Ricci surfaces

TL;DR

This paper studies Ricci surfaces, a special class of 2D Riemannian manifolds with a specific curvature condition, proving local isometric embeddings into Euclidean or Minkowski space and constructing examples with singularities across all genera.

Contribution

It generalizes Ricci's classical result by showing Ricci surfaces can be locally embedded as minimal or maximal surfaces, including at points where curvature vanishes.

Findings

Ricci surfaces can be locally embedded minimally in 3 or maximally in  space.

The theory of closed Ricci surfaces with conical singularities is developed.

Explicit examples of Ricci surfaces are constructed for all genera  and higher.

Abstract

A Ricci surface is a Riemannian 2-manifold $(M,g)$ whose Gaussian curvature $K$ satisfies $K\Delta K+g(dK,dK)+4K^3=0$. 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 $g\geq 2$.