Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds

TL;DR

This paper derives a second variation formula for the scale-invariant Laplace eigenvalue functional on closed manifolds and shows the flat metric on a torus with multiplicity two is not a maximum, extending previous work.

Contribution

It provides a second variation formula for the eigenvalue functional in conformal classes and extends results about maximal metrics to higher dimensions.

Findings

The second variation formula for the eigenvalue functional is established.

The flat metric on a torus with multiplicity two eigenvalue is not a maximum.

Extension of Karpukhin's results to higher dimensions.

Abstract

For a closed Riemannian manifold $(M,g)$ of dimension $n$, let $\lambda_{1}(g)$ be the first positive eigenvalue of the Laplace--Beltrami operator $\Delta_{g}$ and $\mbox{Vol}(M,g)$ the volume of $(M, g)$. Considering the scale-invariant quantity $\lambda_{k}(g)\mbox{Vol}(M,g)^{2/n}$ as a functional over all the metrics in a fixed conformal class, we derive a second variation formula for the functional. As a corollary, we prove that if the canonical flat metric on a torus is such that the multiplicity of $\lambda_{1}$ is two, then the flat metric is not a maximal point of the functional in its conformal class. This is a higher dimensional extension of Karpukhin's very recent work.