Bounds for eigenfunctions of the Laplacian on compact Riemannian manifolds

Suppose that phi is an eigenfunction of -Delta with eigenvalue lambda not equal 0. It is proved that \ \ phi \ \ (infinity) less than or equal to c(1)lambda (n-1/4)\ \ phi \ \ (2), where n is the dimension of M and c(1), 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 \ \ phi (k)\ \ (infinity) greater than or equal to c(2)lambda (n-1/8)(k)\ \ phi \ \ (2.) (C) 2001 Academic Press.