Variations of pseudoconvex domains
A connected Hausdorff space E together with a locally homeomorphic map \(\pi\) of E to \({\mathbb{R}}^ m\) is called an unramified covering domain over the space \({\mathbb{R}}^ m\). Suppose D is such a domain and \(D_ p\) is a sequence of relatively compact subdomains of D such that \(x^ 0\in D_ 1\), \(D_ p\subset D_{p+1}\), \(\cup^{\infty}_{p=1}D_ p=D\), and the boundaries \(\partial D_ p\) of \(D_ p\) in D are real-analytic. By means of the Green functions \(g_ p(x)\) with pole at \(x^ 0\), carried by \(D_ p\), we can introduce \(g(x)=\lim_{p\to \infty}g_ p(x)\), which will be the Green function of D with pole \(x^ 0\). Using the expansion of g, we can also introduce the Robin constant \(\lambda\). Now consider an unramified covering domain \({\mathcal D}\) over \(\Delta \times {\mathbb{C}}^ n\), where \(\Delta\) is the unit disc with center at the origin of the complex plane. Regard the set \({\mathcal D}(t)={\mathcal D}\cap (\{t\}\times {\mathbb{C}}^ n)\), which we call the fiber of \({\mathcal D}\) at \(t\in \Delta\), as a domain of dimension n with parameter \(t\in \Delta\), and write \({\mathcal D}: t\mapsto {\mathcal D}(t)\). A holomorphic map \(\alpha\) of \(\Delta\) into \({\mathcal D}\) such that \(\alpha\) (t)\(\in {\mathcal D}(t)\) for all \(t\in \Delta\) is called a holomorphic section of \({\mathcal D}\) on \(\Delta\). Consider \({\mathcal D}(t)\) as a domain over \({\mathbb{R}}^{2n}\); then we have the Green function g(t,z) of \({\mathcal D}(t)\) with pole \(\alpha\) (t). Thus the Robin constant \(\lambda\) (t) of \({\mathcal D}(t)\) with respect to \(\alpha\) (t) defines a real-valued function. The main result of the author is the following theorem: If \({\mathcal D}\) is a pseudoconvex domain of dimension \(n+1\), then \(\lambda\) (t) is superharmonic on \(\Delta\). Moreover, log(-\(\lambda\) (t)) is subharmonic on \(\Delta\) in the case \(n\geq 2\).
scientific article; zbMATH DE number 3925259