A Ricci surface is a Riemannian 2-manifold (M,g) whose Gaussian curvature
K satisfies KΔK+g(dK,dK)+4K3=0. Every minimal surface isometrically
embedded in R3 is a Ricci surface of non-positive curvature. At the
end of the 19th century Ricci-Curbastro has proved that conversely, every point
x of a Ricci surface has a neighborhood which embeds isometrically in
R3 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 R3 or maximally in R2,1,
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.Comment: 27 pages; final version, to appear in Annali della Scuola Normale
Superiore di Pisa - Classe di Scienz