Kurdyka–Łojasiewicz Functions and Mapping Cylinder Neighborhoods
[Fonctions de Kurdyka–Łojasiewicz et voisinages de cylindres]
Cibotaru, Daniel1 ; Galaz-García, Fernando2
1 Departamento de Matemática, Universidade Federal do Ceará, Fortaleza (Brazil)
2 Department of Mathematical Sciences, Durham University (United Kingdom)
Annales de l'Institut Fourier, Online first, 32 p.

Les fonctions de Kurdyka–Łojasiewicz (KŁ) sont des fonctions à valeurs réelles caractérisées par une inégalité différentielle faisant intervenir la norme de leur gradient. Cette classe de fonctions est assez riche, contenant des objets aussi divers que des fonctions sous-analytiques, transnormales ou de Morse. Nous prouvons que l’ensemble des zéros d’une fonction de Kurdyka–Łojasiewicz admet un voisinage de cylindre. Cela implique, en particulier, que les 2-variétés topologiques sauvagement plongées dans l’espace euclidien de 3-dimensions, telles que les sphères cornues d’Alexander, n’apparaissent pas comme les ensembles des zéros des fonctions KŁ.

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.

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DOI : 10.5802/aif.3656
Classification : 57R99
Keywords: Mapping cylinder neighborhood, relative manifold, Kurdyka–Łojasiewicz function, Alexander horned sphere.
Mot clés : Voisinage cylindrique, varieté relative, fonction de Kurdyka–Łojasiewicz, sphéres cornue d’Alexander.

Cibotaru, Daniel 1 ; Galaz-García, Fernando 2

1 Departamento de Matemática, Universidade Federal do Ceará, Fortaleza (Brazil)
2 Department of Mathematical Sciences, Durham University (United Kingdom) 
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Cibotaru, Daniel; Galaz-García, Fernando. Kurdyka–Łojasiewicz Functions and Mapping Cylinder Neighborhoods. Annales de l'Institut Fourier, Online first, 32 p.

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