Ricci Surfaces

A Ricci surface is a Riemannian 2-manifold (M, g) whose Gaussian curvature K satisfies K∆K+g(dK, dK)+4K = 0. Every minimal surface isometrically embedded in R is a Ricci surface of non-positive curvature. At the end of the 19 century Ricci-Curbastro has proved that conversely, every point x of a Ricci surface has a neighborhood which embeds isometrically in R as a minimal surface, provided K(x) < 0. We prove this result in full generality by showing that Ricci surfaces can be locally isometrically embedded either minimally in R or maximally in R, including near points of vanishing curvature. We then develop the theory of closed Ricci surfaces, possibly with conical singularities, and construct classes of examples in all genera g ≥ 2.