$C^1$ subharmonicity of harmonic spans for certain discontinuously moving Riemann surfaces

Abstract

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 $t\in B$.