  We consider the Ginzburg-Landau heat flow without magnetic effect in a curved
thin domain under the Naumann boundary condition. When the curved thin domain
shrinks to a given closed hypersurface as the thickness of the thin domain
tends to zero, we show that the weighted average of a weak solution to the
thin-domain problem converges weakly on the limit surface under the assumption
that the initial data is of class $L^\infty$ and satisfies some conditions.
Moreover, under the same assumption, we derive a limit equation by
characterizing the limit function as a weak solution, and prove a difference
estimate on the limit surface of an averaged weak solution to the thin-domain
problem and a weak solution to the limit problem explicitly in terms of the
thickness of the thin domain. We also derive a difference estimate in the
curved thin domain of weak solutions to the thin-domain problem and to the
limit problem, but without requiring that the initial data of the thin-domain
problem is of class $L^\infty$.
