  We propose to view hermitian metrics on trivial holomorphic vector bundles
$E\to\Omega$ as noncommutative analogs of functions defined on the base
$\Omega$, and curvature as the notion corresponding to the Laplace operator or
$\partial\overline\partial$. We discuss noncommutative generalizations of basic
results of ordinary potential theory, mean value properties, maximum principle,
Harnack inequality, and the solvability of Dirichlet problems.

Complex Variables;Functional Analysis