\rho_\mathbf{a}(t)\) is obtained for a smooth family \({\mathcal{R}}\) of open Riemann surfaces \(R(t)\) with a complex parameter \(t\) in a disk \(\varDelta \). As an application, we prove that if \({\mathcal{R}}\) is a two-dimensional pseudoconvex domain fibered by open Riemann surfaces of the same topological type, then \(\rho_\mathbf{a}(t)\) is subharmonic on \(\varDelta \). This means the subharmonicity of the diameter of the \(\mathbf{a} \)-directional moduli disk for higher genera when \({\mathcal{R}}\) is pseudoconvex. Cited in 1 Document MSC: 30F99 Riemann surfaces 32T99 Pseudoconvex domains 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions Keywords:pseudoconvexity; moduli; Riemann surfaces; period matrices PDFBibTeX XMLCite \textit{S. Hamano}, Math. Z. 300, No. 1, 979--993 (2022; Zbl 1487.30037) Full Text: DOI References: [1] Ahlfors, L.V., Sario, L.: Riemann Surfaces, Princeton Mathematical Series, vol. 26. Princeton University Press, Princeton (1960). (MR0114911) · Zbl 0196.33801 [2] Gunning, RC; Narasimhan, R., Immersion of open Riemann surfaces, Math. Ann., 174, 103-108 (1967) · Zbl 0179.11402 [3] Hamano, S., Variation formulas for L1-principal functions and application to the simultaneous uniformization problem, Mich. Math. J., 60, 2, 271-288 (2011) · Zbl 1235.30028 [4] Hamano, S., Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans, Math. Z., 284, 1-2, 491-505 (2016) · Zbl 1397.32014 [5] Hamano, S.; Shiba, M.; Yamaguchi, H., Hyperbolic span and pseudoconvexity, Kyoto J. Math., 57, 1, 165-183 (2017) · Zbl 1371.32008 [6] Kusunoki, Y., Theory of Abelian integrals and its applications to conformal mappings, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math., 32, 235-258 (1959) · Zbl 0091.07201 [7] Levenberg, N.; Yamaguchi, H., The metric induced by the Robin function, Mem. Amer. Math. Soc., 92, 448, viii+156 (1991) · Zbl 0742.31003 [8] Oikawa, K., On the prolongation of an open Riemann surface of finite genus, Kôdai Math. Sem. Rep., 9, 34-41 (1957) · Zbl 0078.06901 [9] Sario, L., Nakai, M.: Classification theory of Riemann surfaces, Die Grundlehren der mathematischen Wissenschaften, vol. 164. Springer, New York (1970). (MR0264064) · Zbl 0199.40603 [10] Schiffer, M., The span of multiply connected domains, Duke Math. J., 10, 209-216 (1943) · Zbl 0060.23704 [11] Schmieder, G.; Shiba, M., One-parameter variations of the ideal boundary and compact continuations of a Riemann surface, Analysis (Munich), 18, 2, 125-130 (1998) · Zbl 0917.30027 [12] Shiba, M., The Riemann-Hurwitz relation, parallel slit covering map, and continuation of an open Riemann surface of finite genus, Hiroshima Math. J., 14, 2, 371-399 (1984) · Zbl 0567.30033 [13] Shiba, M., The moduli of compact continuations of an open Riemann surface of genus one, Trans. Am. Math. Soc., 301, 1, 299-311 (1987) · Zbl 0626.30046 [14] Shiba, M., Conformal embeddings of an open Riemann surface into another|a counter of univalent function theory, Interdiscip. Inf. Sci., 25, 1, 23-31 (2019) · Zbl 1456.30080 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.