Quantitative conditions of rectifiability for varifolds
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2449-2506.

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.

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.

DOI: 10.5802/aif.2993
Classification: 28A75, 49Q15
Keywords: quantitative rectifiability, varifolds
Mot clés : rectifiabilité quantitative, varifolds
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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Buet, Blanche. Quantitative conditions of rectifiability for varifolds. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2449-2506. doi : 10.5802/aif.2993. http://archive.numdam.org/articles/10.5802/aif.2993/

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

[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 | Zbl

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

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

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

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

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

[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

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

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

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

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

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

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

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