A refinement of strong multiplicity one for spectra of hyperbolic manifolds

Abstract

Let $\mathcal {M}_1$ and $\mathcal {M}_2$ denote two compact hyperbolic manifolds. Assume that the multiplicities of eigenvalues of the Laplacian acting on $L^2(\mathcal {M}_1)$ and $L^2(\mathcal {M}_2)$ (respectively, multiplicities of lengths of closed geodesics in $\mathcal {M}_1$ and $\mathcal {M}_2$) are the same, except for a possibly infinite exceptional set of eigenvalues (respectively lengths). We define a notion of density for the exceptional set and show that if it is below a certain threshold, the two manifolds must be isospectral.

Locations

• arXiv (Cornell University) - View - PDF
• Transactions of the American Mathematical Society - View - PDF