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January, 2013 $C^1$ subharmonicity of harmonic spans for certain discontinuously moving Riemann surfaces
Sachiko HAMANO
J. Math. Soc. Japan 65(1): 321-341 (January, 2013). DOI: 10.2969/jmsj/06510321


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 $C_j(t)$ $(j=1\ldots,\nu)$ 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 $t\in B$.


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Sachiko HAMANO. "$C^1$ subharmonicity of harmonic spans for certain discontinuously moving Riemann surfaces." J. Math. Soc. Japan 65 (1) 321 - 341, January, 2013.


Published: January, 2013
First available in Project Euclid: 24 January 2013

zbMATH: 1294.30051
MathSciNet: MR3034407
Digital Object Identifier: 10.2969/jmsj/06510321

Primary: 30C85
Secondary: 30F15 , 31C10

Keywords: principal function , pseudoconvexity , Riemann surface , span

Rights: Copyright © 2013 Mathematical Society of Japan


Vol.65 • No. 1 • January, 2013
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