Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds

Abstract

We deal with eigenvalue problems for the Laplacian on noncompact Riemannian manifolds $M$ of finite volume. Sharp conditions ensuring $L^q(M)$ and $L^{\infty }(M)$ bounds for eigenfunctions are exhibited in terms of either the isoperimetric function or the isocapacitary function of $M$.

48 pages, 3 figures