Real and discrete holomorphy : Introduction to an algebraic approach

Sylvain Barré (1) , Abdelghani Zeghib (2)

1 LMAM - Laboratoire de Mathématiques et Applications des Mathématiques, EA 3885
2 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We consider spaces for which there is a notion of harmonicity for complex valued functions defined on them. For instance, this is the case of Riemannian manifolds on one hand, and (metric) graphs on the other hand. We observe that it is then possible to define an "amazing" notion of holomorphic functions on them, and show how rigid it is in some cases.

Mots clés: holomorphic tree harmonic

Domaines: Géométrie métrique [math.MG]; Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]; Variables complexes [math.CV]

Sylvain Barré, Abdelghani Zeghib. Real and discrete holomorphy : Introduction to an algebraic approach. 2007. ⟨hal-00002224v2⟩