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, Online first, 48 p.

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) 
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Lin, Samuel; Schmidt, Benjamin; Sutton, Craig. Geometric structures and the Laplace spectrum, Part I. Annales de l'Institut Fourier, Online first, 48 p.

[1] An, Jinpeng; Yu, Jiu-Kang; Yu, Jun On the dimension datum of a subgroup and its application to isospectral manifolds, J. Differ. Geom., Volume 94 (2013) no. 1, pp. 59-85 http://projecteuclid.org/euclid.jdg/1361889061 | DOI | MR | Zbl

[2] Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry 3-manifold groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2015, xiv+215 pages | DOI | MR | Zbl

[3] Bérard, Pierre H. Spectral geometry: direct and inverse problems, Lecture Notes in Mathematics, 1207, Springer-Verlag, Berlin, 1986, xiv+272 pages | DOI | MR | Zbl

[4] Berger, M. Le spectre des variétés riemanniennes, Rev. Roumaine Math. Pures Appl., Volume 13 (1968), pp. 915-931 | MR | Zbl

[5] Berger, Marcel; Gauduchon, Paul; Mazet, Edmond Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin-New York, 1971, vii+251 pages | MR | Zbl

[6] Bettiol, Renato G.; Lauret, Emilio A.; Piccione, Paolo The first eigenvalue of a homogeneous CROSS, J. Geom. Anal., Volume 32 (2022) no. 3, 76, 63 pages | DOI | MR | Zbl

[7] Borel, Armand Some remarks about Lie groups transitive on spheres and tori, Bull. Amer. Math. Soc., Volume 55 (1949), pp. 580-587 | DOI | MR | Zbl

[8] Buser, Peter Isospectral Riemann surfaces, Ann. Inst. Fourier, Volume 36 (1986) no. 2, pp. 167-192 | MR | Zbl

[9] Chavel, Isaac Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, 115, Academic Press, Inc., Orlando, FL, 1984, xiv+362 pages | MR | Zbl

[10] Doyle, Peter G.; Rossetti, Juan Pablo Tetra and Didi, the cosmic spectral twins, Geom. Topol., Volume 8 (2004), pp. 1227-1242 | DOI | MR | Zbl

[11] Filipkiewicz, R. Four-dimensional geometries, Ph. D. Thesis, Warwick University (1983)

[12] Geng, A. The classification of five-dimensional geometries, Ph. D. Thesis, University of Chicago (2016)

[13] Gordon, Carolyn Isospectral closed Riemannian manifolds which are not locally isometric, J. Differ. Geom., Volume 37 (1993) no. 3, pp. 639-649 http://projecteuclid.org/euclid.jdg/1214453902 | MR | Zbl

[14] Gordon, Carolyn S. Isospectral closed Riemannian manifolds which are not locally isometric. II, Geometry of the spectrum (Seattle, WA, 1993) (Contemp. Math.), Volume 173, Amer. Math. Soc., Providence, RI (1994), pp. 121-131 | DOI | MR | Zbl

[15] Gordon, Carolyn S. Isospectral deformations of metrics on spheres, Invent. Math., Volume 145 (2001) no. 2, pp. 317-331 | DOI | MR | Zbl

[16] Gordon, Carolyn S.; Wilson, Edward N. The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., Volume 33 (1986) no. 2, pp. 253-271 | DOI | MR | Zbl

[17] Gordon, Carolyn S.; Wilson, Edward N. Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Differ. Geom., Volume 47 (1997) no. 3, pp. 504-529 http://projecteuclid.org/euclid.jdg/1214460548 | DOI | MR | Zbl

[18] Guillemin, Victor; Sternberg, Shlomo Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984, xi+468 pages | MR | Zbl

[19] Ikeda, Akira On the spectrum of a Riemannian manifold of positive constant curvature, Osaka Math. J., Volume 17 (1980) no. 1, pp. 75-93 http://projecteuclid.org/euclid.ojm/1200772809 | MR | Zbl

[20] Kac, Mark Can one hear the shape of a drum?, Amer. Math. Monthly, Volume 73 (1966) no. 4, part II, pp. 1-23 | DOI | MR | Zbl

[21] Lin, Samuel; Schmidt, Benjamin; Sutton, Craig Geometric structures and the Laplace spectrum, Part II, Trans. Amer. Math. Soc., Volume 374 (2021) no. 12, pp. 8483-8530 | DOI | MR | Zbl

[22] Milnor, John Curvatures of left invariant metrics on Lie groups, Advances in Math., Volume 21 (1976) no. 3, pp. 293-329 | DOI | MR | Zbl

