Rigidity for capillary liquid drops of nearly circular section with constant vorticity

We consider time-independent solutions with constant vorticity of the free boundary Euler equations for a 3D liquid drop with capillarity.

A rigidity result for the solutions of this problem has been recently proved with variational methods: if a certain quantity involving the vorticity parameter, the capillarity coefficient and the area of the equatorial section of the drop is below a certain value, then the solution has necessarily cylindrical symmetry, the shape of the drop is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity.

In this paper we develop a perturbation analysis of the problem for fluid domains whose equatorial section is close in C2 norm to a disc, and we show that a rigidity result holds also above the threshold obtained with variational methods.