Stability of the Injectivity Radius and the Cut Locus of Submanifolds under Perturbations

Abstract

The continuity of the injectivity radius of a compact manifold under $C^2$ perturbation of the Riemannian metric was originally proved by P. Ehrlich (Composito Math., 1974), and later the proof was simplified by T. Sakai (Math. J. Okayama Univ., 1983). Using this continuity, jointly with J. Itoh and S. Prasad (J. Math. Anal. Appl., 2025), we proved the Hausdorff stability of the cut locus of a point, when both the point and the metric are perturbed. In the present article, we extend both these results to submanifolds. We first show that the injectivity radius of a submanifold depends continuously on the metric. Then, we obtain the Hausdorff stability of the cut locus of the submanifold, under $C^2$ perturbation of the metric. In fact, we allow the submanifold to be perturbed in the Whitney $C^2$ sense as well.