Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions

Let $C$ be a configuration of $n$ ovals in ${\mathbb{S}}^2$. We show that there is a Riemannian metric $g$ over ${\mathbb{S}}^2$ with a Laplacian eigenfunction whose zero set is $C$, and the corresponding eigenvalue is the $k$-th eigenvalue for $n\le k\le {\alpha }_{1}n$. We also have that $\lambda {\mathrm{Vol}}_g\left({\mathbb{S}}^2\right)=\mathrm{\Theta }(n)$. Additionally, assuming $C$ can be drawn as a topological minor of the $m\times m$ grid graph, we show that there is an infinitesimal perturbation of the round metric on ${\mathbb{S}}^2$ and a corresponding Laplacian eigenfunction $f$ with eigenvalue $\mathrm{\Theta }(m^2)$ such that the zero set of $f$ is equivalent to $C$.