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

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; \ $\alpha =\log |P^{\prime }(b)|$ $(\beta =\log |Q^{\prime }(b)|)$ 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 Poincaré distance between $a$ and $b$.

arXiv comment30 pages, 4 figures, MSJ Autumn meeting in Nagoya(Japan) Sep. 20-23, 2010