Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions

TL;DR

This paper demonstrates how to construct Riemannian metrics on the sphere with Laplacian eigenfunctions whose zero sets match prescribed configurations of ovals, linking geometric structures to spectral properties.

Contribution

It introduces methods to realize specific nodal sets as zero sets of Laplacian eigenfunctions on the sphere, including perturbations of the round metric for complex configurations.

Findings

Existence of metrics with prescribed nodal sets for given configurations.

Eigenvalues scale linearly with the number of ovals in the configuration.

Perturbation techniques produce eigenfunctions with complex zero set topologies.

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 $C$.