Finite spectral determination of compact homogeneous manifolds

Nifa, Anass

doi:10.5281/zenodo.23307538

Preprint · v2

Finite spectral determination of compact homogeneous manifolds

Anass Nifa1 October 2026

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Abstract

A finite initial segment of a prescribed scalar Laplace spectrum determines the full spectrum among closed homogeneous manifolds of the same dimension. The realizations are represented by finitely many compact semialgebraic sets of positive invariant cometrics. They are therefore smoothly compact modulo diffeomorphisms and have finitely many diffeomorphism types. Among globally naturally reductive metrics, each spectrum has only finitely many isometry classes. For these metrics, a finite set of scalar eigenvalues also gives a local Lipschitz estimate for the metric modulo diffeomorphisms, simultaneously in every differentiability order. An explicit family on $\operatorname{SU}(2)\times S^1$ has fixed volume and uniform relative convergence of all positive eigenvalues, while its scalar curvature tends to $-\infty$ .

MSC 2020

58J53Primary: Isospectrality
53C30Primary: Differential geometry of homogeneous manifolds
22E46Secondary: Semisimple Lie groups and their representations
14P10Secondary: Semialgebraic sets and related spaces
58J50Secondary: Spectral problems; spectral geometry; scattering theory on manifolds

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.23307538
All versions: 10.5281/zenodo.23225647
Subjects: Laplace spectrum; homogeneous manifold; spectral compactness; semialgebraic set; naturally reductive metric; spectral rigidity
Language: English

Cite this paper

Anass Nifa. Finite spectral determination of compact homogeneous manifolds. Preprint, Zenodo, v2 (2026). doi:10.5281/zenodo.23307538

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References

As listed in the manuscript bibliography.

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[reference-4] [4]
T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Graduate Texts in Mathematics, vol. 98, Springer, New York, 1985.

[reference-5] [5]
D. Corro and J.-B. Kordaß, Short survey on the existence of slices for the space of Riemannian metrics, in Mexican Mathematicians in the World: Trends and Recent Contributions, Contemp. Math., vol. 775, American Mathematical Society, 2021, 65–84, doi:10.1090/conm/775/15589; arXiv:1904.07031v2.
arXiv v2 · DOI

[reference-6] [6]
M. Coste, Real Algebraic Sets, lecture notes, 23 March 2005, https://indico.ictp.it/event/a04204/session/10/contribution/7/material/0/0.pdf.

[reference-7] [7]
X. Dai and G. Wei, Finite part of spectrum and isospectrality, in Geometry of the Spectrum, Contemp. Math., vol. 173, American Mathematical Society, Providence, RI, 1994, 99–107, doi:10.1090/conm/173/01818.
DOI

[reference-8] [8]
D. G. Ebin, The manifold of Riemannian metrics, in Global Analysis (Berkeley, 1968), Proc. Sympos. Pure Math., vol. 15, American Mathematical Society, Providence, RI, 1970, 11–40.

[reference-9] [9]
K. Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. I, Proc. Natl. Acad. Sci. USA 35 (1949), 652–655, doi:10.1073/pnas.35.11.652.
DOI

[reference-10] [10]
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C. S. Gordon, D. Schueth, and C. J. Sutton, Spectral isolation of bi-invariant metrics on compact Lie groups, Ann. Inst. Fourier 60 (2010), no. 5, 1617–1628.
DOI

[reference-12] [12]
C. S. Gordon and C. J. Sutton, Spectral isolation of naturally reductive metrics on simple Lie groups, Math. Z. 266 (2010), 979–995.
DOI

[reference-13] [13]
R. M. Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (1980), 291–302, doi:10.2307/2374240.
DOI

[reference-14] [14]
L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968), 193–218, doi:10.1007/BF02391913.
DOI

[reference-15] [15]
B. Kostant, On differential geometry and homogeneous spaces. II, Proc. Natl. Acad. Sci. USA 42 (1956), no. 6, 354–357, doi:10.1073/pnas.42.6.354.
DOI

[reference-16] [16]
Y. Li, Y. Wu, and J. Zhou, Shortest closed geodesics on three-dimensional isospectral manifolds, preprint, arXiv:2609.28380v1, 23 September 2026.
arXiv v1

[reference-17] [17]
J. Milnor, Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976), 293–329.

[reference-18] [18]
D. Schueth, Isospectral manifolds with different local geometries, J. Reine Angew. Math. 534 (2001), 41–94; arXiv:math/0102123.
arXiv

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R. Storm, Structure theory of naturally reductive spaces, Differential Geom. Appl. 64 (2019), 174–200.
DOI

[reference-20] [20]
C. J. Sutton, Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions, Comment. Math. Helv. 77 (2002), 701–717, doi:10.1007/PL00012438.
DOI

[reference-21] [21]
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DOI

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J. Yu, Some new results on dimension datum, Bull. London Math. Soc. 54 (2022), no. 3, 929–948, doi:10.1112/blms.12605; arXiv:2008.00378v2, 5 February 2021.
arXiv v2 · DOI