Extremal Eigenvalues Of The Conformal Laplacian Under Sire-Xu Normalization

TL;DR

This paper investigates the extremal properties of the $k$-th eigenvalue of the conformal Laplacian on closed Riemannian manifolds under a specific normalization, providing conditions for extremality and existence results.

Contribution

It introduces a new normalization for eigenvalue optimization and analyzes the conditions for extremal eigenvalues, including the case when $k=1$, advancing understanding in spectral geometry.

Findings

Necessary conditions for extremal eigenvalues under Sire-Xu normalization.

Discussion of existence problems for the first eigenvalue.

Insights into variational properties of eigenvalues in conformal classes.

Abstract

Let $(M^n,g)$ be a closed Riemannian manifold of dimension $n\ge 3$. We study the variational properties of the $k$-th eigenvalue functional $\tilde g\in[g] \mapsto \lambda_k(L_{\tilde g})$ under a non-volume normalization proposed by Sire-Xu. We discuss necessary conditions for the existence of extremal eigenvalues under such normalization. Also, we discuss the general existence problem when $k=1$.