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Log-plurisubharmonicity of metric deformations induced by Schiffer and harmonic spans. (English) Zbl 1397.32014
Let \(R\) be a planar open Riemann surface and for every \(\zeta\in R\), let \(s(\zeta)\), \(h(\zeta)\) are respectively the Schiffer span and the harmonic span with respect to \(\zeta\) carried by \(R\), which are closely related to conformal mappings. The purpose of this paper is to study the metric deformations induced by the Schiffer span and the harmonic span on \(R\). The author proves that both metrics \(s(\zeta )|d\zeta|^2\) and \(\frac{\partial^2h(\zeta)}{\partial\zeta\partial\overline\zeta}|d\zeta|^2\) induced on \(R\) coincide, have negative curvature everywhere and are complete. For a complex parameter \(t\), let \(\pi : \widetilde{{\mathcal R}}\rightarrow B\) be a holomorphic family such that \(\widetilde{{\mathcal R}}\) is a \(2\)-dimensional complex manifold, \(\pi\) is a holomorphic projection from \(\widetilde{{\mathcal R}}\) onto a disk \(B\) in \({\mathbb C}_t\), and each fiber \(\widetilde{R}(t)=\pi^{-1}(t)\), \(t\in B\), is irreducible and non-singular in \(\widetilde{{\mathcal R}}\). By setting \(\widetilde{{\mathcal R}}=\bigcup_{t\in B}(t,\widetilde{R}(t))\), let \({\mathcal R}=\bigcup_{t\in B}(t,R(t))\) be a subdomain with \(C^w\) smooth boundary \(\partial{\mathcal R}=\bigcup_{t\in B}(t,\partial R(t))\) in \(\widetilde{{\mathcal R}}\) such that, for \(t\in B\), \(\emptyset \not=R(t)\Subset\widetilde{{\mathcal R}}(t)\). Then \(R(t)\) is a planar bordered Riemann surface in \(\widetilde{{\mathcal R}}(t)\). The author shows that, if \(s(t,\zeta)|d\zeta|^2\) is the metric induced by the Schiffer span \(s(t,\zeta)\) for \((R(t),\zeta)\) and the total space \({\mathcal R}\) is pseudoconvex in \(\widetilde{{\mathcal R}}\), then \(\log s(t,\zeta)\) and \(\log \frac{\partial^2h(t,\zeta)}{\partial\zeta\partial\overline\zeta}\) are plurisubharmonic on \(\widetilde R\).
32U05 Plurisubharmonic functions and generalizations
30F15 Harmonic functions on Riemann surfaces
32G08 Deformations of fiber bundles
Full Text: DOI
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