Hausdorff Stability of the Cut Locus Under $C^2$-Perturbations of the Metric

In this article, we prove the stability with respect to the Hausdorff metric $d_H$ of the cut locus $\mathrm{C}\mathrm{u}\mathrm{t}(p,\mathfrak{g})$ of a point $p$ in a compact Riemannian manifold $(M,\mathfrak{g})$ under $C^2$ perturbation of the metric. Specifically, given a sequence of metrics ${\mathfrak{g}}_i$ on $M$, converging to $\mathfrak{g}$ in the $C^2$ topology, and a sequence of points $p_i$ in $M$, converging to $p$, we show that ${\lim }_i{d}_H(\mathrm{C}\mathrm{u}\mathrm{t}(p_i,{\mathfrak{g}}_i),\mathrm{C}\mathrm{u}\mathrm{t}(p,\mathfrak{g}))=0$. Along the way, we also prove the continuous dependence of the cut time map on the metric.