Cut and singular loci up to codimension 3

Ardoy, Pablo Angulo; Guijarro, Luis

Annales de l'Institut Fourier, Tome 61 (2011) no. 4, pp. 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 $n-3$.

[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