Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912777776398336 |
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| author | Krauz, Yoav |
| author_facet | Krauz, Yoav |
| contents | Let $C$ be a configuration of $n$ ovals in $\mathbb{S}^2$. We show that there is a Riemannian metric $g$ over $\mathbb{S}^2$ with a Laplacian eigenfunction whose zero set is $C$, and the corresponding eigenvalue is the $k$-th eigenvalue for $n\leq k \leq α_1 n$. We also have that $λ\operatorname{Vol}_g\left(\mathbb{S}^2\right) = Θ(n)$.
Additionally, assuming $C$ can be drawn as a topological minor of the $m\times m$ grid graph, we show that there is an infinitesimal perturbation of the round metric on $\mathbb{S}^2$ and a corresponding Laplacian eigenfunction $f$ with eigenvalue $Θ(m^2)$ such that the zero set of $f$ is equivalent to $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_06352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions Krauz, Yoav Spectral Theory Analysis of PDEs Differential Geometry 35P99 (Primary) 53A99 (Secondary) Let $C$ be a configuration of $n$ ovals in $\mathbb{S}^2$. We show that there is a Riemannian metric $g$ over $\mathbb{S}^2$ with a Laplacian eigenfunction whose zero set is $C$, and the corresponding eigenvalue is the $k$-th eigenvalue for $n\leq k \leq α_1 n$. We also have that $λ\operatorname{Vol}_g\left(\mathbb{S}^2\right) = Θ(n)$. Additionally, assuming $C$ can be drawn as a topological minor of the $m\times m$ grid graph, we show that there is an infinitesimal perturbation of the round metric on $\mathbb{S}^2$ and a corresponding Laplacian eigenfunction $f$ with eigenvalue $Θ(m^2)$ such that the zero set of $f$ is equivalent to $C$. |
| title | Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions |
| topic | Spectral Theory Analysis of PDEs Differential Geometry 35P99 (Primary) 53A99 (Secondary) |
| url | https://arxiv.org/abs/2501.06352 |