Stability of \(e^\epsilon\)-Lipschitz and co-Lipschitz maps in Gromov-Hausdorff topology
The notion of Gromov-Hausdorff distance between compact metric spaces and the related Gromov-Hausdorff convergence is a basic one in Riemannian geometry and geometry of more general metric spaces. Since limits of Riemannian manifolds are often not smooth anymore, many concepts and results for Riemannian manifolds have been generalized to suitable metric spaces, for example, Alexandrov spaces. One important problem is concerned with stability of maps between these more general spaces. One result of Perelman states that sufficiently small Gromov-Hausdorff approximations between Alexandrov spaces can be approximated by homeomorphisms. This was reformulated by V. Kapovitch as a special case of a stability result for a submetry from an Alexandrov space to a Riemannian manifold, where the notion of submetry between metric spaces is an analog of the Riemannian submersion between Riemannian manifolds. In this paper, the author proves the stability of \(e^\epsilon\)-Lipschitz and co-Lipschitz maps between Alexandrov spaces and Lipschitz manifolds. One of the motivation for this generalization is that submetries is rare but \(e^\epsilon\)-Lipschitz and co-Lipschitz maps occur naturally.
scientific article; zbMATH DE number 6084081
scientific article