Riemannian manifolds with maximal eigenfunction growth

On any compact Riemannian manifold (M, g) of dimension n, the L²-normalized eigenfunctions {φλ} satisfy ∥φλ∥∞ ≤ Cλ^(n-1)/2, where -Δφλ = λ²φλ. The bound is sharp in the class of all (M, g) since it is obtained by zonal spherical harmonics on the standard n-sphere Sⁿ. But, of course, it is not sharp for many Riemannian manifolds, for example, for flat tori ℝⁿ/Γ. We say that Sⁿ, but not ℝⁿ/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_x*M. We show that if (M, g) is real analytic, this puts topological restrictions on M; for example, only M = S² or M = ℝP² (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.