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Erschienen in:
2021 | OriginalPaper | Buchkapitel
High-Energy Eigenfunctions of the Laplacian on the Torus and the Sphere with Nodal Sets of Complicated Topology
verfasst von : A. Enciso, D. Peralta-Salas, F. Torres de Lizaur
Erschienen in: Nonlinear Partial Differential Equations for Future Applications
Verlag: Springer Singapore
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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.