  This paper concerns the variational description of prestrained materials, in
the context of dimension reduction for thin films $\Omega^h=\omega\times
(-\frac{h}{2}, \frac{h}{2})$. Given a Riemann metric $G$ on $\Omega^1$, we
study the question of what is the infimum of the averaged pointwise deficit of
a given immersion from being an orientation-preserving isometric immersion of
$G_{\mid \Omega^h}$ on $\Omega^h,$ over all weakly regular immersions. This
deficit is measured by the non-Euclidean energies $\mathcal{E}^h$, which can be
seen as modifications of the classical nonlinear three-dimensional elasticity.
  Building on our previous results, we complete the scaling analysis of
$\mathcal{E}^h$ and the derivation of $\Gamma$-limits of the scaled energies
$h^{-2n}\mathcal{E}^h$, for all $n\geq 1$. We show the energy quantisation in
the sense that the even powers $2n$ of $h$ are indeed the only possible ones
(all of them are also attained). For each $n$, we identify the equivalent
conditions for the validity of the corresponding scaling, in terms of the
vanishing of appropriate Riemann curvatures of $G$ to certain orders, and in
terms of the matched isometry expansions. We also establish the asymptotic
behaviour of the minimizing immersions as $h\to 0$.
