Local flexibility for open partial differential relations
- 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 that have very restrictive local properties on open dense subsets. This shows, for instance, that given any K is an element of Double-struck capital R every manifold of dimension at least 2 carries a complete C-1,C- 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.
Author details: | Christian BärORCiDGND, Bernhard HankeORCiDGND |
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DOI: | https://doi.org/10.1002/cpa.21982 |
ISSN: | 0010-3640 |
ISSN: | 1097-0312 |
Title of parent work (English): | Communications on pure and applied mathematics / issued by the Courant Institute of Mathematical Sciences, New York Univ. |
Publisher: | Wiley |
Place of publishing: | Hoboken |
Publication type: | Article |
Language: | English |
Date of first publication: | 2021/02/21 |
Publication year: | 2022 |
Release date: | 2023/10/02 |
Volume: | 75 |
Issue: | 6 |
Number of pages: | 39 |
First page: | 1377 |
Last Page: | 1415 |
Funding institution: | Deutsche Forschungsgemeinschaft German Research Foundation (DFG) [SPP 2026] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Hybrid Open-Access |
License (German): | CC-BY - Namensnennung 4.0 International |