SCALING LAWS FOR NON-EUCLIDEAN PLATES AND THE W2,2 ISOMETRIC 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 Gamma-convergence under the proper scaling. As a corollary, we obtain new necessary and sufficient conditions for existence of a W-2,W-2 isometric immersion of a given 2d metric into R-3.