Poletsky, Evgeny A. Plurisubharmonic functions as solutions of variational problems. (English) Zbl 0739.32015 Several complex variables and complex geometry, Proc. Summer Res. Inst., Santa Cruz/CA (USA) 1989, Proc. Symp. Pure Math. 52, Part 1, 163-171 (1991). [For the entire collection see Zbl 0732.00007.]Let \(A(\overline U,D)\) be the space of all mappings of the unit disk \(U\) into a domain \(D\subset\mathbb{C}^ n\) that are holomorphic in a neighbourhood of \(\overline U\). A notion of holomorphic current is introduced as a mapping of \(A(\overline U,D)\) into the set of subharmonic functions in \(U\), which is invariant under proper holomorphic changes of coordinates. Some variational problems for holomorphic currents are studied. In particular, the following result is obtained. For an upper semicontinuous function \(\varphi\) on \(D\) the function \[ u(z)\overset\text{def}= \inf\left\{(1/2\pi)\int_ 0^{2\pi}\varphi(f(e^{i\theta}))d\theta:\;f\in A(\overline U, D),\;f(0)=z\right\} \] is plurisubharmonic in \(D\); furthermore \(u(z)\) is the supremum of plurisubharmonic functions \(V\leq\varphi\). Reviewer: A.Yu.Rashkovsky (Khar’kov) Cited in 2 ReviewsCited in 27 Documents MSC: 32U05 Plurisubharmonic functions and generalizations 31C05 Harmonic, subharmonic, superharmonic functions on other spaces 32A10 Holomorphic functions of several complex variables Keywords:plurisubharmonic function; holomorphic mapping Citations:Zbl 0732.00007 PDFBibTeX XMLCite \textit{E. A. Poletsky}, Proc. Symp. Pure Math. None, 163--171 (1991; Zbl 0739.32015)