Discrete Riemann Surfaces

Christian Mercat (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

We detail the theory of Discrete Riemann Surfaces. It takes place on a cellular decomposition of a surface, together with its Poincaré dual, equipped with a discrete conformal structure. A lot of theorems of the continuous theory follow through to the discrete case, we define the discrete analogs of period matrices, Riemann's bilinear relations, exponential of constant argument and series. We present the notion of criticality and its relationship with integrability.

Mots clés: discrete Riemann surfaces; discrete geometry; integrable systems

Domaines: Variables complexes [math.CV]; Géométrie différentielle [math.DG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Christian Mercat. Discrete Riemann Surfaces. Athanase Papadopoulos. Handbook of Teichmüller theory. Vol. I, Eur. Math. Soc., Zürich, pp.541--575, 2007, IRMA Lect. Math. Theor. Phys., 11, 978-3-03719-029-6. ⟨10.4171/029⟩. ⟨hal-00250897⟩