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.

Reçu le :
Révisé le :
Accepté le :
Première publication :
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) 
@unpublished{AIF_0__0_0_A111_0,
     author = {Cibotaru, Daniel and Galaz-Garc{\'\i}a, Fernando},
     title = {Kurdyka{\textendash}{\L}ojasiewicz {Functions} and {Mapping} {Cylinder} {Neighborhoods}},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3656},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Cibotaru, Daniel
AU  - Galaz-García, Fernando
TI  - Kurdyka–Łojasiewicz Functions and Mapping Cylinder Neighborhoods
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3656
LA  - en
ID  - AIF_0__0_0_A111_0
ER  - 
%0 Unpublished Work
%A Cibotaru, Daniel
%A Galaz-García, Fernando
%T Kurdyka–Łojasiewicz Functions and Mapping Cylinder Neighborhoods
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3656
%G en
%F AIF_0__0_0_A111_0
Cibotaru, Daniel; Galaz-García, Fernando. Kurdyka–Łojasiewicz Functions and Mapping Cylinder Neighborhoods. Annales de l'Institut Fourier, Online first, 32 p.

[1] Alexander, James Waddell An Example of a Simply Connected Surface Bounding a Region which is not Simply Connected, Proc. Natl. Acad. Sci. USA, Volume 10 (1924) no. 1, pp. 8-10 | DOI | Zbl

[2] Alexander, James Waddell On the subdivision of a 3-Space by a polyhedron, Proc. Natl. Acad. Sci. USA, Volume 10 (1924) no. 1, pp. 6-8 | DOI | Zbl

[3] Bolte, Jérôme; Daniilidis, Aris; Ley, Olivier; Mazet, Laurent Characterizations of Łojasiewicz inequalities: subgradient flows, talweg, convexity, Trans. Am. Math. Soc., Volume 362 (2010) no. 6, pp. 3319-3363 | DOI | MR | Zbl

[4] Bolte, Jérôme; Pauwels, Edouard Curiosities and counterexamples in smooth convex optimization (2020) (to appear in Math. Program., https://arxiv.org/abs/2001.07999)

[5] Brown, Morton Locally flat embeddings in topological manifolds, Ann. Math., Volume 75 (1962) no. 2, pp. 331-341 | DOI

[6] Burgess, Cecil E.; Cannon, James W. Embeddings of surfaces in E 3 , Rocky Mt. J. Math., Volume 1 (1971) no. 2, pp. 259-344 | DOI | MR | Zbl

[7] Daniilidis, Aris; Haddou, Mounir; Ley, Olivier A convex function satisfying the Łojasiewicz inequality but failing the gradient conjecture both at zero and infinity (2021) (to appear in Bull. Lond. Math. Soc., https://arxiv.org/abs/2102.05342)

[8] Davis, James F.; Kirk, Paul Lecture notes in algebraic topology, Graduate Studies in Mathematics, 35, American Mathematical Society, 2001, xvi+367 pages | DOI | MR | Zbl

[9] Fritsch, Rudolf; Piccinini, Renzo A. Cellular structures in topology, Cambridge Studies in Advanced Mathematics, 19, Cambridge University Press, 1990, xii+326 pages | DOI | MR | Zbl

[10] Hajłasz, Piotr; Zhou, Xiaodan Sobolev embedding of a sphere containing an arbitrary Cantor set in the image, Geom. Dedicata, Volume 184 (2016), pp. 159-173 | DOI | MR | Zbl

[11] Hirsch, Morris W.; Smale, Stephen; Devaney, Robert L. Differential equations, dynamical systems, and an introduction to chaos, Pure and Applied Mathematics, 60, Elsevier/Academic Press, 2004, xiv+417 pages | MR | Zbl

[12] Hu, Sze-tsen Theory of retracts, Wayne State University Press, 1965, 234 pages | MR | Zbl

[13] Huang, Sen-Zhong Gradient inequalities with applications to asymptotic behavior and stability of gradient-like systems, Mathematical Surveys and Monographs, 126, American Mathematical Society, 2006, viii+184 pages | DOI | MR | Zbl

[14] Kurdyka, Krzysztof On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier, Volume 48 (1998) no. 3, pp. 769-783 | DOI | Numdam | MR | Zbl

[15] Kwun, Kyung Whan; Raymond, Frank Mapping cylinder neighborhoods, Mich. Math. J., Volume 10 (1963), pp. 353-357 | DOI | MR | Zbl

[16] Łojasiewicz, Stanisław Ensembles semi-analytiques (1965) (preprint IHES, available at http://perso.univ-rennes1.fr/michel.coste/Lojasiewicz.pdf)

[17] Milnor, John Singular points of complex hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press; University of Tokyo Press, 1968, iii+122 pages | DOI | MR | Zbl

[18] Milnor, John Differential topology forty-six years later, Notices Am. Math. Soc., Volume 58 (2011) no. 6, pp. 804-809 | MR | Zbl

[19] Moise, Edwin E. Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. Math., Volume 56 (1952), pp. 96-114 | DOI | MR | Zbl

[20] Nicholson, Victor Mapping cylinder neighborhoods, Trans. Am. Math. Soc., Volume 143 (1969), pp. 259-268 | DOI | MR | Zbl

[21] Palis, Jacob Jr; de Melo, Welington Geometric theory of dynamical systems. An introduction. Translated from the Portuguese by A. K. Manning, Springer, 1982, xii+198 pages | DOI | MR | Zbl

[22] Petersen, Peter Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, 2016, xviii+499 pages | DOI | MR | Zbl

[23] Quinn, Frank Ends of maps. I, Ann. Math., Volume 110 (1979) no. 2, pp. 275-331 | DOI | MR | Zbl

[24] Quinn, Frank Lectures on controlled topology: mapping cylinder neighborhoods, Topology of high-dimensional manifolds, No. 1, 2 (Trieste, 2001) (ICTP Lecture Notes), Volume 9, Abdus Salam International Centre for Theoretical Physics, 2002, pp. 461-489 | MR | Zbl

[25] Rockafellar, R. Tyrrell; Wets, Roger J.-B. Variational analysis, Grundlehren der Mathematischen Wissenschaften, 317, Springer, 1998, xiv+733 pages | DOI | MR | Zbl

[26] Spanier, Edwin H. Algebraic topology, Springer, 1981, xvi+528 pages (corrected reprint) | MR

[27] Strøm, Arne Note on cofibrations, Math. Scand., Volume 19 (1966), pp. 11-14 | DOI | MR | Zbl

[28] Wang, Qi Ming Isoparametric functions on Riemannian manifolds. I, Math. Ann., Volume 277 (1987) no. 4, pp. 639-646 | DOI | MR | Zbl

Cité par Sources :