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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 $$\widetilde R$$.
##### MSC:
 32U05 Plurisubharmonic functions and generalizations 30F15 Harmonic functions on Riemann surfaces 32G08 Deformations of fiber bundles
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