The supremum of conformally covariant eigenvalues in a conformal class

TL;DR

This paper proves that within any conformal class of a compact Riemannian manifold, one can find metrics of volume 1 making the first positive eigenvalues of the conformal Laplacian and Dirac operator arbitrarily large, highlighting extreme spectral behavior.

Contribution

It establishes the unboundedness of the first positive eigenvalues of key geometric operators within conformal classes, extending to both Laplacian and Dirac operators.

Findings

Existence of conformal metrics with arbitrarily large eigenvalues

Results apply to manifolds of dimension >2 for Laplacian

Results apply to spin manifolds of dimension >1 for Dirac operator

Abstract

Let (M,g) be a compact Riemannian manifold of dimension >2. We show that there is a metric h conformal to g and of volume 1 such that the first positive eigenvalue the conformal Laplacian with repect to h is arbitrarily large. A similar statement is proven for the first positive eigenvalue of the Dirac operator on a spin manifold of dimension >1.