Abstract
We propose to view hermitian metrics on trivial holomorphic vector bundles E → Ω as noncommutative analogs of functions defined on the base Ω, and curvature as the notion corresponding to the Laplace operator or ∂∂̅. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.
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Lempert, L. Noncommutative potential theory. Anal Math 43, 603–627 (2017). https://doi.org/10.1007/s10476-017-0505-x
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DOI: https://doi.org/10.1007/s10476-017-0505-x