Kurdyka–Łojasiewicz functions and mapping cylinder neighborhoods

Cibotaru, Daniel; Galaz-García, Fernando

Abstract

Kurdyka–Łojasiewicz (KŁ) functions are real-valued functions characterized by a differential inequality involving the norm of their gradient. This class of functions is quite rich, containing objects as diverse as subanalytic, transnormal or Morse functions. We prove that the zero locus of a Kurdyka–Łojasiewicz function admits a mapping cylinder neighborhood. This implies, in particular, that wildly embedded topological 2-manifolds in 3-dimensional Euclidean space, such as Alexander horned spheres, do not arise as the zero loci of KŁ functions.

Citation

Cibotaru, D., & Galaz-García, F. (online). Kurdyka–Łojasiewicz functions and mapping cylinder neighborhoods. Annales de l'Institut Fourier, https://doi.org/10.5802/aif.3656

• Journal Article Type: Article
• Acceptance Date: Sep 23, 2022
• Online Publication Date: Jul 3, 2024
• Deposit Date: Sep 26, 2022
• Publicly Available Date: Jul 31, 2024
• Journal: Annales de l'Institut Fourier
• Print ISSN: 0373-0956
• Electronic ISSN: 1777-5310
• Publisher: Association des Annales de l'Institut Fourier
• Peer Reviewed: Peer Reviewed
• DOI: https://doi.org/10.5802/aif.3656
• Public URL: https://durham-repository.worktribe.com/output/1191190