Rotating drops with helicoidal symmetry

Bennett Palmer and Oscar M. Perdomo

Vol. 273 (2015), No. 2, 413–441

DOI: 10.2140/pjm.2015.273.413

Abstract

It is known that, if we ignore gravitational forces, the shape of an equilibrium drop in ${ℝ}^3$ rotating about the $z$-axis is a surface that satisfies the equation $2H={\Lambda }_0-\frac{1}{2}a{R}^2$, where $H$ is the mean curvature and $R$ is the distance from a point in the surface to the $z$-axis. We consider helicoidal immersions in ${ℝ}^3$ that satisfy the rotating drop equation. We prove the existence of properly immersed solutions that contain the $z$-axis. We also show the existence of several families of embedded examples. We describe the set of possible solutions and we show that most of these solutions are not properly immersed and are dense in the region bounded by two concentric cylinders. We show that all properly immersed solutions, besides being invariant under a one-parameter helicoidal group, are invariant under a cyclic group of rotations of the variables $x$ and $y$.

The second variation of energy for the volume constrained problem with Dirichlet boundary conditions is also studied.

Keywords

rotating drops, mean curvature, helicoidal surfaces

Mathematical Subject Classification 2000

Primary: 53C43, 53C42, 53C10

Milestones

Received: 4 March 2014

Revised: 21 May 2014

Accepted: 23 May 2014

Published: 23 December 2014