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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)].

MSC:
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
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