Equilibrium of surfaces in a vertical force field (Q2053766)

The authors investigate surfaces of \(\mathbb{R}^3 \) whose mean curvature \(H\) satisfies \(H=\dot{\varphi}\langle N,e_3\rangle\), where \(N\) is the Gauss maps, \(e_3=(0,0,1)\), \(\varphi=\varphi(z)\) and \(\dot{\varphi}\) is the derivative of \(\varphi\) with respect to the third coordinate of \(\mathbb{R}^3\). Here \((x,y,z)\) are the standard Cartesian coordinates of \(\mathbb{R}^3\). These surfaces are called \(\varphi\)-minimal surfaces. In the paper, there is a full classification of complete flat-embedded \(\varphi\)-minimal surfaces if \(\varphi\) is strictly monotone. Also, they characterize rotational \(\varphi\)-minimal surfaces by its behavior at infinity when \(\varphi\) has a quadratic growth.