Equi-singularity of real families and Lipschitz–Killing curvature densities at infinity

Dutertre, Nicolas; Grandjean, Vincent

doi:10.5802/aif.3607

Dutertre, Nicolas ¹; Grandjean, Vincent ²

¹ Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers (France)
² Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis – SC, (Brasil)

Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2379-2423.

Abstract
Résumé

Fix an o-minimal structure expanding the ordered field of real numbers. Let ${(W_{y})}_{y \in ℝ^{s}}$ be a definable family of closed subsets of $ℝ^{n}$ whose total space $W = \cup_{y} W_{y} \times y$ is a closed connected $C^{2}$ definable sub-manifold of $ℝ^{n} \times ℝ^{s}$ . Let $φ : W \to ℝ^{s}$ be the restriction of the projection to the second factor.

After defining $K (φ)$ , the set of generalized critical values of $φ$ , showing that they are closed and definable of positive codimension in $ℝ^{s}$ , contain the bifurcation values of $φ$ and are stable under generic plane sections, we prove that all the Lipschitz–Killing curvature densities at infinity $y \mapsto κ_{i}^{\infty} (W_{y})$ are continuous functions over $ℝ^{s} ∖ K (φ)$ . When $W$ is a $C^{2}$ definable hypersurface of $ℝ^{n} \times ℝ^{s}$ , we further obtain that the symmetric principal curvature densities at infinity $y \mapsto σ_{i}^{\infty} (W_{y})$ are continuous functions over $ℝ^{s} ∖ K (φ)$ .

On fixe une structure o-minimale qui étend le corps ordonné des nombres réels. Soit ${(W_{y})}_{y \in ℝ^{s}}$ une famille définissable de sous-ensembles fermés de $ℝ^{n}$ dont l’espace total $W = \cup_{y} W_{y} \times y$ est une sous-variété définissable connexe et fermée de $ℝ^{n} \times ℝ^{s}$ de classe $C^{2}$ . Soit $φ : W \to ℝ^{s}$ la restriction de la projection sur le second facteur.

Après avoir défini $K (φ)$ , l’ensemble des valeurs critiques généralisées de $φ$ , montré qu’elles forment un sous-ensemble définissable fermé de codimension non-nulle de $ℝ^{s}$ , contiennent les valeurs de bifurcations de $φ$ et sont stables par section plane générique, nous montrons que toutes les densités à l’infini des courbures de Lipschitz–Killing $y \mapsto κ_{i}^{\infty} (W_{y})$ sont des fonctions continues sur $ℝ^{s} ∖ K (φ)$ . Quand $W$ est une hypersurface définissable de $ℝ^{n} \times ℝ^{s}$ de classe $C^{2}$ , nous obtenons de plus que les densités à l’infini des courbures symétriques principales $y \mapsto σ_{i}^{\infty} (W_{y})$ sont des fonctions continues sur $ℝ^{s} ∖ K (φ)$ .

Received: 2021-12-23
Revised: 2023-03-17
Accepted: 2023-05-15
Published online: 2024-10-11

DOI: 10.5802/aif.3607

Classification: 14P10, 03C64, 57R70
Keywords: Real equi-singularity, Lipschitz–Killing curvatures, the Malgrange–Rabier condition.
Mot clés : Équisingularité réelle, courbures de Lipschitz–Killing, condition de Malgrange–Rabier.

Author's affiliations:

Dutertre, Nicolas ¹; Grandjean, Vincent ²

¹ Univ Angers, CNRS, LAREMA, SFR MATHSTIC, F-49000 Angers (France)
² Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis – SC, (Brasil)

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Dutertre, Nicolas and Grandjean, Vincent},
     title = {Equi-singularity of real families and {Lipschitz{\textendash}Killing} curvature densities at infinity},
     journal = {Annales de l'Institut Fourier},
     pages = {2379--2423},
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Dutertre, Nicolas; Grandjean, Vincent. Equi-singularity of real families and Lipschitz–Killing curvature densities at infinity. Annales de l'Institut Fourier, Volume 74 (2024) no. 6, pp. 2379-2423. doi : 10.5802/aif.3607. https://aif.centre-mersenne.org/articles/10.5802/aif.3607/

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