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 go thru Zoll metrics, gt = eft with ft = 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 Ut = eit√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 Ck 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 σQ-1 dμ where dμ is the Liouville measure on S*M. We complete the calculation by determining σQ-1. It was conjectured [W.2][Ku] that up to universal constants, σQ-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.
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