We classify rotational surfaces in the three-dimensional Euclidean space whose Gaussian curvature $K$ satisfies \begin{equation*} K\Delta K - \|\nabla K\|^2-4K^3 = 0. \end{equation*} These surfaces are referred to as rotational Ricci surfaces. As an application, we show that there is a one-parameter family of such surfaces that is free boundary in the unit Euclidean three-ball. In addition, this family interpolates a vertical geodesic and the critical catenoid.PDF: Rotational Ricci surfaces.pdf
