Comme application du flot de raccourcissement des courbes, nous prouverons une inégalité sur la courbure affine minimale d’une courbe fermée simple lisse dans le plan euclidien.
As an application of the affine curve shortening flow, we will prove an inequality for minimum affine curvature of a smooth simple closed curve in the Euclidean plane.
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@article{CRMATH_2020__358_2_139_0, author = {Yang, Yunlong}, title = {An inequality for the minimum affine curvature of a plane curve}, journal = {Comptes Rendus. Math\'ematique}, pages = {139--142}, publisher = {Acad\'emie des sciences, Paris}, volume = {358}, number = {2}, year = {2020}, doi = {10.5802/crmath.19}, language = {en}, url = {http://www.numdam.org/articles/10.5802/crmath.19/} }
TY - JOUR AU - Yang, Yunlong TI - An inequality for the minimum affine curvature of a plane curve JO - Comptes Rendus. Mathématique PY - 2020 SP - 139 EP - 142 VL - 358 IS - 2 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.19/ DO - 10.5802/crmath.19 LA - en ID - CRMATH_2020__358_2_139_0 ER -
%0 Journal Article %A Yang, Yunlong %T An inequality for the minimum affine curvature of a plane curve %J Comptes Rendus. Mathématique %D 2020 %P 139-142 %V 358 %N 2 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.19/ %R 10.5802/crmath.19 %G en %F CRMATH_2020__358_2_139_0
Yang, Yunlong. An inequality for the minimum affine curvature of a plane curve. Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 139-142. doi : 10.5802/crmath.19. http://www.numdam.org/articles/10.5802/crmath.19/
[1] Self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differ. Equ., Volume 13 (2001) no. 3, pp. 311-337 | MR | Zbl
[2] Axioms and fundamental equations of image processing, Arch. Ration. Mech. Anal., Volume 132 (1993) no. 3, pp. 199-257 | DOI | MR | Zbl
[3] Contraction of convex hypersurfaces by their affine normal, J. Differ. Geom., Volume 43 (1996) no. 2, pp. 207-230 | DOI | MR | Zbl
[4] The affine curve-lengthening flow, J. Reine Angew. Math., Volume 506 (1999), pp. 48-83 | MR | Zbl
[5] On the affine heat equation for non-convex curves, J. Am. Math. Soc., Volume 11 (1998) no. 3, pp. 601-634 | DOI | MR | Zbl
[6] The Curve Shortening Problem, Chapman & Hall/CRC, 2001 | Zbl
[7] On the maximal mean curvature of a smooth surface, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 9, pp. 891-895 | DOI | MR | Zbl
[8] The heat equation shrinking convex plane curves, J. Differ. Geom., Volume 23 (1986), pp. 69-96 | DOI | MR | Zbl
[9] The heat equation shrinks embedded plane curves to round points, J. Differ. Geom., Volume 26 (1987), pp. 285-314 | DOI | MR | Zbl
[10] Four-manifolds with positive curvature operator, J. Differ. Geom., Volume 24 (1986), pp. 153-179 | DOI | MR | Zbl
[11] A reverse isoperimetric inequality, stability and extremal theorems for plane curves with bounded curvature, Rocky Mt. J. Math., Volume 25 (1995) no. 2, pp. 635-684 | DOI | MR | Zbl
[12] Centro-affine curvature flows on centrally symmetric convex curves, Trans. Am. Math. Soc., Volume 366 (2014) no. 11, pp. 5671-5692 | DOI | MR | Zbl
[13] -periodic self-similar solutions for the anisotropic affine curve shortening problem, Calc. Var. Partial Differ. Equ., Volume 41 (2011) no. 3-4, pp. 535-565 | DOI | MR | Zbl
[14] Differential invariant signatures and flows in computer vision: A symmetry group approach, Geometry-Driven Diffusion in Computer Vision (Computational Imaging and Vision), Volume 1, Springer, 1994, pp. 205-306
[15] An inequality for the maximum curvature through a geometric flow, Arch. Math., Volume 105 (2015) no. 3, pp. 297-300 | DOI | MR | Zbl
[16] Mean curvature bounds and eigenvalues of Robin Laplacians, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 2, pp. 1947-1961 | DOI | MR | Zbl
[17] On the largest possible circle imbedded in a given closed curve, Dokl. Akad. Nauk SSSR, Volume 127 (1959), pp. 1170-1172 | MR | Zbl
[18] On affine plane curve evolution, J. Funct. Anal., Volume 119 (1994) no. 1, pp. 79-120 | DOI | MR | Zbl
[19] Affine Differential Geometry, Gordon and Breach, Science Publishers, 1983 | Zbl
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