Inverse Spectral Problems for Collapsing Manifolds II: Quantitative Stability of Reconstruction for Orbifolds

TL;DR

This paper establishes a quantitative stability result for inverse spectral problems on collapsing 1-dimensional Riemannian orbifolds, improving understanding of how spectral data determines orbifold geometry.

Contribution

It provides a quantitative stability result for inverse spectral problems on collapsing 1-dimensional orbifolds, extending prior uniqueness and stability results.

Findings

Quantitative stability for spectral inverse problems on orbifolds.

Improved unique continuation for wave operators without curvature derivative assumptions.

Applicability to collapsing 1-dimensional Riemannian orbifolds.

Abstract

We consider the inverse problem of determining the metric-measure structure of collapsing manifolds from local measurements of spectral data. In the part I of the paper, we proved the uniqueness of the inverse problem and a continuity result for the stability in the closure of Riemannian manifolds with bounded diameter and sectional curvature in the measured Gromov-Hausdorff topology. In this paper we show that when the collapse of dimension is $1$-dimensional, it is possible to obtain quantitative stability of the inverse problem for Riemannian orbifolds. The proof is based on an improved version of the quantitative unique continuation for the wave operator on Riemannian manifolds by removing assumptions on the covariant derivatives of the curvature tensor.