Riemannian manifolds with maximal eigenfunction growth

On any compact Riemannian manifold $(M, g)$ of dimension n, the L 2-normalized eigenfunctions \φ λ\ satisfy φ λ ∞≤ Cλ (n-1)/2, where -Δφ λ=λ 2φ λ. 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 n. But, of course, it is not sharp for many Riemannian manifolds, for example, for flat tori R n/Γ. We say that S n, but not R n/Γ, is a Riemannian manifold with maximal eigenfunction growth. The problem that 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 L x 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; for example, only M=S 2 or M=RP 2 (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 L x has positive measure for an open set of x but which does not have maximal eigenfunction growth; thus, it disproves a naive converse to the main result.