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 $n-3$.

Classification : 35F30, 53C60, 53B40
Keywords: Cut locus, Hamilton-Jacobi equations, focal points
Mot clés : cut locus, équations de Hamilton-Jacobi, points focaux

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