Extending Whitney's extension theorem: nonlinear function spaces
Abstract
We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains , with non-smooth boundary, in possibly non-compact manifolds. Assuming is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.
Cite
@article{arxiv.1801.04126,
title = {Extending Whitney's extension theorem: nonlinear function spaces},
author = {David Michael Roberts and Alexander Schmeding},
journal= {arXiv preprint arXiv:1801.04126},
year = {2022}
}
Comments
37 pages, 1 colour figure. v2 small edits, correction to Definition A.3, which makes no impact on proofs or results. Version submitted for publication. v3 small changes in response to referee comments, title extended. v4 crucial gap filled, results not affected. v5 final version to appear in Annales de l'Institut Fourier