The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian
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arXiv
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| Natura: | Preprint |
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2021
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| _version_ | 1866912046691385344 |
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| author | Bourque, Maxime Fortier Petri, Bram |
| author_facet | Bourque, Maxime Fortier Petri, Bram |
| contents | We prove that Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$, with multiplicity equal to $8$. We also obtain partial results in genus $2$, where we find that the maximum multiplicity is between $3$ and $6$. Along the way, we show that for every $g\geq 2$, there exists some $δ_g>0$ such that the multiplicity of any eigenvalue of the Laplacian on a closed hyperbolic surface of genus $g$ in the interval $[0,1/4+δ_g]$ is at most $2g-1$ despite the fact that this interval can contain arbitrarily many eigenvalues. This extends a result of Otal to a larger interval but with a weaker bound, which nevertheless improves upon the general upper bound of Sévennec. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_14699 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian Bourque, Maxime Fortier Petri, Bram Geometric Topology Differential Geometry Spectral Theory We prove that Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian among all closed hyperbolic surfaces of genus $3$, with multiplicity equal to $8$. We also obtain partial results in genus $2$, where we find that the maximum multiplicity is between $3$ and $6$. Along the way, we show that for every $g\geq 2$, there exists some $δ_g>0$ such that the multiplicity of any eigenvalue of the Laplacian on a closed hyperbolic surface of genus $g$ in the interval $[0,1/4+δ_g]$ is at most $2g-1$ despite the fact that this interval can contain arbitrarily many eigenvalues. This extends a result of Otal to a larger interval but with a weaker bound, which nevertheless improves upon the general upper bound of Sévennec. |
| title | The Klein quartic maximizes the multiplicity of the first positive eigenvalue of the Laplacian |
| topic | Geometric Topology Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2111.14699 |