×

zbMATH — the first resource for mathematics

Variation formulas for principal functions. II: Applications to variation for harmonic spans. (English) Zbl 1234.32008
Summary: A domain \(D\subset\mathbb{C}_{z}\) admits a circular slit mapping \(P(z)\) for \(a,b\in D\) such that \(P(z)-1/(z-a)\) is regular at \(a\) and \(P(b)=0\). We call \(p(z)=\log|P(z)|\) the \(L_{1}\)-principal function and \(\alpha =\log|P'(b)|\) the \(L_{1}\)-constant, and similarly, the radial slit mapping \(Q(z)\) implies a \(L_{0}\)-principal function \(q(z)\) and a \(L_{0}\)-constant \(\beta\). We call \(s=\alpha-\beta \) the harmonic span for \((D,a,b)\). We show the geometric meaning of \(s\). S. Hamano [Mich. Math. J. 60, No. 2, 271–288 (2011; Zbl 1235.30028)] showed a variation formula for the \(L_{1}\)-constant \(\alpha(t)\) for the moving domain \(D(t)\) in \(\mathbb{C}_{z}\) with \(t\in B:=\{t\in \mathbb{C}:|t|<\rho\}\). We show a corresponding formula for the \(L_{0}\)-constant \(\beta(t)\) for \(D(t)\) and combine these to prove that, if the total space \(\mathcal{D}=\bigcup_{t\in B}(t,D(t))\) is pseudoconvex in \(B\times\mathbb{C}_{z}\), then \(s(t)\) is subharmonic on \(B\). As a direct application, we get the subharmonicity of \(\log \cosh d(t)\) on \(B\), where \(d(t)\) is the Poincaré distance between \(a\) and \(b\) on \(D(t)\).

MSC:
32T99 Pseudoconvex domains
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L.V. Ahlfors and L. Sario, Riemann Surfaces , Princeton Math. Ser. 26 , Princeton University Press, Princeton, 1960. · Zbl 0196.33801
[2] E. Bedford and B. Gaveau, Envelopes of holomorphy of certain 2-spheres in \Bbb C 2 , Amer. J. Math. 105 (1983), 975-1009. · Zbl 0535.32008 · doi:10.2307/2374301
[3] H. Behnke, Die Kanten singuärer Mannigfaltigkeiten , Abh. Math. Semin. Univ. Hambg. 4 (1926), 347-365. · JFM 52.0343.01
[4] M. Brunella, Subharmonic variation of the leafwise Poincaré metric , Invent. Math. 152 (2003), 119-148. · Zbl 1029.32014 · doi:10.1007/s00222-002-0269-0
[5] L. Ford, Automorphic Functions , 2nd ed., Chelsea Publishing, New York, 1951. · JFM 55.0810.04
[6] H. Grunsky, Neue Abschätzungen zur konformen Abbildung ein- und mehrfach zusammenhängender Beriche , Schr. Sem. Univ. Berlin 1 (1932), 95-140. · Zbl 0005.36204 · eudml:204228
[7] R. Gunning and R. Narasimhan, Immersion of open Riemann surfaces , Math. Ann. 174 (1967), 103-108. · Zbl 0179.11402 · doi:10.1007/BF01360812 · eudml:161628
[8] S. Hamano, A lemma on C 1 subharmonicity of the harmonic spans for the discontinuously moving Riemann surfaces , preprint to appear in J. Math. Soc. Japan. · Zbl 1294.30051
[9] S. Hamano, Variation formulas for L 1 -principal functions and application to simultaneous uniformization problem , Michigan Math. J. 60 (2011), 271-288. · Zbl 1235.30028
[10] S. Hamano, Variation formulas for principal functions, III: Applications to variation for Schiffer spans , · Zbl 1234.32008
[11] N. Levenberg and H. Yamaguchi, The metric induced by the Robin function , Mem. Amer. Math. Soc. 448 (1991), 1-155. · Zbl 0742.31003
[12] F. Maitani and H. Yamaguchi, Variation of Bergman metrics on Riemann surfaces , Math. Ann. 330 (2004), 477-489. · Zbl 1077.32006 · doi:10.1007/s00208-004-0556-8
[13] M. Nakai and L. Sario, Classification Theory of Riemann Surfaces , Grundlehren Math. Wiss. 164 , Springer, New York, 1970. · Zbl 0199.40603
[14] Y. Nishimura, Immersion analytique d’une famille de surfaces de Riemann ouverts , Publ. Res. Inst. Math. Sci. 14 (1978), 643-654. · Zbl 0434.32021 · doi:10.2977/prims/1195188831
[15] T. Nishino, Function Theory in Several Complex Variables , Transl. Math. Monogr. 193 , Amer. Math. Soc., Providence, 2001. · Zbl 0972.32001
[16] M. Schiffer, The span of multiply connected domains , Duke Math. J. 10 (1943), 209-216. · Zbl 0060.23704 · doi:10.1215/S0012-7094-43-01019-1
[17] H. Yamaguchi, Variations of pseudoconvex domains over \Bbb C n , Michigan Math. J. 36 (1989), 415-457. · Zbl 0692.31004 · doi:10.1307/mmj/1029004011
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.