Fine Structure of Zoll Spectra

Abstract

This paper is concerned with the wave equation on a Zoll manifold, i.e. a compact Riemannian manifold (M, g) all of whose geodesics are closed. By a result of V. Guillemin, such metrics exist in abundance onS2: for any odd functionfonS2there exists a one-parameter deformation of the canonical metricgothru Zoll metrics,gt=eftwithft=tf+…. And, as has been known for a long time [DG][CV.1][HoIV][W.1,2], the periodicity of the geodesic flow leads to a near periodicity in the wave groupUt=eitΔ. Precisely, there exists a unitary equivalenceΔ=R+Q−1where [R, Q−1]=0, where the spectrum ofRlies on an arithmetic progression and whereQ−1is a pseudodifferential operator of order −1. Hence the eigenvalues fall into clusterCkaround spec(R) with widthsO(k−1). The main result of this paper is to determine the limit distribution of the eigenvalues in the clusters. It was long ago shown by Weinsten [loc. cit] that (when properly normalized) the limit distribution has the formσQ−1dμwheredμis the Liouville measure onS*M. We complete the calculation by determiningσQ−1. It was conjectured [W.2][Ku] that up to universal constants,σQ−1=R(τ) whereRis the Radon transform of (M, g) andτis the scalar curvature [W.2][Ku]; but it turns out there is a second term involving the Jacobi fields. In addition, we show that the cluster projections define almost isometric minimal immersions of (M, g) into spheres and that in the case of maximally degenerate Zoll metrics the errors are of rapid decay.