Local Flexibility for Open Partial Differential Relations

Abstract

AbstractWe show that local deformations, near closed subsets, of solutions to open partial differential relations can be extended to global deformations, provided all but the highest derivatives stay constant along the subset. The applicability of this general result is illustrated by a number of examples, dealing with convex embeddings of hypersurfaces, differential forms, and lapse functions in Lorentzian geometry.The main application is a general approximation result by sections that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K ∈ ℝ every manifold of dimension at least 2 carries a complete C1, 1‐metric which, on a dense open subset, is smooth with constant sectional curvature K. Of course, this is impossible for C2‐metrics in general. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.