Constructing Riemannian metrics with prescribed nodal sets for Laplacian eigenfunctions

TL;DR

The paper shows that any configuration of non-intersecting ovals on $\mathbb{S}^2$ can be realized as the nodal set of a Laplacian eigenfunction by appropriately deforming the metric, with quantitative control on the eigenvalue scale and curvature. The authors introduce a three-block construction and an interpolation framework that converts the metric design problem into a vector-field problem, enabling extension of local eigenfunction data to a global metric. They also develop a blueprint-based reduction to handle level-set data and provide a perturbation result: if the oval configuration is grid-drawable, an infinitesimal round-metric perturbation yields an eigenfunction with the prescribed nodal pattern and a quadratic eigenvalue growth in the grid size. Overall, the work constructs explicit metrics realizing rich nodal topologies on $\mathbb{S}^2$ and gives sharp-linear-scale eigenvalue bounds tied to the nodal graph complexity.

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 $n\leq k \leq α_1 n$. We also have that $λ\operatorname{Vol}_g\left(\mathbb{S}^2\right) = Θ(n)$. 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 $Θ(m^2)$ such that the zero set of $f$ is equivalent to $C$.

Figures (8)

• Figure 1: An example of a configuration $C$ of $5$ ovals and a drawing of $C$ in the $5\times 5$ grid graph
• Figure 2: The region $M$ (in gray) used to build the third simple block type, and level lines of the function $f_0$. The blue line is the level line $f_0=0$ (which will become the Dirichlet-type boundary component); the red lines are the level lines $f_0=1$ (which will become the Neumann-type boundary components).
• Figure 3: Example of a blueprint $\pi:X\to\mathbb{R}$ with $\operatorname{Sing}{X}=\left\{p_1,p_2,p_3,p_4\right\}$. This $X$ is built from two copies of $\mathbb{R}$ by identifying the two copies of $\left(0,1\right)\subset \mathbb{R}$ together.
• Figure 4: Values of $d_C(p)$ given how $C$ looks like in a neighborhood of $p$. Here a full line represents points that belong to $C$; a dashed line represents points that do not belong to $C$.
• Figure 5: Perturbation of a vertex

Theorems & Definitions (44)

• theorem 2.0.1
• definition 1
• theorem 2.0.2
• lemma 1
• definition 2
• lemma 2
• proof : lemma \ref{['lemma:blocks']}
• theorem 6.2.1
• lemma 3
• proof : End of proof of lemma \ref{['lemma:blocks']}