Cut and singular loci up to codimension 3

TL;DR

This paper provides a detailed classification of the structure of cut loci and singular sets in Hamilton-Jacobi equations, extending existing local descriptions to include points up to Hausdorff dimension n-3.

Contribution

It introduces a new classification of points in the cut locus and singular sets, refining the understanding beyond previous local descriptions.

Findings

Classification of all points up to Hausdorff dimension n-3

Detailed description of cut loci structure

Applications to singular sets of Hamilton-Jacobi equations

Abstract

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 clasification of all points up to a set of Hausdorff dimension $n-3$.