Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds

For a closed Riemannian manifold $(M,g)$ of dimension $n$, let ${\lambda }_1(g)$ be the first positive eigenvalue of the Laplace–Beltrami operator ${\mathrm{\Delta }}_g$ and $Vol(M,g)$ the volume of $(M,g)$. Considering the scale-invariant quantity ${\lambda }_k(g)Vol(M,g{)}^{2\mathrm{/}n}$ 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.