On the stability of the cut locus

In RdRd we consider a Riemannian metric, gg, and an open bounded subset, ΩΩ. We study the stability of the cut locus associated with ΩΩ and gg w.r.t. perturbations both of the set ΩΩ and of the metric gg. In order to have the stability of the cut locus, we assume C2C2 regularity of the data, the metrics and the sets (in the case of sets with C1,1C1,1 boundaries, the cut locus may be unstable). We prove that to C2C2 perturbations both of the set and of the metric correspond small changes of the cut locus w.r.t. the Hausdorff distance, i.e. the cut locus is stable in the C2C2 category. We give some examples showing that C1C1 perturbations may lead to large variations of the cut locus.