Equivariant inverse spectral theory and toric orbifolds
Advances in Mathematics
Let O-2n be a symplectic toric orbifold with a fixed T-n-action and with a tonic Kahler metric g. In  we explored whether, when O is a manifold, the equivariant spectrum of the Laplace Delta(g) operator on C-infinity(O) determines O up to symplectomorphism. In the setting of tonic orbifolds we shmilicantly improve upon our previous results and show that a generic tone orbifold is determined by its equivariant spectrum, up to two possibilities. This involves developing the asymptotic expansion of the heat trace on an orbifold in the presence of an isometry. We also show that the equivariant spectrum determines whether the toric Kahler metric has constant scalar curvature. (C) 2012 Elsevier Inc. All rights reserved.
Dryden, Emily B.; Guillemin, Victor; and Sena-Dias, Rosa. "Equivariant inverse spectral theory and toric orbifolds." Advances in Mathematics (2012) : 1271-1290.