Given $a,b\in\mathbb{ R}$ and $\Phi\in C^1(\mathbb{ S}^2)$, we study immersed oriented surfaces $\Sigma$ in the Euclidean 3-space $\mathbb{ R}^3$ whose mean curvature $H$ and Gauss curvature $K$ satisfy $2aH+bK=\Phi(N)$, where $N:\Sigma\rightarrow\mathbb{ S}^2$ is the Gauss map. This theory widely generalize some of paramount importance such as the ones constant mean and Gauss curvature surfaces, linear Weingarten surfaces and self-translating solitons of the mean curvature flow. Under mild assumptions on the prescribed function $\Phi$, we exhibit a classification result for rotational surfaces in the case that the underlying fully nonlinear PDE that governs these surfaces is elliptic or hyperbolic.
Categories: math.DG