  In this paper we consider three-dimensional steady water waves with
vorticity, under the action of gravity and surface tension; in particular we
consider so-called Beltrami flows, for which the velocity field and the
vorticity are collinear. We discuss a recent variational formulation of the
problem which involves a generalisation of the classical Dirichlet-Neumann
operator. We study this operator in detail, extending some well-known results
for the classical Dirichlet-Neumann operator, such as the Taylor expansion in
homogeneous powers of the wave profile, the computation of its differential and
the asymptotic expansion of its associated symbol. A new formulation of the
problem as a single equation for the wave profile is also presented and
discussed in a similar vein. As an application of these results we rigorously
prove existence of doubly periodic gravity-capillary steady waves and construct
approximate doubly periodic gravity steady waves.
