Nested nodal loops for sums of Laplace eigenfunctions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913142116712448 |
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| author | Koirala, Robert |
| author_facet | Koirala, Robert |
| contents | We study nested loops in zero sets of sums of Laplace eigenfunctions on closed surfaces. In the real-analytic category, answering a question of Logunov, we prove a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. We show that this analyticity hypothesis is sharp: on a smooth sphere, a linear combination of eigenfunctions with eigenvalues \(0\) and \(2\) can have infinitely many rooted double nests. We also answer a question of Logunov and Nadirashvili by constructing a planar biharmonic function whose nodal set contains a double nest, and we prove a quantitative bound for entire biharmonic functions of polynomial growth. The biharmonic construction gives a nodal-set manifestation of the failure of the Boggio--Hadamard conjecture from the 1900s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18705 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nested nodal loops for sums of Laplace eigenfunctions Koirala, Robert Analysis of PDEs Differential Geometry Spectral Theory 35B05, 35P05, 58J50, 35J30, 31A30, 14P25 We study nested loops in zero sets of sums of Laplace eigenfunctions on closed surfaces. In the real-analytic category, answering a question of Logunov, we prove a uniform bound for the number of rooted double nests in terms of the surface, the root, and the spectral cutoff. We show that this analyticity hypothesis is sharp: on a smooth sphere, a linear combination of eigenfunctions with eigenvalues \(0\) and \(2\) can have infinitely many rooted double nests. We also answer a question of Logunov and Nadirashvili by constructing a planar biharmonic function whose nodal set contains a double nest, and we prove a quantitative bound for entire biharmonic functions of polynomial growth. The biharmonic construction gives a nodal-set manifestation of the failure of the Boggio--Hadamard conjecture from the 1900s. |
| title | Nested nodal loops for sums of Laplace eigenfunctions |
| topic | Analysis of PDEs Differential Geometry Spectral Theory 35B05, 35P05, 58J50, 35J30, 31A30, 14P25 |
| url | https://arxiv.org/abs/2605.18705 |