  In this paper, we use frequency decomposition techniques to give a direct
proof of global existence and regularity for the Navier-Stokes equations on
two-dimensional Riemannian manifolds without boundary. Our techniques are
inspired by an approach of Mattingly and Sinai [15] which was developed in the
context of periodic boundary conditions on a flat background, and which is
based on a maximum principle for Fourier coefficients.
  The extension to general manifolds requires several new ideas, connected to
the less favorable spectral localization properties in our setting. Our
arguments make use of frequency projection operators, multilinear estimates
that originated in the study of the non-linear Schr\"odinger equation, and
ideas from microlocal analysis.
