  H.-O. Bae and H.J. Choe, in a 1997 paper, established a regularity criteria
for the incompressible Navier-Stokes equations in the whole space $\R^3$ based
on two velocity components. Recently, one of the present authors extended this
result to the half-space case $\R^3_+\,.$ Further, this author in collaboration
with J. Bemelmans and J. Brand extended the result to cylindrical domains under
physical slip boundary conditions. In this note we obtain a similar result in
the case of smooth arbitrary boundaries, but under a distinct, apparently very
similar, slip boundary condition. They coincide just on flat portions of the
boundary. Otherwise, a reciprocal reduction between the two results looks not
obvious, as shown in the last section below.
