Noncommutative potential theory

Abstract

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.