Local Flexibility for Open Partial Differential Relations

TL;DR

This paper proves that local deformations of solutions to open partial differential relations can be extended globally under certain conditions, enabling new approximation results for metrics with prescribed curvature properties.

Contribution

It introduces a general method for extending local deformations to global solutions in PDE relations, with applications in geometry and metric approximation.

Findings

Global deformations can be constructed from local ones near closed subsets.

Manifolds can admit dense open subsets with highly restricted local properties.

Existence of complete $C^{1,1}$-metrics with prescribed curvature on dense subsets.

Abstract

We 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 which have very restrictive local properties an open dense subsets. This shows, for instance, that given any $K \in \mathbb{R}$ every manifold of dimension at least two carries a complete $C^{1,1}$-metric which, on a dense open subset, is smooth with constant sectional curvature $K$. Of course this is impossible for $C^2$-metrics in general.