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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.26274 |
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Table of 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.