Generalized Ricci Surfaces

We consider smooth Riemannian surfaces whose curvature K satisfies the relation Δ log | K - c | = a K + b away from points where K = c for some ( a , b , c ) ∈ R 3 , which we call generalized Ricci surfaces. We prove some isometric immersion theorems allowing points where K = c using properties of log-harmonic functions. For instance, we obtain a characterization of Riemannian surfaces that locally admit minimal isometric immersions, possibly with umbilical points, into a 3-dimensional Riemannian manifold of constant sectional curvature. We also give an application to convex affine spheres. Finally, we study compact generalized Ricci surfaces: we obtain topological obstructions and construct examples.