Fine structure of zoll spectra

This paper is concerned with the wave equation on a Zoll manifold, i.e. a compact Riemannian manifold (M, g) all of whose geodesies are closed. By a result of V. Guillemin, such metrics exist in abundance on S²: for any odd function f on S² there exists a one-parameter deformation of the canonical metric g_o thru Zoll metrics, g_t = e^ft with f_t = 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 group U_t = e^it√Δ. Precisely, there exists a unitary equivalence √Δ = R + Q_-1 where [R, Q_-1] = 0, where the spectrum of R lies on an arithmetic progression and where Q_-1 is a pseudodifferential operator of order -1. Hence the eigenvalues fall into cluster C_k around spec(R) with widths O(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 form σ_Q-1 dμ where dμ is the Liouville measure on S*M. We complete the calculation by determining σ_Q-1. It was conjectured [W.2][Ku] that up to universal constants, σ_Q-1 =R(τ) where R is the Radon transform of (M, g) and τ is 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.