# A note on eigenvalue bounds for non-compact manifolds

## TL;DR

This paper establishes upper bounds for Laplace eigenvalues on certain non-compact negatively curved manifolds, linking spectral properties to geometric data, and contrasting with graph Laplacians where such bounds do not hold.

## Contribution

It provides new eigenvalue bounds for non-compact negatively curved manifolds, extending spectral theory in geometric analysis.

## Key findings

- Eigenvalues are bounded above by a quadratic function of the index k.
- The bounds depend explicitly on geometric data of the manifold.
- Contrast with graph Laplacians where such bounds are not generally valid.

## Abstract

In this article we prove upper bounds for the Laplace eigenvalues $\lambda_k$ below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of $k^2$ and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to $-\infty$, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.