Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912970862231552 |
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| author | Narita, Kazumasa |
| author_facet | Narita, Kazumasa |
| contents | For a closed Riemannian manifold $(M,g)$ of dimension $n$, let $λ_{1}(g)$ be the first positive eigenvalue of the Laplace--Beltrami operator $Δ_{g}$ and $\mbox{Vol}(M,g)$ the volume of $(M, g)$. Considering the scale-invariant quantity $λ_{k}(g)\mbox{Vol}(M,g)^{2/n}$ 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 $λ_{1}$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16324 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds Narita, Kazumasa Differential Geometry Spectral Theory 58J50 For a closed Riemannian manifold $(M,g)$ of dimension $n$, let $λ_{1}(g)$ be the first positive eigenvalue of the Laplace--Beltrami operator $Δ_{g}$ and $\mbox{Vol}(M,g)$ the volume of $(M, g)$. Considering the scale-invariant quantity $λ_{k}(g)\mbox{Vol}(M,g)^{2/n}$ 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 $λ_{1}$ 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. |
| title | Second Variation Formula for the Laplace Eigenvalue Functional on Closed Manifolds |
| topic | Differential Geometry Spectral Theory 58J50 |
| url | https://arxiv.org/abs/2603.16324 |