On the stability of the cut locus. (May 2016) - Albano, Paolo
Abstract: In R d we consider a Riemannian metric, g, and an open bounded subset, Ω . We study the stability of the cut locus associated with Ω and g w.r.t. perturbations both of the set Ω and of the metric g . In order to have the stability of the cut locus, we assume C 2 regularity of the data, the metrics and the sets (in the case of sets with C 1, 1 boundaries, the cut locus may be unstable). We prove that to C 2 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 C 2 category. We give some examples showing that C 1 perturbations may lead to large variations of the cut locus.