  We consider the Navier-Stokes equations in a three-dimensional curved thin
domain around a given closed surface under Navier's slip boundary conditions.
When the thickness of the thin domain is sufficiently small, we establish the
global existence of a strong solution for large data. We also show several
estimates for the strong solution with constants explicitly depending on the
thickness of the thin domain. The proofs of these results are based on a
standard energy method and a good product estimate for the convection and
viscous terms following from a detailed study of average operators in the thin
direction. We use the average operators to decompose a three-dimensional vector
field on the thin domain into the almost two-dimensional average part and the
residual part, and derive good estimates for them which play an important role
in the proof of the product estimate.
