Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions

Abstract

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 $n\leq k \leq \alpha_1 n$. We also have that $\lambda\operatorname{Vol}_g\left(\mathbb{S}^2\right) = \Theta(n)$. 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 $\Theta(m^2)$ such that the zero set of $f$ is equivalent to $C$.