Variation formulas for principal functions (II) Applications to variation for harmonic spans

TL;DR

This paper derives variation formulas for harmonic spans in complex domains and demonstrates their subharmonicity under pseudoconvexity, linking conformal mappings with geometric properties like the Poincaré distance.

Contribution

It extends second-order variation formulas to the $L_0$-constant and establishes subharmonicity of harmonic spans in pseudoconvex families of domains.

Findings

The harmonic span $s(t)$ is subharmonic on $B$ when the total space is pseudoconvex.

The variation formula for the $L_0$-constant is derived and combined with existing formulas.

The subharmonicity of $ ext{log} ext{cosh} d(t)$ is proved, relating to the Poincaré distance.

Abstract

For a domain $D$ in $\mathbb{C}_z$ with smooth boundary and for $a,b\in D, a\ne b$, we have the circular (radial) slit mapping $P(z)(Q(z))$ on $D$ such that $P(z)- \frac{1}{z-a}\ (Q(z)- \frac{1}{z-a})$ is regular at $a$ and $P(b)(Q(b))=0$, and we call $p(z)=\log |P(z)|\ (q(z)=\log|Q(z)|)$ the $L_1$-($L_0$-)principal function; \ $\alpha =\log|P'(b)|$ $(\beta =\log|Q'(b)|)$ the $L_1$-($L_0$-)constant, and \ $s=\alpha - \beta $ the harmonic span, for $D$. S.\,Hamano in \cite{hamano-2} showed the variation formula of the second order for the $L_1$-const. $\alpha (t)$ for the moving domain $D(t)$ in $\mathbb{C}_z$ with $t \in B:=\{t\in \mathbb{C}: |t|<\rho\}$. We show the corresponding formula for the $L_0$-const. $\beta (t)$ for $D(t)$, and combine these formulas to obtain, if the total space ${\mathcal D}=\cup_{t\in B}(t, D(t)) $ is pseudoconvex in $ B \times \mathbb{C}_z$, then $s(t)$ is subharmonic on $B$. Since the geometric meaning of $s(t)$ is showed, this fact gives one of the relations between the conformal mappings on each fiber $D(t), t\in B$ and the pseudoconvexity of ${\mathcal D}$. As a simple application we obtain the subharmonicity of $\log \cosh d(t)$ on $B$, where $d(t)$ is the Poincar\'e distance between $a$ and $b$.