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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 \(V\leq\varphi\).

MSC:

32U05 Plurisubharmonic functions and generalizations
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
32A10 Holomorphic functions of several complex variables

Citations:

Zbl 0732.00007
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