Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

TL;DR

This paper establishes precise curvature conditions for Riemannian manifolds to have dense eigenvalues embedded in the Laplacian spectrum, and constructs examples with sharp curvature bounds.

Contribution

It provides sharp criteria on radial curvature for manifolds with embedded eigenvalues and constructs explicit examples with dense embedded spectra.

Findings

Criteria for radial curvature ensuring embedded eigenvalues

Construction of manifolds with dense embedded point spectrum

Sharp curvature bounds for such manifolds

Abstract

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.