  In the series of this paper and the forthcoming papers [41,42] we study the
Navier-Stokes equations in a three-dimensional curved thin domain around a
given closed surface under Navier's slip boundary conditions. We focus on the
study of the Stokes operator for the curved thin domain in this paper. The
uniform norm equivalence for the Stokes operator and a uniform difference
estimate for the Stokes and Laplace operators are established in which
constants are independent of the thickness of the curved thin domain. To prove
these results we show a uniform Korn inequality and a uniform a priori estimate
for the vector Laplace operator on the curved thin domain based on a careful
analysis of vector fields and surface quantities on the boundary. We also
present examples of curved thin domains and vector fields for which the uniform
Korn inequality is not valid but a standard Korn inequality holds with a
constant that blows up as the thickness of a thin domain tends to zero.
