Recovering $S^1$-invariant metrics on $S^2$ from the equivariant spectrum

TL;DR

This paper demonstrates that the asymptotic equivariant spectrum uniquely determines $S^1$-invariant metrics on $S^2$, extending previous results by incorporating higher order spectral data.

Contribution

It generalizes prior inverse spectral results by utilizing higher order terms in the asymptotic expansion of the spectral measure for $S^1$-invariant metrics on $S^2$.

Findings

Unique recovery of $S^1$-invariant metrics from asymptotic equivariant spectrum.

Extension of previous inverse spectral results to broader class of metrics.

Use of higher order asymptotic terms in spectral measure analysis.

Abstract

We prove an inverse spectral result for $S^1$-invariant metrics on $S^2$ based on the so-called asymptotic equivariant spectrum. This is roughly the spectrum together with large weights of the $S^1$ action on the eigenspaces. Our result generalizes an inverse spectral result of the first and last named authors, together with Victor Guillemin, concerning $S^1$-invariant metrics on $S^2$ which are invariant under the antipodal map. We use higher order terms in the asymptotic expansion of a natural spectral measure associated with the Laplacian and the $S^1$ action.