Ricci surfaces

A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\mathrm{\Delta }K+g(dK,dK)+4K^3=0$. Every minimal surface isometrically embedded in ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$ or maximally in ${\mathbb{R}}^{2,1}$, 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 $g\ge 2$.