# Noncompact complete Riemannian manifolds with singular continuous spectrum embedded into the essential spectrum of the Laplacian, I. The hyperbolic case

## TL;DR

This paper constructs asymptotically hyperbolic Riemannian manifolds with specific spectral properties, embedding singular continuous spectrum into the Laplacian's essential spectrum, advancing understanding of spectral theory in geometric analysis.

## Contribution

It introduces a method to construct manifolds with embedded singular continuous spectrum within the Laplacian's essential spectrum, particularly in the hyperbolic setting.

## Key findings

- Manifolds with embedded singular continuous spectrum are constructed.
- The manifolds are asymptotically hyperbolic with sharp curvature bounds.
- The spectral properties of the Laplacian on these manifolds are characterized.

## Abstract

We construct Riemannian manifolds with singular continuous spectrum embedded in the absolutely continuous spectrum of the Laplacian. Our manifolds are asymptotically hyperbolic with sharp curvature bounds.