Quantitative conditions of rectifiability for varifolds

Buet, Blanche

doi:10.5802/aif.2993

Quantitative conditions of rectifiability for varifolds
[Conditions quantitative de rectifiabilité dans l’espace des varifolds]

Buet, Blanche ¹

¹ Université Lyon 1 Institut Camille Jordan 69622 Villeurbanne cedex (France)

Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2449-2506.

Résumé
Abstract

L’objet de ce travail est d’énoncer des conditions quantitatives garantissant la rectifiabilité de la limite d’une suite de varifolds qui ne sont pas nécessairement rectifiables. Dans ce but, on définit, dans l’espace des varifolds, des fonctionnelles $ℰ_{i}$ de telle sorte que : si ${sup}_{i} ℰ_{i} (V_{i}) < + \infty$ et si, aux échelles $β_{i} \to 0$ , la densité $d$ –dimensionnelle de $V_{i}$ vérifie un contrôle uniforme, alors $V = {lim}_{i} V_{i}$ est $d$ –rectifiable.

Ce travail participe à la mise en place d’un cadre théorique pour l’approximation des courbes, surfaces ou de façon plus générale, des ensembles $d$ –rectifiables minimisant des fonctionnelles géométriques, par des objets “discrets” (approximations volumiques, nuages de points etc.) minimisant des fonctionnelles géométriques discrétisées.

Our purpose is to state quantitative conditions ensuring the rectifiability of a $d$ –varifold $V$ obtained as the limit of a sequence of $d$ –varifolds ${(V_{i})}_{i}$ which need not to be rectifiable. More specifically, we introduce a sequence ${\{ℰ_{i}\}}_{i}$ of functionals defined on $d$ –varifolds, such that if ${sup}_{i} ℰ_{i} (V_{i}) < + \infty$ and $V_{i}$ satisfies a uniform density estimate at some scale $β_{i}$ , then $V = {lim}_{i} V_{i}$ is $d$ –rectifiable.

The main motivation of this work is to set up a theoretical framework where curves, surfaces, or even more general $d$ –rectifiable sets minimizing geometrical functionals (like the length for curves or the area for surfaces), can be approximated by “discrete” objects (volumetric approximations, pixelizations, point clouds etc.) minimizing some suitable “discrete” functionals.

DOI : 10.5802/aif.2993

Classification : 28A75, 49Q15
Keywords: quantitative rectifiability, varifolds
Mot clés : rectifiabilité quantitative, varifolds

Affiliations des auteurs :

Buet, Blanche ¹

¹ Université Lyon 1 Institut Camille Jordan 69622 Villeurbanne cedex (France)

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     title = {Quantitative conditions of rectifiability for varifolds},
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     pages = {2449--2506},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Buet, Blanche. Quantitative conditions of rectifiability for varifolds. Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2449-2506. doi : 10.5802/aif.2993. http://www.numdam.org/articles/10.5802/aif.2993/

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