Zero and negative eigenvalues of the conformal Laplacian

TL;DR

This paper investigates the spectral properties of the conformal Laplacian, demonstrating that zero eigenvalues are non-generic and exploring the behavior of negative eigenvalues in metric sequences.

Contribution

It establishes that zero is typically not an eigenvalue of the conformal Laplacian and analyzes the non-compactness phenomena related to negative eigenvalues.

Findings

Zero is not an eigenvalue for generic metrics

Non-compactness occurs with sequences of metrics having increasing negative eigenvalues

Insights into the spectral stability of the conformal Laplacian

Abstract

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.