  In Ginzburg-Landau theory, a strong magnetic field is responsible for the
breakdown of superconductivity. This work is concerned with the identification
of the region where superconductivity persists, in a thin shell superconductor
modeled by a compact surface $\mathcal M\subset\mathbb R^3$, as the intensity
$h$ of the external magnetic field is raised above $H_{c1}$. Using a mean field
reduction approach devised by Sandier and Serfaty as the Ginzburg-Landau
parameter $\kappa$ goes to infinity, we are led to studying a two-sided
obstacle problem. We show that superconductivity survives in a neighborhood of
size $(H_{c1}/h)^{1/3}$ of the zero locus of the normal component $H$ of the
field. We also describe intermediate regimes, focusing first on a symmetric
model problem. In the general case, we prove that a striking phenomenon we call
freezing of the boundary takes place: one component of the superconductivity
region is insensitive to small changes in the field.
