  We are concerned with the dimension reduction analysis for thin
three-dimensional elastic films, prestrained via Riemannian metrics with weak
curvatures.
  For the prestrain inducing the incompatible version of the F\"oppl-von
K\'arm\'an equations, we find the $\Gamma$-limits of the rescaled energies,
identify the optimal energy scaling laws, and display the equivalent conditions
for optimality in terms of both the prestrain components and the curvatures of
the related Riemannian metrics. When the stretching-inducing prestrain carries
no in-plane modes, we discover similarities with the previously described
shallow shell models.
  In higher prestrain regimes, we prove new energy upper bounds by constructing
deformations as the Kirchhoff-Love extensions of the highly perturbative,
H\"older-regular solutions to the Monge-Ampere equation obtained by means of
convex integration.
