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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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[1] Ahlfors, L.V., Sario, L.: Riemann Surfaces. Princeton University Press, Princeton (1960) · Zbl 0196.33801
[2] Berndtsson, B, Subharmonicity of the Bergman kernel and some other functions associated to pseudoconvex domains, Ann. Inst. Fourier, 56, 1633-1662, (2006) · Zbl 1120.32021
[3] Berndtsson, B., Lempert, L.: A proof of the Ohsawa-Takegoshi theorem with sharp estimates, to appear in J. Math. Soc. Jpn · Zbl 1360.32006
[4] Guan, Q; Zhou, X, A solution of an \(L^2\) extension problem with an optimal estimate and applications, Ann. Math., 181, 1139-1208, (2015) · Zbl 1348.32008
[5] Gunning, R; Narasimhan, R, Immersion of open Riemann surfaces, Math. Ann., 174, 103-108, (1967) · Zbl 0179.11402
[6] Hamano, S, Variation formulas for \(L_1\)-principal functions and the application to simultaneous uniformization problem, Michigan Math. J., 60, 271-288, (2011) · Zbl 1235.30028
[7] Hamano, S, Schiffer functions on domains in \({\mathbb{C}}^n\), RIMS Kôkyûroku Bessatsu, B43, 17-28, (2013) · Zbl 1298.32010
[8] Hamano, S, Uniformity of holomorphic families of non-homeomorphic planar Riemann surfaces, Ann. Pol. Math., 111, 165-182, (2014) · Zbl 1300.32026
[9] Hamano, S; Maitani, F; Yamaguchi, H, Variation formulas for principal functions (II) applications to variation for the harmonic spans, Nagoya Math. J., 204, 19-56, (2011) · Zbl 1234.32008
[10] Levenberg, L; Yamaguchi, H, The metric induced by the Robin function, Mem. AMS., 448, 1-155, (1991) · Zbl 0742.31003
[11] Maitani, F; Yamaguchi, H, Variation of Bergman metrics on Riemann surfaces, Math. Ann., 330, 477-489, (2004) · Zbl 1077.32006
[12] Schiffer, M, The span of multiply connected domains, Duke Math. J., 10, 209-216, (1943) · Zbl 0060.23704
[13] Schiffer, M, The kernel function of an orthogonal system, Duke Math. J., 13, 529-540, (1946) · Zbl 0060.23708
[14] Suita, N, Capacities and kernels on Riemann surfaces, Arch. Ration. Mech. Anal., 46, 212-217, (1972) · Zbl 0245.30014
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