Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza-Klein \(3\)-folds

This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian \(3\)-manifolds, namely nontrivial principal \(S^1\) bundles \(P \to X\) over Riemann surfaces equipped with certain \(S^1\) invariant metrics, the Kaluza-Klein metrics. We prove for generic Ka...

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Bibliographic Details
Published in:arXiv.org 2019-10
Main Authors: Jung, Junehyuk, Zelditch, Steve
Format: Article
Language:English
Subjects:
Domains
Eigenvectors
Invariants
Riemann surfaces
Toruses
Online Access:Get full text
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Summary:This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian \(3\)-manifolds, namely nontrivial principal \(S^1\) bundles \(P \to X\) over Riemann surfaces equipped with certain \(S^1\) invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the \(S^1\) action. We also construct an explicit orthonormal eigenbasis on the flat \(3\)-torus \(\mathbb{T}^3\) for which every non-constant eigenfunction belonging to the basis has two nodal domains.
ISSN:2331-8422