  In this paper we prove global regularity for the full water waves system in 3
dimensions for small data, under the influence of both gravity and surface
tension. This problem presents essential difficulties which were absent in all
of the earlier global regularity results for other water wave models. To
construct global solutions we use a combination of energy estimates and
matching dispersive estimates. There is a significant new difficulty in proving
energy estimates in our problem, namely the combination of slow pointwise decay
of solutions (no better than $|t|^{-5/6}$) and the presence of a large,
codimension 1, set of quadratic time-resonances. To deal with such a situation
we propose here a new mechanism, which exploits a non-degeneracy property of
the time-resonant hypersurfaces and some special structure of the quadratic
part of the nonlinearity, connected to the conserved energy of the system. The
dispersive estimates rely on analysis of the Duhamel formula in the Fourier
space. The main contributions come from the set of space-time resonances, which
is a large set of dimension 1. To control the corresponding bilinear
interactions we use Harmonic Analysis techniques, such as orthogonality
arguments in the Fourier space and atomic decompositions of functions. Most
importantly, we construct and use a refined norm which is well adapted to the
geometry of the problem.
