Unbounded Topology of Nodal Sets of Harmonic Functions

For every integer $n\ge 3$, every $1\le \mathrm{\ell }\le n-2$, and every sufficiently large integer $m$, we construct harmonic functions $u_{m,\mathrm{\ell }}$ 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 $\{u_{m,\mathrm{\ell }}=0\}\cap B_{1\mathrm{/}2}(0)$ 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.