  We consider the Neumann type problem of the heat equation in a moving thin
domain around a given closed moving hypersurface. The main result of this paper
is an error estimate in the sup-norm for classical solutions to the thin domain
problem and a limit equation on the moving hypersurface which appears in the
thin-film limit of the heat equation. To prove the error estimate, we show a
uniform a priori estimate for a classical solution to the thin domain problem
based on the maximum principle. Moreover, we construct a suitable approximate
solution to the thin domain problem from a classical solution to the limit
equation based on an asymptotic expansion of the thin domain problem and apply
the uniform a priori estimate to the difference of the approximate solution and
a classical solution to the thin domain problem.
