  We construct global curves of rotational traveling wave solutions to the $2D$
water wave equations on a compact domain. The real analytic interface is
subject to surface tension, while gravitational effects are ignored. In
contrast to the rotational surface waves, the fluid flow follows the
incompressible, irrotational Euler equations. This model can provide a
description for tiny water droplets in breaking waves and white caps. The
primary tool we use is global bifurcation theory, via a conformal formulation
of the problem. The obtained fluid domains have $m$-fold discrete rotational
symmetry, as well as a reflection symmetry.
