Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913826804334592 |
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| author | Karpukhin, Mikhail Petrides, Romain Stern, Daniel |
| author_facet | Karpukhin, Mikhail Petrides, Romain Stern, Daniel |
| contents | Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof of existence in the orientable case complementing that of [Pet24b], thus resolving the long-standing existence problem for $λ_1$-maximizing metrics on closed surfaces of any topology. Namely, we prove by contradiction that the supremum $Λ_1(M)$ of the normalized first eigenvalue over all metrics on $M$ obeys the strict monotonicity $Λ_1(M\#\mathbb{RP}^2)>Λ_1(M)$ and $Λ_1(M\#\mathbb{T}^2)>Λ_1(M)$ under the attachment of cross-caps and handles, via a substantial refinement of techniques introduced in [KKMS24]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05293 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces Karpukhin, Mikhail Petrides, Romain Stern, Daniel Differential Geometry Analysis of PDEs Spectral Theory Building on seminal work of Nadirashvili and previous work of the authors, we prove the existence of metrics maximizing the area-normalized first eigenvalue of the Laplacian on every closed nonorientable surface, and give a simple new proof of existence in the orientable case complementing that of [Pet24b], thus resolving the long-standing existence problem for $λ_1$-maximizing metrics on closed surfaces of any topology. Namely, we prove by contradiction that the supremum $Λ_1(M)$ of the normalized first eigenvalue over all metrics on $M$ obeys the strict monotonicity $Λ_1(M\#\mathbb{RP}^2)>Λ_1(M)$ and $Λ_1(M\#\mathbb{T}^2)>Λ_1(M)$ under the attachment of cross-caps and handles, via a substantial refinement of techniques introduced in [KKMS24]. |
| title | Existence of metrics maximizing the first Laplace eigenvalue on closed surfaces |
| topic | Differential Geometry Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2505.05293 |