no. 1
Curvature measures, normal cycles and asymptotic cones
Sun, Xiang1 ; Morvan, Jean-Marie2
1 Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia
2 University Claude Bernard Lyon P1, France, C.N.R.S. U.M.R. 5028 Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia
Actes des rencontres du CIRM, Courbure discrète : théorie et applications, 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 : 10.5802/acirm.50
Classification : 00X99
Keywords: curvature measure, shape operator, surfaces, normal cycle, asymptotic cones

Sun, Xiang 1 ; Morvan, Jean-Marie 2

1 Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia
2 University Claude Bernard Lyon P1, France, C.N.R.S. U.M.R. 5028 Visual Computing Center King Abdullah University of Science and Technology Saudi Arabia 
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Sun, Xiang; Morvan, Jean-Marie. Curvature measures, normal cycles and asymptotic cones. Actes des rencontres du CIRM, Courbure discrète : théorie et applications, Tome 3 (2013) no. 1, pp. 3-10. doi: 10.5802/acirm.50

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

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

[6] Morvan, Jean-Marie Generalized curvatures, 2, Springer, 2008 | Zbl | MR

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

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

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