Spectral and geometric bounds on 2-orbifold diffeomorphism type

TL;DR

This paper demonstrates that families of 2-orbifolds with certain spectral and geometric bounds are limited to finitely many diffeomorphism types, linking spectral data to geometric constraints.

Contribution

It establishes a finiteness result for 2-orbifolds based on spectral and geometric bounds, connecting spectral data to orbifold classification.

Findings

Finite orbifold diffeomorphism types under spectral bounds

Finiteness of orbifold types with curvature, volume, and diameter bounds

Spectral data implies geometric bounds

Abstract

We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2-orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.