Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian

TL;DR

This paper investigates the variational properties of the second eigenvalue of the conformal Laplacian on closed Riemannian manifolds, establishing existence of maximizers and characterizing their geometric and analytical features.

Contribution

It proves the existence of metrics that maximize the second eigenvalue of the conformal Laplacian and characterizes these maximizers as solutions to either the Yamabe equation or harmonic maps to spheres.

Findings

Existence of a generalized metric maximizing the second eigenvalue.

Maximal metrics correspond to nodal solutions or harmonic maps.

Constructed examples of both types of maximal metrics.

Abstract

Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. Assume $[g]$ is a conformal class for which the Conformal Laplacian $L_g$ has at least two negative eigenvalues. We show the existence of a (generalized) metric that maximizes the second eigenvalue of $L_g$ over all conformal metrics (the first eigenvalue is maximized by the Yamabe metric). We also show that a maximal metric defines either a nodal solution of the Yamabe equation, or a harmonic map to a sphere. Moreover, we construct examples of each possibility.