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

Abstract

Let $\Sigma$ be an oriented compact hypersurface in the round sphere ${\mathbb{S}}^n$ or in the flat torus ${\mathbb{T}}^n$, $n\ge 3$. 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.

14 pages. arXiv admin note: text overlap with arXiv:1712.10310, arXiv:1505.01605