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

2018/10/19 by Enciso, Alberto, Peralta-Salas, Daniel, de Lizaur, Francisco Torres · 1 citation
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1810.09277

Abstract

Let Σ be an oriented compact hypersurface in the round sphere \mathbbSⁿ or in the flat torus \mathbbTⁿ, n≥ 3. In the case of the torus, Σ is further assumed to be contained in a contractible subset of \mathbbTⁿ. We show that for any sufficiently large enough odd integer N there exists an eigenfunctions ψ of the Laplacian on \mathbbSⁿ or \mathbbTⁿ satisfying Δψ=-λψ (with λ=N(N+n-1) or N² on \mathbbSⁿ or \mathbbTⁿ, respectively), and with a connected component of the nodal set of ψ given by~Σ, up to an ambient diffeomorphism.

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