A monotonicity result for volumes of holomorphic images
We quote from the introduction: ``This paper studies the Euclidean \(2k\)-dimensional volumes of parametrized holomorphic images of certain pseudoconvex domains in complex Euclidean space \(\mathbb{C}^{k}\). Its motivation arises from studying a more specific situation -- namely, the classification problem for proper holomorphic mappings between balls in complex Euclidean spaces (usually of different dimensions). We provide two versions of a monotonicity result. The first applies for mappings from balls and eggs; its proof involves the computation of explicit integrals. The second applies for mappings from more general domains, but requires some regularity assumptions on the mappings at the boundary. By way of introduction, first suppose that \(k=1\), that \(f\) is holomorphic on the closed unit disc \(B_{1}\), and that \(f^{\prime}\) is square integrable. Then the area of the parametrized image of the unit disc is \(\pi\|f^{\prime}\|_{L^{2}}^{2}\). In particular, when \(f(z)=z^{m}\), the area (with multiplicity taken into account) is \(m\pi\). Furthermore, one can easily see that \(\|(zf)^{\prime}\|_{L^{2}}^{2} \geq \|f^{\prime}\|_{L^{2}}^{2}\), with strict inequality unless \(f=0\), so the area is increased when \(f\) is replaced by \(zf\). In this paper, we generalize these results to the more complicated situation in higher dimensions and show how they fit into the classification problem.''
scientific article; zbMATH DE number 5373795