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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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##### References:
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