Cut and singular loci up to codimension 3  [ Cut-loci et lieux singuliers jusqu’à codimension 3 ]
Annales de l'Institut Fourier, Tome 61 (2011) no. 4, p. 1655-1681
Le cut locus d’une variété Finslerienne peut être non-triangulable, mais une description locale à tous les points sauf pour un ensemble de dimension de Hausdorff n-2 est bien connu. Nous donnons une nouvelle description de la structure de ces ensembles, avec des applications directes pour les ensembles des points singuliers de certaines équations de Hamilton-Jacobi. Nous donnons une classification de tous les points sauf pour un ensemble de dimension de Hausdorff n-3.
We give a new and detailed description of the structure of cut loci, with direct applications to the singular sets of some Hamilton-Jacobi equations. These sets may be non-triangulable, but a local description at all points except for a set of Hausdorff dimension n-2 is well known. We go further in this direction by giving a classification of all points up to a set of Hausdorff dimension n-3.
DOI : https://doi.org/10.5802/aif.2655
Classification:  35F30,  53C60,  53B40
Mots clés: cut locus, équations de Hamilton-Jacobi, points focaux
@article{AIF_2011__61_4_1655_0,
     author = {Ardoy, Pablo Angulo and Guijarro, Luis},
     title = {Cut and singular loci up to codimension 3},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     pages = {1655-1681},
     doi = {10.5802/aif.2655},
     mrnumber = {2951748},
     zbl = {1242.35095},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2011__61_4_1655_0}
}
Ardoy, Pablo Angulo; Guijarro, Luis. Cut and singular loci up to codimension 3. Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 1655-1681. doi : 10.5802/aif.2655. http://www.numdam.org/item/AIF_2011__61_4_1655_0/

[1] Alberti, G.; Ambrosio, L.; Cannarsa, P. On the singularities of convex functions, Comm. Pure Appl. Math., Tome 76 (1992), pp. 421-435 | MR 1185029 | Zbl 0784.49011

[2] Ardoy, P. A.; Guijarro, L. Balanced split sets and Hamilton Jacobi equations (http://arxiv.org/abs/0807.2046, (2008-2009) (to appear in Calc. Var. Partial Differential Equations))

[3] Barden, D.; Le, H. Some consequences of the nature of the distance function on the cut locus in a riemannian manifold, J. London Math. Soc. (2), Tome 56 (1997) no. 2, pp. 369-383 | Article | MR 1489143 | Zbl 0892.53021

[4] Buchner, M. A. The structure of the cut locus in dimension less than or equal to six, Compositio Math., Tome 37 (1978) no. 1, pp. 103-119 | Numdam | MR 501100 | Zbl 0407.58008

[5] Cannarsa, P.; Sinestrari, C. Semiconcave functions, Hamilton-Jacobi equations, and optimal control, Birkhäuser Boston, Boston, Progress in Nonlinear Differential Equations and Their Applications, Tome 58 (2004) | MR 2041617 | Zbl 1095.49003

[6] Federer, H. Geometric measure theory, Springer-Verlag New York Inc., New York, Progress in Nonlinear Differential Equations and Their Applications, Tome 153 (1969) | MR 257325 | Zbl 0874.49001

[7] Gluck, H.; Singer, D. Scattering of Geodesic Fields, I, Annals of Mathematics, Tome 108 (1978) no. 2, pp. 347-372 | Article | MR 506991 | Zbl 0399.58011

[8] Hebda, J. Parallel translation of curvature along geodesics, Trans. Amer. Math. Soc., Tome 299 (1987), pp. 559-572 | Article | MR 869221 | Zbl 0615.53026

[9] Itoh, J.; Tanaka, M. The dimension of a cut locus on a smooth Riemannian manifold, Tohoku Math. J. (2), Tome 50 (1998) no. 4, pp. 571-575 | Article | MR 1653438 | Zbl 0939.53029

[10] Itoh, J.; Tanaka, M. The Lipschitz continuity of the distance function to the cut locus, Transactions of the A.M.S., Tome 353 (2000) no. 1, pp. 21-40 | Article | MR 1695025 | Zbl 0971.53031

[11] Li, Yy.; Nirenberg, L. The distance function to the boundary, Finsler geometry, and the singular set of viscosity solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math., Tome 58 (2005) no. 1, pp. 85-146 | Article | MR 2094267 | Zbl 1062.49021

[12] Lions, P. L. Generalized Solutions of Hamilton-Jacobi Equations, Pitman, Boston, MA Tome 69 (1982) | MR 667669 | Zbl 0497.35001

[13] Mantegazza, C.; Mennucci, A. C. Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds, Appl. Math. Optim., Tome 47 (2003) no. 2, pp. 1-25 | MR 1941909 | Zbl 1048.49021

[14] Mennucci, A.C. Regularity And Variationality Of Solutions To Hamilton-Jacobi Equations. Part I: Regularity (2nd Edition), ESAIM Control Optim. Calc. Var., Tome 13 (2007) no. 2, pp. 413-417 | Article | Numdam | MR 2306644 | Zbl 1121.49028

[15] Milnor, J. Morse theory, Princeton University Press, Princeton, N.J., Annals of Mathematics Studies, Tome 51 (1963) | MR 163331 | Zbl 0108.10401

[16] Warner, F. W. The conjugate locus of a Riemannian manifold, Amer. J. of Math., Tome 87 (1965), pp. 573-604 | Article | MR 208534 | Zbl 0129.36002