Recovering $S^1$-Invariant Metrics on $S^2$ from the Equivariant Spectrum
We prove an inverse spectral result for S1-invariant metrics on S2 based on the so-called asymptotic equivariant spectrum. This is roughly the spectrum together with large weights of the S1-action on the eigenspaces. Our result generalizes an inverse spectral result from  concerning S1-invariant metrics on S2 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 S1-action.
International Mathematics Research Notices
Dryden, Emily; Sena-Dias, Rosa; and Macedo, Diana. "Recovering $S^1$-Invariant Metrics on $S^2$ from the Equivariant Spectrum." International Mathematics Research Notices (2016) : 4882-4902.