Negative eingenvalues of the conformal Laplacian

TL;DR

This paper investigates the possible number of negative eigenvalues of the conformal Laplacian on closed manifolds, proving the existence of metrics with any number of negative eigenvalues above a certain minimum, and discussing bounds for this minimum.

Contribution

It establishes that for any number above the minimal non-positive eigenvalues, there exists a metric with exactly that many negative eigenvalues of the conformal Laplacian, and explores bounds for this minimal number.

Findings

Existence of metrics with any number of negative eigenvalues above a minimum

Upper bounds for the minimal number of non-positive eigenvalues

Characterization of the spectrum of the conformal Laplacian

Abstract

Let $M$ be a closed differentiable manifold of dimension at least $3$. Let $\Lambda_0 (M)$ be the minimun number of non-positive eigenvalues that the conformal Laplacian of a metric on $M$ can have. We prove that for any $k$ greater than or equal to $\Lambda_0 (M)$, there exists a Riemannian metric on $M$ such that its conformal Laplacian has exactly $k$ negative eigenvalues. Also, we discuss upper bounds for $\Lambda_0 (M)$.