  We consider the global bifurcation problem for spatially periodic traveling
waves for two-dimensional gravity-capillary vortex sheets. The two fluids have
arbitrary constant, non-negative densities (not both zero), the gravity
parameter can be positive, negative, or zero, and the surface tension parameter
is positive. Thus, included in the parameter set are the cases of pure
capillary water waves and gravity-capillary water waves. Our choice of
coordinates allows for the possibility that the fluid interface is not a graph
over the horizontal. We use a technical reformulation which converts the
traveling wave equations into a system of the form "identity plus compact."
Rabinowitz' global bifurcation theorem is applied and the final conclusion is
the existence of either a closed loop of solutions, or an unbounded set of
nontrivial traveling wave solutions which contains waves which may move
arbitrarily fast, become arbitrarily long, form singularities in the vorticity
or curvature, or whose interfaces self-intersect.
