Riemannian Manifolds with Maximal Eigenfunction Growth

On any compact Riemannian manifold (M,g) of dimension n, the Lnormalized eigenfunctions {φλ} satisfy ||φλ||∞ ≤ Cλ n−1 2 where −∆φλ = λ 2φλ. The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere S. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori R/Γ. We say that S, but not R/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the (M, g) with maximal eigenfunction growth. Our main result is that such an (M, g) must have a point x where the set Lx of geodesic loops at x has positive measure in S xM . We show that if (M, g) is real analytic, this puts topological restrictions on M , e.g. only M = S (topologically) in dimension 2 can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any M fail to have maximal eigenfunction growth. In addition, we construct an example of (M, g) for which Lx has positive measure for an open set of x but which does not have maximal eigenfunction growth, thus disproving a naive converse to the main result.