Equivariant Isospectrality and Isospectral Deformations of Metrics on Spherical Orbifolds

Most known examples of isospectral manifolds can be constructed through variations of Sunada’s method or Gordon’s torus method. In this paper we explore these two techniques in the framework of equivariant isospectrality. We begin by establishing an equivariant version of Sunada’s technique and then we observe that many examples arising from the torus method are equivariantly isospectral. Using these observations we notice that many of the known torus method examples give rise to other (locally non-isometric) isospectral manifolds, and we also construct continuous families of isospectral metrics on orbifolds. In particular, for each finite subgroup Γ of the 2-torus there exists a spherical orbifold (resp. orbifold with boundary) O of dimension n ≥ 8 (resp. n ≥ 9) which has points with Γ-isotropy and admits a continuous family of locally non-isometric isospectral metrics (resp. Dirichlet and Neumann isospectral metrics).