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- W4309131505 abstract "We prove that the Beltrami operator on closed oriented 3-manifolds, i.e. the square root of the Hodge Laplacian on its coclosed spectrum, generically has 1-dimensional eigenspaces, even along !-parameter families of $mathcal{C}^k$ Riemannian metrics, where $kgeq 2$. We show further that the Hodge Laplacian in dimension 3 has two possible sources for nonsimple eigenspaces along generic 1-parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the Beltrami operator cross, or a closed and a coclosed eigenvalues cross. We provide examples for both of these phenomena. In order to prove our results we generalize a method of Teytel, allowing us to compute the meagre codimension of the set of Riemannian metrics for which the Beltrami operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension $3$ is a meagre codimension 1 property with respect to the $mathcal{C}^k$ topology (as proven by Enciso and Peralta-Salas), it is not a meagre codimension 2 property." @default.
- W4309131505 created "2022-11-23" @default.
- W4309131505 creator A5085295444 @default.
- W4309131505 date "2022-11-13" @default.
- W4309131505 modified "2023-10-14" @default.
- W4309131505 title "In Dimension 3 the Beltrami Operator has simple Spectrum along generic 1-Parameter Families of Riemannian metrics, but the Hodge Laplacian does not" @default.
- W4309131505 doi "https://doi.org/10.48550/arxiv.2211.06916" @default.
- W4309131505 hasPublicationYear "2022" @default.
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