Noncommutative potential theory

TL;DR

This paper develops a noncommutative potential theory by viewing hermitian metrics as analogs of functions and curvature as a Laplace-like operator, extending classical results to a noncommutative setting.

Contribution

It introduces a framework for noncommutative potential theory, generalizing classical potential theory results to hermitian metrics on vector bundles.

Findings

Established noncommutative mean value properties

Proved maximum principle and Harnack inequality in the noncommutative context

Demonstrated solvability of Dirichlet problems for noncommutative potentials

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.