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Kurdyka–Łojasiewicz functions and mapping cylinder neighborhoods

Cibotaru, Daniel; Galaz-García, Fernando


Daniel Cibotaru


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.

Journal Article Type Article
Acceptance Date Sep 23, 2022
Deposit Date Sep 26, 2022
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
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