  We study the 2D Ginzburg-Landau theory for a type-II superconductor in an
applied magnetic field varying between the second and third critical value. In
this regime the order parameter minimizing the GL energy is concentrated along
the boundary of the sample and is well approximated to leading order by a
simplified 1D profile in the direction perpendicular to the boundary. Motivated
by a conjecture of Xing-Bin Pan, we address the question of whether this
approximation can hold uniformly in the boundary region. We prove that this is
indeed the case as a corollary of a refined, second order energy expansion
including contributions due to the curvature of the sample. Local variations of
the GL order parameter are controlled by the second order term of this energy
expansion, which allows us to prove the desired uniformity of the surface
superconductivity layer.
