Equivariant inverse spectral theory and toric orbifolds

Abstract

Let O[superscript 2n] be a symplectic toric orbifold with a fixed T[superscript n]-action and with a toric Kähler metric g. In [10] we explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator Δ[subscript g] on C[superscript ∞](O) determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results and show that a generic toric 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 Kähler metric has constant scalar curvature.