Fine Level Set Structure of Flat Isometric Immersions

A result by Pogorelov asserts that C-1 isometric immersions u of a bounded domain S subset of R-2 into R-3 whose normal takes values in a set of zero area enjoy the following regularity property: the gradient f := del u is 'developable' in the sense that the nondegenerate level sets of f consist of straight line segments intersecting the boundary of S at both endpoints. Motivated by applications in nonlinear elasticity, we study the level set structure of such f when S is an arbitrary bounded Lipschitz domain. We show that f can be approximated by uniformly bounded maps with a simplified level set structure. We also show that the domain S can be decomposed (up to a controlled remainder) into finitely many subdomains, each of which admits a global line of curvature parametrization.