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Curvature measures, normal cycles and asymptotic cones
Actes des rencontres du CIRM, Tome 3 (2013) no. 1, pp. 3-10.

The purpose of this article is to give an overview of the theory of the normal cycle and to show how to use it to define a curvature measures on singular surfaces embedded in an (oriented) Euclidean space 𝔼 3 . In particular, we will introduce the notion of asymptotic cone associated to a Borel subset of 𝔼 3 , generalizing the asymptotic directions defined at each point of a smooth surface. For simplicity, we restrict our singular subsets to polyhedra of the 3-dimensional Euclidean space 𝔼 3 . The coherence of the theory lies in a convergence theorem: If a sequence of polyhedra (P n ) tends (for a suitable topology) to a smooth surface S, then the sequence of curvature measures of (P n ) tends to the curvature measures of S. Details on the first part of these pages can be found in [6].

Publié le :
DOI : https://doi.org/10.5802/acirm.50
Classification : 00X99
Mots clés : curvature measure, shape operator, surfaces, normal cycle, asymptotic cones 
@article{ACIRM_2013__3_1_3_0,
     author = {Xiang Sun and Jean-Marie Morvan},
     title = {Curvature measures, normal cycles and asymptotic cones},
     journal = {Actes des rencontres du CIRM},
     pages = {3--10},
     publisher = {CIRM},
     volume = {3},
     number = {1},
     year = {2013},
     doi = {10.5802/acirm.50},
     language = {en},
     url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.50/}
}
Sun, Xiang; Morvan, Jean-Marie. Curvature measures, normal cycles and asymptotic cones. Actes des rencontres du CIRM, Tome 3 (2013) no. 1, pp. 3-10. doi : 10.5802/acirm.50. https://acirm.centre-mersenne.org/articles/10.5802/acirm.50/

[1] David Cohen-Steiner; Jean-Marie Morvan Restricted delaunay triangulations and normal cycle, Proceedings of the nineteenth annual symposium on Computational geometry (2003), pp. 312-321

[2] David Cohen-Steiner; Jean-Marie Morvan 4 Differential Geometry on Discrete Surfaces, Effective computational geometry for curves and surfaces (2006)

[3] David Cohen-Steiner; Jean-Marie Morvan Second fundamental measure of geometric sets and local approximation of curvatures, Journal of Differential Geometry, Tome 74 (2006) no. 3, pp. 363-394

[4] Joseph HG Fu Monge-Ampère Functions 1, Indiana Univ. Math. J., Tome 38 (1989), pp. 745-771

[5] Joseph HG Fu Convergence of curvatures in secant approximations, Journal of Differential Geometry, Tome 37 (1993) no. 1, pp. 177-190

[6] Jean-Marie Morvan Generalized curvatures Tome 2, Springer, 2008

[7] Peter Wintgen Normal cycle and integral curvature for polyhedra in Riemannian manifolds, Differential Geometry. North-Holland Publishing Co., Amsterdam-New York (1982)

[8] Martina Zähle Integral and current representation of Federer’s curvature measures, Archiv der Mathematik, Tome 46 (1986) no. 6, pp. 557-567