Bounds for eigenfunctions of the Laplacian on noncompact Riemannian manifolds

TL;DR

This paper establishes sharp conditions under which eigenfunctions of the Laplacian on noncompact, finite-volume Riemannian manifolds are bounded in various L^q spaces, using geometric functions like isoperimetric and isocapacitary functions.

Contribution

It provides new sharp criteria linking geometric properties of the manifold to bounds on Laplacian eigenfunctions, extending previous results to noncompact settings.

Findings

Derived sharp conditions for L^q and L^ bounds

Connected geometric functions to eigenfunction bounds

Extended analysis to noncompact finite-volume 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 $M$.