Real and discrete holomorphy: Introduction to an algebraic approach. (English) Zbl 1146.31007

Barré, Sylvain; Zeghib, Abdelghani

J. Math. Pures Appl. (9) 87, No. 5, 495-513 (2007).

A complex-valued (even more generally a ring-valued) function \(\varphi\) defined on a Riemannian manifold or a graph \(X\) is here said to be holomorphic if \(\varphi\) and \(\varphi^2\) are harmonic on \(X\). Some properties of such holomorphic functions (such as holomorphic extensions, holomorphic morphisms, holomorphic classification etc.) are considered extensively on the bi-infinite tree of valency 3, with indications of extensions to other types of graphs. The question of finding a real function \(g\) on \(X\) associated to a known real function \(f\) on \(X\) so that \(\varphi= f+ ig\) is holomorphic on \(X\) is considered, with examples of existence and non-existence of this conjugate function \(g\).

Reviewer: Victor Anandam (Kerala)

MSC:

31C05

Harmonic, subharmonic, superharmonic functions on other spaces

Keywords:

holomorphy on graphs and Riemannian manifolds; harmonic morphisms

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