Rotational Ricci surfaces

Abstract

We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature K satisfies

$$\begin{aligned} K\Delta K - \Vert \nabla K\Vert ^2-4K^3 = 0. \end{aligned}$$

These surfaces are referred to as rotational Ricci surfaces. As an application, we show that there is a one-parameter family of such surfaces meeting the boundary of the unit Euclidean three-ball orthogonally. In addition, we show that this family interpolates a vertical geodesic and the critical catenoid.