Unbounded Topology of Nodal Sets of Harmonic Functions
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918523105705984 |
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| author | Koirala, Robert |
| author_facet | Koirala, Robert |
| contents | For every integer \(n\ge 3\), every \(1\le \ell\le n-2\), and every sufficiently large integer \(m\), we construct harmonic functions \(u_{m,\ell}\) on the unit ball \(B_1(0)\subset\mathbb{R}^n\) such that the frequency is bounded independently of \(m\), every point of the nodal set \(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\) is regular, but the Betti numbers satisfy
\begin{align*}
b_\ell\bigl(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\bigr)\ge 2m.
\end{align*} Thus bounded frequency, even together with regularity of the nodal set, does not imply a uniform topological bound. In particular, these examples give counterexamples to the claimed global Betti-number bound of Lin and Liu. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unbounded Topology of Nodal Sets of Harmonic Functions Koirala, Robert Analysis of PDEs Differential Geometry Geometric Topology 35J05, 35B05, 57R19 For every integer \(n\ge 3\), every \(1\le \ell\le n-2\), and every sufficiently large integer \(m\), we construct harmonic functions \(u_{m,\ell}\) on the unit ball \(B_1(0)\subset\mathbb{R}^n\) such that the frequency is bounded independently of \(m\), every point of the nodal set \(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\) is regular, but the Betti numbers satisfy \begin{align*} b_\ell\bigl(\{u_{m,\ell}=0\}\cap B_{1/2}(0)\bigr)\ge 2m. \end{align*} Thus bounded frequency, even together with regularity of the nodal set, does not imply a uniform topological bound. In particular, these examples give counterexamples to the claimed global Betti-number bound of Lin and Liu. |
| title | Unbounded Topology of Nodal Sets of Harmonic Functions |
| topic | Analysis of PDEs Differential Geometry Geometric Topology 35J05, 35B05, 57R19 |
| url | https://arxiv.org/abs/2605.26274 |