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Variation of Bergman metrics on Riemann surfaces. (English) Zbl 1077.32006
Let \(\pi : \mathcal R\to\mathbb B\) be a holomorphic family of Riemann surfaces \(R(t) =\pi^{-1}(t)\), \(t\in\mathbb B\), where \(\mathcal R\) is a 2-dimensional analytic space and \(\mathbb B\) is a disk in \(\mathbb C\) (such that \(R(t)\), \(t\in\mathbb B\), is irreducible. For the non-exceptional Riemann surface \(R(t)\), we have the Bergman metric \(K(t,\zeta)| d\zeta| ^2\). The Riemann surface \(R\) is said to be exceptional, if \(R\) is conformally equivalent to \(\mathbb P^1\setminus e\) such that \(e\) is of logarithmic capacity \(0\). Put \(K(t,\zeta)\equiv 0\) for the exceptional \(R(t)\). In this note, the behavior of \(K(t,\zeta)\) on \(\mathcal R\) is studied. The authors first establish the variation formula for \(\frac{\partial^2K(t,\zeta)}{\partial t\partial\overline t}\) in the case of the unramified domain \(\mathcal D\) over \(B\times\mathbb C_z\) with smooth boundary, where \(K(t,\zeta)| dt| ^2\) denotes the Bergman metric on the fiber \(D(t)\) of \(\mathcal D\) for \(t\in\mathbb B\). This formula directly implies that \(\log K(t,\zeta)\) is plurisubharmonic on \(\mathcal D\) in the case when \(\mathcal D\) is pseudoconvex. Secondly, the authors generalize this property for the holomorphic family \(\pi : \mathcal R \to\mathbb B\) such that \(\mathcal R\) is a Stein space. Using this generalized theorem, the authors finally show the uniformization condition for the following Stein space \(\mathcal R\): If the set of the points \(t\in\mathbb B\) such that \(R(t)\) is exceptional is of positive capacity in \(\mathbb B\), then \(\mathcal R\) is biholomorphic to a univalent domain in \(\mathbb B\times\mathbb P_w\) by the transformation of the form \(t =\pi(p)\), \(w = f(p)\) for \(p\in\mathcal R\). As one of the special cases, this involves Nishino’s Fundamental Lemma on p. 253 in T. Nishino [J. Math. Kyoto Univ. 9, 221–274 (1969; Zbl 0192.43703)].

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
31C10 Pluriharmonic and plurisubharmonic functions
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
30F15 Harmonic functions on Riemann surfaces
Full Text: DOI
[1] Hadamard, J.: Mémoire sur les problèmes de fonctions holomorphes relatifs à l?équilibre des plaques élastiques éncastrées. Mémoires Presentés par Divers Savants al?Académie de Sciences 33, (1907)
[2] Hamano, H., Yamaguchi, H.: A note on variation of Bergman metrics on Riemann surfaces under pseudoconvexity. To appear · Zbl 1063.30040
[3] Levenberg, N., Yamaguchi, H.: The metric induced by the Robin function. Mem. AMS. 448, 1-155 (1991) · Zbl 0742.31003
[4] Nishino, T.: Nouvelles recherches sur les fonctions entières de plusieurs variables complexes (II). Fonctions entières qui se réduisent à celles d?une variable. J. Math. of Kyoto Univ. 9, 221-274 (1969) · Zbl 0192.43703
[5] Nishino, T.: Function theory in several complex variables. Trans Math. Monographs AMS 193, (2001) · Zbl 0972.32001
[6] Schiffer, M.: The kernel function of an orthogonal system. Duke M.J. 13, 529-540 (1946) · Zbl 0060.23708
[7] Suita, N.: Capacities and kernels on Riemann surfaces. Arch. Rational Mech. Anal. 46, 212-217 (1972) · Zbl 0245.30014
[8] Weyl, H.: Die Idee der Riemannschen Fläche. Teubner, Leipzig, 1940 · JFM 44.0492.01
[9] Yamaguchi, H.: Parabolicité d?une fonction entière. J. Math. Kyoto Univ. 16, 71-92 (1976) · Zbl 0326.32007
[10] Yamaguchi, H.: Variations of pseudoconvex domains over Michigan Math. J. 36, 415-457 (1989) · Zbl 0692.31004
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