Rotating drops with helicoidal symmetry (Q2346665)

A surface is a helicoidal rotating drop if it is a helicoidal immersion in \(\mathbb{R}^3\) whose axis of symmetry is the \(z\)-axis and which is a solution of the equation \(2H = \Lambda_0 - a R^2/ 2\), where \(H\) is the mean curvature of the surface, \(R\) is the distance from the point in the surface to the \(z\)-axis and \(a\) is a real number. After describing all helicoidal rotating drops, the existence of properly immersed solutions that contain the \(z\)-axis is proved and several families of embedded and properly immersed examples are given. All properly immersed solutions are invariant under a one-parameter helicoidal group, and under a cyclic group of rotations of the variables \(x\) and \(y\). For a helicoidal surface which is not a round cylinder, a necessary condition for the stability of the part of the surface between horizontal planes separated by a distance \(h\) is provided.