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\(C^1\) subharmonicity of harmonic spans for certain discontinuously moving Riemann surfaces. (English) Zbl 1294.30051

Summary: We showed in [3] and [4] the variation formulas for Schiffer spans and harmonic spans of the moving domain \(D(t)\) in \(\mathbb{C}_z\) with parameter \(t\in B= \{t\in\mathbb{C}_t: |t|<\rho\}\), respectively, such that each \(\partial D(t)\) consists of a finite number of \(C^\omega\) contours \(C_j(t)\) \((j= 1,\dots,\nu)\) in \(\mathbb{C}_z\) and each \(C_j(t)\) varies \(C^\omega\) smoothly with \(t\in B\). This implied that, if the total space \({\mathcal D}=\bigcup_{t\in B}(t, D(t))\) is pseudoconvex in \(B\times \mathbb{C}_z\), then the Schiffer span is logarithmically subharmonic and the harmonic span is subharmonic on \(B\), respectively, so that we showed those applications.
In this paper, we give the indispensable condition for generalizing these results to Stein manifolds. Precisely, we study the general variation under pseudoconvexity, i.e., the variation of domains \({\mathcal D}: t\in B\to D(t)\) is pseudoconvex in \(B\times\mathbb{C}_z\) but neither each \(\partial D(t)\) is smooth nor the variation is smooth for \(t\in B\).

MSC:

30C85 Capacity and harmonic measure in the complex plane
30F15 Harmonic functions on Riemann surfaces
31C10 Pluriharmonic and plurisubharmonic functions
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References:

[1] S. Hamano, Variation formulas for \(L_1\)-principal functions and application to the simultaneous uniformization problem, Michigan Math. J., 60 (2011), 271-288. · Zbl 1235.30028
[2] S. Hamano, Variation formulas for principal functions and harmonic spans, RIMS Kôkyûroku, 1694 (2010), 121-130.
[3] S. Hamano, Variation formulas for principal functions, III: Applications to variation for Schiffer spans (submitted).
[4] S. Hamano, F. Maitani and H. Yamaguchi, Variation formulas for principal functions, II: Applications to variation for harmonic spans, Nagoya Math. J., 204 (2011), 19-56. · Zbl 1234.32008
[5] F. Maitani and H. Yamaguchi, Variation of Bergman metrics on Riemann surfaces, Math. Ann., 330 (2004), 477-489. · Zbl 1077.32006
[6] L. Sario and M. Nakai, Classification theory of Riemann surfaces, Grundlehren Math. Wiss., 164 , Springer-Verlag, 1970. · Zbl 0199.40603
[7] H. Yamaguchi, Variations of pseudoconvex domains over \(\mathbb{C}^n\), Michigan Math. J., 36 (1989), 415-457. · Zbl 0692.31004
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