[23] Minakshisundaram, S.; Pleijel, Å. Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math., Volume 1 (1949), pp. 242-256 | DOI | MR | Zbl

[24] Montgomery, Deane; Samelson, Hans Transformation groups of spheres, Ann. Math. (2), Volume 44 (1943), pp. 454-470 | DOI | MR | Zbl

[25] Polterovich, Iosif Heat invariants of Riemannian manifolds, Israel J. Math., Volume 119 (2000), pp. 239-252 | DOI | MR | Zbl

[26] Proctor, Emily Isospectral metrics and potentials on classical compact simple Lie groups, Michigan Math. J., Volume 53 (2005) no. 2, pp. 305-318 | DOI | MR | Zbl

[27] Raymond, Frank; Vasquez, Alphonse T. 3-manifolds whose universal coverings are Lie groups, Topology Appl., Volume 12 (1981) no. 2, pp. 161-179 | DOI | MR | Zbl

[28] Reid, Alan W. Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds, Duke Math. J., Volume 65 (1992) no. 2, pp. 215-228 | DOI | MR | Zbl

[29] Rossetti, J. P.; Conway, J. H. Hearing the platycosms, Math. Res. Lett., Volume 13 (2006) no. 2-3, pp. 475-494 | DOI | MR | Zbl

[30] Sakai, Takashi On eigen-values of Laplacian and curvature of Riemannian manifold, Tôhoku Math. J. (2), Volume 23 (1971), pp. 589-603 | DOI | MR | Zbl

[31] Sakai, Takashi Riemannian geometry, Translations of Mathematical Monographs, 149, American Mathematical Society, Providence, RI, 1996, xiv+358 pages | DOI | MR | Zbl

[32] Schiemann, Alexander Ternary positive definite quadratic forms are determined by their theta series, Math. Ann., Volume 308 (1997) no. 3, pp. 507-517 | DOI | MR | Zbl

[33] Schueth, Dorothee Continuous families of isospectral metrics on simply connected manifolds, Ann. Math. (2), Volume 149 (1999) no. 1, pp. 287-308 | DOI | MR | Zbl

[34] Schueth, Dorothee Isospectral manifolds with different local geometries, J. Reine Angew. Math., Volume 534 (2001), pp. 41-94 | DOI | MR | Zbl

[35] Schueth, Dorothee Isospectral metrics on five-dimensional spheres, J. Differ. Geom., Volume 58 (2001) no. 1, pp. 87-111 http://projecteuclid.org/euclid.jdg/1090348283 | DOI | MR | Zbl

[36] Scott, Peter The geometries of 3-manifolds, Bull. London Math. Soc., Volume 15 (1983) no. 5, pp. 401-487 | DOI | MR | Zbl

[37] Sekigawa, Kouei On some 3-dimensional curvature homogeneous spaces, Tensor (N.S.), Volume 31 (1977) no. 1, pp. 87-97 | MR | Zbl

[38] Singer, I. M. Infinitesimally homogeneous spaces, Comm. Pure Appl. Math., Volume 13 (1960), pp. 685-697 | DOI | MR | Zbl

[39] Sunada, Toshikazu Riemannian coverings and isospectral manifolds, Ann. Math. (2), Volume 121 (1985) no. 1, pp. 169-186 | DOI | MR | Zbl

[40] Sutton, Craig J. Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions, Comment. Math. Helv., Volume 77 (2002) no. 4, pp. 701-717 | DOI | MR | Zbl

[41] Szabó, Z. I. Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal., Volume 9 (1999) no. 1, pp. 185-214 | DOI | MR | Zbl

[42] Tanno, Shǔkichi Eigenvalues of the Laplacian of Riemannian manifolds, Tôhoku Math. J. (2), Volume 25 (1973), pp. 391-403 | DOI | MR | Zbl

[43] Tanno, Shûkichi A characterization of the canonical spheres by the spectrum, Math. Z., Volume 175 (1980) no. 3, pp. 267-274 | DOI | MR | Zbl

[44] Thurston, William P. Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.), Volume 6 (1982) no. 3, pp. 357-381 | DOI | MR | Zbl

[45] Vignéras, Marie-France Variétés riemanniennes isospectrales et non isométriques, Ann. Math. (2), Volume 112 (1980) no. 1, pp. 21-32 | DOI | MR | Zbl

[46] Ziller, W. Homogeneous Einstein metrics on spheres and projective spaces, Math. Ann., Volume 259 (1982) no. 3, pp. 351-358 | DOI | MR | Zbl

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