  We study steady axisymmetric water waves with general vorticity and swirl,
subject to the influence of surface tension. Explicit solutions to such a water
wave problem are static configurations where the surface is an unduloid, that
is, a periodic surface of revolution with constant mean curvature. We prove
that to any such configuration there connects a global continuum of non-static
solutions by means of a global implicit function theorem. To prove this, the
key is strict monotonicity of a certain function describing the mean curvature
of an unduloid and involving complete elliptic integrals. From this point of
view, this paper is an interesting interplay between water waves, geometry, and
properties of elliptic integrals.
