Spectral rigidity for spherically symmetric manifolds with boundary

Hoop, Maarten V. de; Ilmavirta, Joonas; Katsnelson, Vitaly

doi:10.1016/j.matpur.2021.12.009

by Maarten V. de Hoop, Joonas Ilmavirta, Vitaly Katsnelson

Abstract:

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. We then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided the Herglotz condition and a generic geometrical condition are satisfied. Combining the two results shows that the Neumann spectrum of the Laplace--Beltrami operator is rigid in this class of manifolds with boundary.

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Reference:

Spectral rigidity for spherically symmetric manifolds with boundary (Maarten V. de Hoop, Joonas Ilmavirta, Vitaly Katsnelson), Journal de mathématiques pures et appliquées, volume 160, pp. 54–98, 2022. [show abstract] [hide abstract] We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace–Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. We then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided the Herglotz condition and a generic geometrical condition are satisfied. Combining the two results shows that the Neumann spectrum of the Laplace–Beltrami operator is rigid in this class of manifolds with boundary. [arXiv]

Bibtex Entry:

@article{spherical-rigidity,
	author = {Maarten V. de Hoop and Joonas Ilmavirta and Vitaly Katsnelson},
	title = {{Spectral rigidity for spherically symmetric manifolds with boundary}},
	journal = {Journal de math\'{e}matiques pures et appliqu\'{e}es},
	month = april,
	year = {2022},
	volume = 160,
	pages = {54--98},
	arxiv = {1705.10434},
        gsid = {10231635494873617296},
        doi = {10.1016/j.matpur.2021.12.009},
	url={http://users.jyu.fi/~jojapeil/pub/spherical-rigidity.pdf},
	abstract = {We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the length spectrum. The trace formula also applies for the toroidal modes of the free oscillations in the earth. We then prove that the length spectrum is rigid: Deformations preserving the length spectrum and spherical symmetry are necessarily trivial in any dimension, provided the Herglotz condition and a generic geometrical condition are satisfied. Combining the two results shows that the Neumann spectrum of the Laplace--Beltrami operator is rigid in this class of manifolds with boundary.}
}