An improved upper bound for the second eigenvalue on tori
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911599530344448 |
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| author | Kang, Fan |
| author_facet | Kang, Fan |
| contents | In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $λ_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for $λ_2(T_{a,b},g)$ on any flat torus $T_{a, b}$ with $(a, b)\in \mathbb{R}^2$. Our bound improves the general estimate $λ_2(T_{a, b},g)\le 4A_c(T_{a, b}, [g])$ in the case of the torus. As applications, we derive a uniform upper bound $λ_2(T,g)< \frac{16π^2}{\sqrt{3}}$ for any torus $T$ and any metric $g$, and reduce the Kao-Lai-Osting conjecture to proving an upper bound for $λ_2(T_{a,b},g)$ on the specific family of flat tori $T_{a,b}$ with $0\leq a\leq \frac12$ and $\sqrt{1-a^2}\leq b\leq 1.76$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05846 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An improved upper bound for the second eigenvalue on tori Kang, Fan Differential Geometry Spectral Theory In this paper, we study the maximization problem of the second non-zero Laplace eigenvalue $λ_2(T,g)$ on a torus $T$, among all unit-area metrics in a fixed conformal class. First, we obtain a new upper bound for $λ_2(T_{a,b},g)$ on any flat torus $T_{a, b}$ with $(a, b)\in \mathbb{R}^2$. Our bound improves the general estimate $λ_2(T_{a, b},g)\le 4A_c(T_{a, b}, [g])$ in the case of the torus. As applications, we derive a uniform upper bound $λ_2(T,g)< \frac{16π^2}{\sqrt{3}}$ for any torus $T$ and any metric $g$, and reduce the Kao-Lai-Osting conjecture to proving an upper bound for $λ_2(T_{a,b},g)$ on the specific family of flat tori $T_{a,b}$ with $0\leq a\leq \frac12$ and $\sqrt{1-a^2}\leq b\leq 1.76$. |
| title | An improved upper bound for the second eigenvalue on tori |
| topic | Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2506.05846 |