Variation formulas for \(L_{1}\)-principal functions and application to simultaneous uniformization problem (Q2276079)

Let \(\widetilde R= \bigcup_{t\in\Delta}\widetilde R(t)\) be an unramified domain over \(\Delta\times\mathbb{C}\) and \(R= \bigcup_{t\in\Delta} R(t)\) a subdomain with \(\partial R= \bigcup_{t\in\Delta}\partial R(t)\) real analytic in \(\widetilde R\) consisting of \(\nu+1\) components \(C_0,C_1,\dots, C_\nu\). Let \(u(t,z)\) be an appropriately normalized Green function of \(R(t)\) for every \(t\in\Delta\). Write \[ u(t,z)= \ln{1\over|z|}+ \gamma(t)+ h(t,z) \] in a neighborhood of a zero section \(O:\Delta\to R\), which is supposed to exist. The author proves several results, one of them being the following: If \(R\) is pseudoconvex over \(\Delta\times\mathbb{C}\) then \(\gamma\) is a real analytic superharmonic function on \(\Delta\).