Cut and singular loci up to codimension 3
Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1655-1681.

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.

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.

DOI: 10.5802/aif.2655
Classification: 35F30, 53C60, 53B40
Keywords: Cut locus, Hamilton-Jacobi equations, focal points
Mot clés : cut locus, équations de Hamilton-Jacobi, points focaux
Ardoy, Pablo Angulo 1; Guijarro, Luis 2

1 Universidad Autónoma de Madrid Departamento de Matemáticas Facultad de Ciencias Campus de Cantoblanco 28049 Madrid (Spain)
2 Department of Mathematics Universidad Autónoma de Madrid. Please complete ICMAT CSIC-UAM-UCM-UC3M
@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},
     pages = {1655--1681},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {4},
     year = {2011},
     doi = {10.5802/aif.2655},
     zbl = {1242.35095},
     mrnumber = {2951748},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2655/}
}
TY  - JOUR
AU  - Ardoy, Pablo Angulo
AU  - Guijarro, Luis
TI  - Cut and singular loci up to codimension 3
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 1655
EP  - 1681
VL  - 61
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2655/
DO  - 10.5802/aif.2655
LA  - en
ID  - AIF_2011__61_4_1655_0
ER  - 
%0 Journal Article
%A Ardoy, Pablo Angulo
%A Guijarro, Luis
%T Cut and singular loci up to codimension 3
%J Annales de l'Institut Fourier
%D 2011
%P 1655-1681
%V 61
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2655/
%R 10.5802/aif.2655
%G en
%F AIF_2011__61_4_1655_0
Ardoy, Pablo Angulo; Guijarro, Luis. Cut and singular loci up to codimension 3. Annales de l'Institut Fourier, Volume 61 (2011) no. 4, pp. 1655-1681. doi : 10.5802/aif.2655. http://www.numdam.org/articles/10.5802/aif.2655/

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

[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), Volume 56 (1997) no. 2, pp. 369-383 | DOI | MR | Zbl

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

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

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

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

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

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

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

[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., Volume 58 (2005) no. 1, pp. 85-146 | DOI | MR | Zbl

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

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

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

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

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

Cited by Sources: