A general collapsing result for families of stratified Riemannian metrics on orbifolds

Abstract

This paper proves a general collapsing result for families of stratified Riemannian metrics $\widehat{g}^μ$ on a compact orbifold $E$, subject to suitable limiting conditions on the metrics $\widehat{g}^μ$ as $μ\to \infty$. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of $\left(E,\widehat{g}^μ\right)$ and thus allows for Gromov-Hausdorff limits of $\left(E,\widehat{g}^μ\right)$ which have strictly lower dimension than $E$. The paper also introduces and studies a new class of stratified fibrations between orbifolds, termed weak submersions, and new classes of geometric structures on orbifolds, termed stratified Riemannian metrics, stratified Riemannian semi-metrics and stratified quasi-Finslerian structures, all of which play a key role in the proof of the main theorem.