High-energy eigenfunctions of the Laplacian on the torus and the sphere with nodal sets of complicated topology

TL;DR

This paper constructs high-energy Laplacian eigenfunctions on spheres and tori with nodal sets that have complex topologies, matching any given hypersurface, demonstrating rich geometric structures of eigenfunctions.

Contribution

It shows that for large odd integers, there exist eigenfunctions with nodal sets diffeomorphic to any specified hypersurface in spheres or tori, revealing intricate topological possibilities.

Findings

Existence of eigenfunctions with prescribed nodal topology

Construction of eigenfunctions with complex nodal set components

Demonstration of topological richness in high-energy eigenfunctions

Abstract

Let $\Sigma$ be an oriented compact hypersurface in the round sphere $\mathbb{S}^n$ or in the flat torus $\mathbb{T}^n$, $n\geq 3$. In the case of the torus, $\Sigma$ is further assumed to be contained in a contractible subset of $\mathbb{T}^n$. We show that for any sufficiently large enough odd integer $N$ there exists an eigenfunctions $\psi$ of the Laplacian on $\mathbb{S}^n$ or $\mathbb{T}^n$ satisfying $\Delta \psi=-\lambda \psi$ (with $\lambda=N(N+n-1)$ or $N^2$ on $\mathbb{S}^n$ or $\mathbb{T}^n$, respectively), and with a connected component of the nodal set of $\psi$ given by~$\Sigma$, up to an ambient diffeomorphism.