no. 2
Geometric structures and the Laplace spectrum, Part I
[Structures géométriques et le spectre du Laplacien, Première partie]
Lin, Samuel1 ; Schmidt, Benjamin2 ; Sutton, Craig3
1 University of Oklahoma Department of Mathematics Norman, OK 73071 (USA)
2 Michigan State University Department of Mathematics E. Lansing, MI 48824 (USA)
3 Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)
Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 867-914.

Inspiré par le rôle central que jouent les structures géométriques dans notre compréhension de la taxonomie des trois-variétés, on cherche dans quelle mesure les trois-variétés compactes localement homogènes sont caractérisées par leur spectre. En utilisant les quatre premiers invariants de la chaleur, on démontre que parmi les variétés riemanniennes compactes localement homogènes de dimensions trois, munies de structures modelées sur six des huit géométries de Thurston, leurs spectres déterminent le revêtement universel. Cette classe de variétés inclut toutes les variétés compactes localement symétriques de dimension trois. Notre résultat est optimal, en raison de l’existence de trois-variétés hyperboliques isopectrales. De plus, nous montrons que tout espace modelé sur l’espace symétrique 𝕊 2 ×𝔼 ou Nil, et équipé d’une métrique invariante à gauche arbitraire, est uniquement déterminé par son spectre parmi tous les espaces localement homogènes. Ces résultats découlent d’observations plus générales concernant les huit géométries tridimensionnelles «  métriquement maximales  », qui suggérent que la géométrie locale est «  audible  » parmi les trois variétés compactes localement homogènes.

Inspired by the central role geometric structures play in our understanding of the taxonomy of three-manifolds, we initiate the exploration of the extent to which compact locally homogeneous three-manifolds are characterized up to universal Riemannian cover by their spectra. Using the first four heat invariants, we conclude that within the universe of compact locally homogeneous Riemannian manifolds, closed three-manifolds equipped with geometric structures modeled on six of the eight Thurston geometries are determined up to universal Riemannian cover by their spectra, a result that includes all compact locally symmetric three-manifolds and is optimal due to the existence of isospectral hyperbolic three-manifolds, for example. Furthermore, we show that any space modeled on the symmetric space 𝕊 2 ×𝔼 or Nil equipped with an arbitrary left-invariant metric is uniquely determined by its spectrum among all locally homogeneous spaces. These results follow from more general observations, regarding the eight “metrically maximal” three-dimensional geometries, that strongly suggest local geometry is “audible” among compact locally homogeneous three-manifolds.

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DOI : 10.5802/aif.3602
Classification : 58J53, 53C30, 58J50
Keywords: Laplace spectrum, heat invariants, geometric structures, three-manifolds
Mot clés : Spectre du Laplacien, invariants de la chaleur, structures géométriques, variétés de dimension trois.

Lin, Samuel 1 ; Schmidt, Benjamin 2 ; Sutton, Craig 3

1 University of Oklahoma Department of Mathematics Norman, OK 73071 (USA)
2 Michigan State University Department of Mathematics E. Lansing, MI 48824 (USA)
3 Dartmouth College Department of Mathematics Hanover, NH 03755 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Lin, Samuel; Schmidt, Benjamin; Sutton, Craig. Geometric structures and the Laplace spectrum, Part I. Annales de l'Institut Fourier, Tome 74 (2024) no. 2, pp. 867-914. doi : 10.5802/aif.3602. https://aif.centre-mersenne.org/articles/10.5802/aif.3602/

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