Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910001933582336 |
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| author | Morozov, Egor |
| author_facet | Morozov, Egor |
| contents | We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the $k$-th normalized Laplace-Beltrami eigenvalue functional $\barλ_k$ in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain $\mathbb S^1$-equivariant harmonic maps $\mathbb T^2\to\mathbb S^3$. For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for $\barλ_1$ in this conformal class with the value of $\barλ_1$ being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for $\barλ_1$ in its conformal class, then it is $\mathbb S^1$-equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in $\mathbb S^3$ called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_19196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class Morozov, Egor Differential Geometry Spectral Theory 58E11, 58E20, 58J50 We study the problem of maximizing the first Laplace-Beltrami eigenvalue normalized by area in a conformal class on a torus. By a result of Nadirashvili, El Soufi, and Ilias, critical metrics for the $k$-th normalized Laplace-Beltrami eigenvalue functional $\barλ_k$ in a conformal class correspond to harmonic maps to spheres. In this paper we construct certain $\mathbb S^1$-equivariant harmonic maps $\mathbb T^2\to\mathbb S^3$. For each non-rhombic conformal class on a torus, one of these maps corresponds to a rotationally symmetric critical metric for $\barλ_1$ in this conformal class with the value of $\barλ_1$ being greater than that of the flat metric. This refines a recent result by Karpukhin that answers a question by El Soufi, Ilias, and Ros. Also, we are able to show that if a rotationally invariant metric on a rectangular torus is maximal for $\barλ_1$ in its conformal class, then it is $\mathbb S^1$-equivariant and coincides (up to a scalar factor) with the above metric. Finally, we show that a family of minimal tori in $\mathbb S^3$ called Otsuki tori fits naturally into our family. This gives an explicit parametrization of Otsuki tori in terms of elliptic integrals. |
| title | Rotationally symmetric critical metrics for Laplace eigenvalues on tori in a conformal class |
| topic | Differential Geometry Spectral Theory 58E11, 58E20, 58J50 |
| url | https://arxiv.org/abs/2601.19196 |