The length of a cut locus on a surface and Ambrose's problem
The paper is devoted to the proof of two theorems. Theorem A. Suppose \((M,g)\) is a complete surface with a Riemannian metric of class \(C^2\). Then any compact subset of the cut locus of \(p\in M\) has finite one-dimensional Hausdorff measure. The result fails if the differentiability class of the metric is weakened. Theorem B. There is a \(C^{1,1}\) metric on \(S^2\) so that there is a point \(p\in S^2\) whose cut locus has infinite total length (one-dimensional Hausdorff measure). Theorem A was proved earlier by the reviewer [J. Differ. Geom. 40, No. 3, 621-642 (1994; Zbl 0823.53031)] in the \(C^\infty\) case, but by a more complicated argument. Combining Theorem A with [the reviewer, Trans. Am. Math. Soc. 299, 559-572 (1987; Zbl 0615.53026)] solves Ambrose's problem in dimension 2.
scientific article; zbMATH DE number 955321