Scaling laws for non-Euclidean plates and theW2,2isometric immersions of Riemannian metrics

Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ -convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W 2,2 isometric immersion of a given 2 d metric into .