Extremal Eigenvalues Of The Conformal Laplacian Under Sire-Xu Normalization

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 $k=1$.

References in corpus (2)

• Laplacian eigenvalues functionals and metric deformations on compact manifolds
• Variational Properties Of The Second Eigenvalue Of The Conformal Laplacian