Bounds for Eigenfunctions of the Laplacian on Compact Riemannian Manifolds

Suppose that φ is an eigenfunction of −Δ with eigenvalue λ≠0. It is proved that ‖φ‖∞⩽c1λn−14 ‖φ‖2, where n is the dimension of M and c1 depends only upon a bound for the absolute value of the sectional curvature of M and a lower bound for the injectivity radius of M. It is then shown that if M admits an isometric circle action, and the metric is generic, one has exceptional sequences of eigenfunctions satisfying the complementary bounds ‖φk‖∞⩾c2λn−18k ‖φ‖2.