A Whitney extension problem for manifolds
The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set E⊂Rn, how can we tell if there exists a d-dimensional, Cm-smooth manifold M⊃E? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like...
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Veröffentlicht in: | Journal of functional analysis 2025-03, Vol.288 (5), p.110753, Article 110753 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | The purpose of this paper is to address a manifold-based version of Whitney's extension problem: Given a compact set E⊂Rn, how can we tell if there exists a d-dimensional, Cm-smooth manifold M⊃E? We provide an answer for compact manifolds with boundary in terms of a Glaeser refinement much like that used in the solution of the classical Whitney extension problem and a topological condition. This condition is the existence of a continuous selection for Grassmannian-valued functions, meant to reflect the collection of possible tangent spaces. We demonstrate the necessity of this condition in general and its non-redundancy in an example, while also showing it need not be checked when d=1. |
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ISSN: | 0022-1236 |
DOI: | 10.1016/j.jfa.2024.110753 |