Orbifold finiteness under geometric and spectral constraints

TL;DR

This paper proves finiteness results for classes of Riemannian orbifolds under geometric and spectral constraints, showing only finitely many orbifold types exist within these bounds.

Contribution

It establishes new finiteness theorems for orbifolds with bounds on curvature, volume, diameter, and spectral data, extending previous results to broader classes.

Findings

Finiteness of orbifolds with bounded curvature, volume, and diameter.

Finiteness of isospectral orbifolds with bounded sectional curvature.

Results hold up to orbifold homeomorphism.

Abstract

The class of Riemannian orbifolds of dimension n defined by a lower bound on the sectional curvature and the volume and an upper bound on the diameter has only finitely many members up to orbifold homeomorphism. Furthermore, any class of isospectral Riemannian orbifolds with a lower bound on the sectional curvature is finite up to orbifold homeomorphism.