  We consider the Navier-Stokes equations with Navier's slip boundary
conditions in a three-dimensional curved thin domain around a given closed
surface. Under suitable assumptions we show that the average in the thin
direction of a strong solution to the bulk Navier-Stokes equations converges
weakly in appropriate function spaces on the limit surface as the thickness of
the thin domain tends to zero. Moreover, we characterize the limit as a weak
solution to limit equations, which are the damped and weighted Navier-Stokes
equations on the limit surface. We also prove the strong convergence of the
average of a strong solution to the bulk equations towards a weak solution to
the limit equations by showing estimates for the difference between them. In
some special case our limit equations agree with the Navier-Stokes equations on
a Riemannian manifold in which the viscous term contains the Ricci curvature.
This is the first result on a rigorous derivation of the surface Navier-Stokes
equations on a general closed surface by the thin-film limit.
