Metric structure of cut loci in surfaces and Ambrose's problem
James J. Hebda · Journal of Differential Geometry · 1994
It is shown that every compact subset of the cut locus of a point in a complete, two-dimensional Riemannian manifold has finite one-dimensional Hausdorff measure.When combined with a result from an earlier paper, this completes the solution of the two-dimensional case of a long-standing problem of Ambrose.