On the critical points of Steklov eigenfunctions
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910642961645568 |
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| author | Battaglia, Luca Pistoia, Angela Provenzano, Luigi |
| author_facet | Battaglia, Luca Pistoia, Angela Provenzano, Luigi |
| contents | We consider the critical points of Steklov eigenfunctions on a compact, smooth $n$-dimensional Riemannian manifold $M$ with boundary $\partial M$. For generic metrics on $M$ we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to $\partial M$, and the Euler characteristic of $M$. In dimension $2$ this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of $M$ and of the number of sign changes of $u$ on $\partial M$. In the case of the second Steklov eigenfunction on a genus $0$ surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on $M$ Steklov eigenfunctions are Morse functions in $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01190 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the critical points of Steklov eigenfunctions Battaglia, Luca Pistoia, Angela Provenzano, Luigi Analysis of PDEs Differential Geometry Spectral Theory We consider the critical points of Steklov eigenfunctions on a compact, smooth $n$-dimensional Riemannian manifold $M$ with boundary $\partial M$. For generic metrics on $M$ we establish an identity which relates the sum of the indexes of a Steklov eigenfunction, the sum of the indexes of its restriction to $\partial M$, and the Euler characteristic of $M$. In dimension $2$ this identity gives a precise count of the interior critical points of a Steklov eigenfunction in terms of the Euler characteristic of $M$ and of the number of sign changes of $u$ on $\partial M$. In the case of the second Steklov eigenfunction on a genus $0$ surface, the identity holds for any metric. As a by-product of the main result, we show that for generic metrics on $M$ Steklov eigenfunctions are Morse functions in $M$. |
| title | On the critical points of Steklov eigenfunctions |
| topic | Analysis of PDEs Differential Geometry Spectral Theory |
| url | https://arxiv.org/abs/2402.01190 |