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Conference Paper

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

##### External Resource

https://doi.org/10.1007/978-981-33-4822-6_7

(Publisher version)

##### Fulltext (public)

1810.09277.pdf

(Preprint), 201KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Enciso, A., Peralta-Salas, D., & Torres de Lizaur, F. (2021). High-energy eigenfunctions
of the Laplacian on the torus and the sphere with nodal sets of complicated topology. In S. Koike (*Nonlinear Partial Differential Equations for Future Applications: Sendai, Japan, July 10–28 and October
2–6, 2017* (pp. 245-261). Singapore: Springer.

Cite as: http://hdl.handle.net/21.11116/0000-0008-B537-E

##### 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.