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

2023/08/09 by Mayther, Laurence H.
#28A75 (Secondary) #53C15 #53C23 #53C60 #57R18 #58A35 (Primary) 28A20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2308.05016

Abstract

This paper proves a general collapsing result for families of stratified Riemannian metrics \widehatg^μ on a compact orbifold E, subject to suitable limiting conditions on the metrics \widehatg^μ as μ→ ∞. The result is distinct from similar theorems in the literature since it does not require bounds on curvature or injectivity radius of (E,\widehatg^μ) and thus allows for Gromov-Hausdorff limits of (E,\widehatg^μ) 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.

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