  In this paper, we prove global regularity, scattering, and the non-existence
of small traveling waves for the $3D$ finite depth capillary waves system for
small initial data. The non-existence of small traveling waves shows a
fundamental difference between the capillary waves ($\sigma=1, g=0$) and the
gravity-capillary waves ($\sigma=1,$ $ 0< g< 3$) in the finite depth setting.
As, for the later case, there exists arbitrary small $L^2$ traveling waves.
  Different from the water waves system in the infinite depth setting, the
quadratic terms of the same system in the finite depth setting are worse due to
the absence of null structure inside the Dirichlet-Neumann operator. In the
finite depth setting, the capillary waves system has the worst quadratic terms
among the water waves systems with all possible values of gravity effect
constant and surface tension coefficient. It loses favorable cancellations not
only in the High $\times$ Low type interaction but also in the High $\times $
High type interaction.
  In the worst scenario, the best decay rate of the nonlinear solution that one
could expect is $ (1+t) ^ {-1/2} $, because the $3D$ finite depth capillary
waves system lacks null structures and there exists $Q(u, \bar{u})$ type
quadratic term, which causes a very large time resonance set and the definite
growth of the associated profile. As a result, the problematic terms are not
only the quadratic terms but also the cubic terms.
  To prove global regularity for the capillary waves system, we identify a good
variable to prove the dispersion estimate, fully exploit the hidden structures
inside the capillary waves system, and use a novel method to control the
weighted norms.
