×

Variation formulas for \(L_{1}\)-principal functions and application to simultaneous uniformization problem. (English) Zbl 1235.30028

Let \(\widetilde R= \bigcup_{t\in\Delta}\widetilde R(t)\) be an unramified domain over \(\Delta\times\mathbb{C}\) and \(R= \bigcup_{t\in\Delta} R(t)\) a subdomain with \(\partial R= \bigcup_{t\in\Delta}\partial R(t)\) real analytic in \(\widetilde R\) consisting of \(\nu+1\) components \(C_0,C_1,\dots, C_\nu\). Let \(u(t,z)\) be an appropriately normalized Green function of \(R(t)\) for every \(t\in\Delta\). Write \[ u(t,z)= \ln{1\over|z|}+ \gamma(t)+ h(t,z) \] in a neighborhood of a zero section \(O:\Delta\to R\), which is supposed to exist. The author proves several results, one of them being the following:
If \(R\) is pseudoconvex over \(\Delta\times\mathbb{C}\) then \(\gamma\) is a real analytic superharmonic function on \(\Delta\).

MSC:

30F15 Harmonic functions on Riemann surfaces
32U05 Plurisubharmonic functions and generalizations
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30F10 Compact Riemann surfaces and uniformization
32D05 Domains of holomorphy
32G10 Deformations of submanifolds and subspaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. Ahlfors and L. Sario, Riemann surfaces, Princeton Math. Ser., 26, Princeton Univ. Press, Princeton, NJ, 1960. · Zbl 0196.33801
[2] L. R. Ford, Automorphic functions, Chelsea, New York, 1972. · JFM 55.0810.04
[3] N. Levenberg and H. Yamaguchi, The metric induced by the Robin function, Mem. Amer. Math. Soc. 448 (1991), 1-155. · Zbl 0742.31003
[4] F. Maitani and H. Yamaguchi, Variation of Bergman metrics on Riemann surfaces, Math. Ann. 330 (2004), 477-489. · Zbl 1077.32006
[5] Y. Nishimura, Immersion analytique d’une famille de surfaces de Riemann ouvertes, Publ. Res. Inst. Math. Sci. 14 (1978), 643-654. · Zbl 0434.32021
[6] T. Nishino, Nouvelles recherches sur les fonctions entières de plusieurs variables complexes (II). Fonctions entières qui se réduisent à celles d’une variable, J. Math. Kyoto Univ. 9 (1969), 221-274. · Zbl 0192.43703
[7] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Grundlehren Math. Wiss., 164, Springer-Verlag, Berlin, 1970. · Zbl 0199.40603
[8] H. Yamaguchi, Variations de surfaces de Riemann, C. R. Acad. Sci. Paris Ser. A-B 286 (1978), 1121-1124. · Zbl 0395.30035
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.