Quantitative conditions of rectifiability for varifolds  [ Conditions quantitative de rectifiabilité dans l’espace des varifolds ]
Annales de l'Institut Fourier, Tome 65 (2015) no. 6, pp. 2449-2506.

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 i de telle sorte que : si sup i i (V i )<+ et si, aux échelles β i 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 i i of functionals defined on d–varifolds, such that if sup i i (V i )<+ 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.

Reçu le : 2014-08-14
Révisé le : 2015-01-29
Accepté le : 2015-02-11
Publié le : 2015-12-08
DOI : https://doi.org/10.5802/aif.2993
Classification : 28A75,  49Q15
Mots clés: rectifiabilité quantitative, varifolds
@article{AIF_2015__65_6_2449_0,
     author = {Buet, Blanche},
     title = {Quantitative conditions of rectifiability for varifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2449--2506},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {6},
     year = {2015},
     doi = {10.5802/aif.2993},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2015__65_6_2449_0/}
}
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. https://aif.centre-mersenne.org/item/AIF_2015__65_6_2449_0/

[1] Allard, William K. On the first variation of a varifold: boundary behavior, Ann. of Math. (2), Tome 101 (1975), pp. 418-446 | MR 397520 | Zbl 0319.49026

[2] Ambrosio, L.; Fusco, N.; Pallara, D. Functions of bounded variation and free discontinuity problems, Oxford mathematical monographs, Clarendon Press, Oxford, New York, 2000 http://opac.inria.fr/record=b1096464 (Autres tirages : 2006) | MR 1857292 | Zbl 0957.49001

[3] Brakke, Kenneth A. The motion of a surface by its mean curvature, Mathematical Notes, Tome 20, Princeton University Press, Princeton, N.J., 1978, i+252 pages | MR 485012 | Zbl 0386.53047

[4] David, G.; Semmes, S. Singular integrals and rectifiable sets in Rn: Au-dela des graphes lipschitziens Tome 193, Société mathématique de France, 1991 | Zbl 0743.49018

[5] David, G.; Semmes, S. Analysis of and on uniformly rectifiable sets Tome 38, Mathematical Surveys and Monographs, 1993 | MR 1251061 | Zbl 0832.42008

[6] David, G.; Semmes, S. Quantitative Rectifiability and Lipschitz Mappings, Transactions of the American Mathematical Society, Tome 337 (1993) no. 2, pp. 855-889 | MR 1132876 | Zbl 0792.49029

[7] Dorronsoro, J. R. A characterization of potential spaces, Proceedings of A.M.S., Tome 95 (1985) no. 1, pp. 21-31 | MR 796440 | Zbl 0577.46035

[8] Evans, L. C.; Gariepy, R. F. Measure theory and fine properties of functions, Studies in advanced mathematics, CRC Press, Boca Raton (Fla.), 1992 http://opac.inria.fr/record=b1089059 | MR 1158660

[9] Jones, P. W. Rectifiable sets and the traveling salesman problem, Inventiones Mathematicae, Tome 102 (1990) no. 1, pp. 1-15 | MR 1069238 | Zbl 0731.30018

[10] Mattila, P. Cauchy Singular Integrals and Rectifiability of Measures in the Plane, Advances in Mathematics, Tome 115 (1995) no. 1, pp. 1 -34 | MR 1351323 | Zbl 0842.30029

[11] Menne, U. Decay estimates for the quadratic tilt-excess of integral varifolds, Arch. Ration. Mech. Anal., Tome 204 (2012) no. 1, pp. 1-83 | Article | MR 2898736 | Zbl 1252.49071

[12] Okikiolu, K. Characterization of subsets of rectifiable curves in Rn, Journal of the London Mathematical Society, Tome 2 (1992) no. 2, pp. 336-348 | MR 1182488 | Zbl 0758.57020

[13] Pajot, H. Conditions quantitatives de rectifiabilité, Bull. Soc. Math. France, Tome 125 (1997) no. 1, pp. 15-53 | Numdam | MR 1459297 | Zbl 0890.28004

[14] Simon, L. Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, Tome 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983, vii+272 pages | MR 756417 | Zbl 0546.49019