Maximizing higher eigenvalues in dimensions three and above
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913938750308352 |
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| author | Vinokurov, Denis |
| author_facet | Vinokurov, Denis |
| contents | We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$.
For dimensions $3 \leq m \leq 6$, we generalize the work of Karpukhin and Stern on the first eigenvalue, showing that the maximizing measures are realized by smooth harmonic maps into finite-dimensional spheres.
For $m \geq 7$, the maximizing measures are again induced by harmonic maps, which may now exhibit singularities. We prove that $m-7$ is the optimal upper bound for the Hausdorff dimension of the singular set. More precisely, for any $m \geq 7$, there exist maximizing harmonic maps on the $m$-dimensional sphere whose singular sets have any prescribed integer dimension up to $m - 7$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_09328 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximizing higher eigenvalues in dimensions three and above Vinokurov, Denis Spectral Theory Analysis of PDEs Differential Geometry 58J50 (Primary) 58E20, 53C43 (Secondary) We study the problem of maximizing the $k$-th eigenvalue functional over the class of absolutely continuous measures on a closed Riemannian manifold of dimension $m\geq 3$. For dimensions $3 \leq m \leq 6$, we generalize the work of Karpukhin and Stern on the first eigenvalue, showing that the maximizing measures are realized by smooth harmonic maps into finite-dimensional spheres. For $m \geq 7$, the maximizing measures are again induced by harmonic maps, which may now exhibit singularities. We prove that $m-7$ is the optimal upper bound for the Hausdorff dimension of the singular set. More precisely, for any $m \geq 7$, there exist maximizing harmonic maps on the $m$-dimensional sphere whose singular sets have any prescribed integer dimension up to $m - 7$. |
| title | Maximizing higher eigenvalues in dimensions three and above |
| topic | Spectral Theory Analysis of PDEs Differential Geometry 58J50 (Primary) 58E20, 53C43 (Secondary) |
| url | https://arxiv.org/abs/2506.09328 |