First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908568146411520 |
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| author | Matinpour, Elham |
| author_facet | Matinpour, Elham |
| contents | Consider $\mathbb{R}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $Σ$ be a complete embedded self-shrinker in $\mathbb{R}^3$ with the induced metric and weighted measure, and let $λ_1$ denote the first eigenvalue of the drift Laplacian in the weighted $L^2$ space. Inspired by Choe and Soret's estimate of the first eigenvalue of the Laplacian on symmetric minimal surfaces in $\mathbb{S}^3$, we prove that $λ_1$= 1/2 for self-shrinkers invariant under the dihedral group $\mathbb{D}_{g+1}$ or the prismatic group $\mathbb{D}_{g+1}\times \mathbb{Z}_2$. In particular, this holds for known self-shrinkers confirming a universal spectral property tied to their symmetry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25617 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$ Matinpour, Elham Differential Geometry Consider $\mathbb{R}^3$ equipped with the Euclidean metric and the Gaussian measure. Let $Σ$ be a complete embedded self-shrinker in $\mathbb{R}^3$ with the induced metric and weighted measure, and let $λ_1$ denote the first eigenvalue of the drift Laplacian in the weighted $L^2$ space. Inspired by Choe and Soret's estimate of the first eigenvalue of the Laplacian on symmetric minimal surfaces in $\mathbb{S}^3$, we prove that $λ_1$= 1/2 for self-shrinkers invariant under the dihedral group $\mathbb{D}_{g+1}$ or the prismatic group $\mathbb{D}_{g+1}\times \mathbb{Z}_2$. In particular, this holds for known self-shrinkers confirming a universal spectral property tied to their symmetry. |
| title | First eigenvalue and nodal domains of the drift Laplacian on symmetric self-shrinkers in $\mathbb{R}^3$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2509.25617 |