Local spectral sections and topology of eigenspaces

Nifa, Anass

doi:10.5281/zenodo.22859622

Preprint · v1

Local spectral sections and topology of eigenspaces

Anass Nifa15 April 2026

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Abstract

Let $\lambda>0$ be an isolated Laplace eigenvalue on a closed connected smooth manifold of dimension at least two. We construct a local right inverse of its symmetric spectral-cluster matrix in the reference conformal class. The inverse is $1/2$ -Hölder continuous and smooth off the initial matrix. The exponent is optimal at a conformally unstable reference. Metrics at which the cluster derivative is surjective approximate the reference while preserving both $\lambda$ and its multiplicity. This proves weak conformal stability. A separate signature bound gives unrestricted strong stability through multiplicity seven, with failure possible at eight. In either the full metric space or a conformal class, the fixed-cluster multiplicity locus is a $C^1$ embedded submanifold precisely at strongly stable points.

The same second-variation argument gives local matrix sections for $\Delta_g+V$ under scalar potentials supported in any prescribed nonempty interior open set, with the metric and the Dirichlet or Neumann realization fixed. Here the reference eigenvalue may be any real number. The analytic step is a sign-changing principal symbol for every nonzero contracted cokernel Hessian; each resulting quadratic form has infinite positive and negative index. Finite-dimensional regular-zero arguments then give exact realizations.

The local sections realize Grassmannians as retracts of internal spectral-gap spaces. Localized, volume-preserving metric variations on round spheres yield prescribed spectral subbundles; a pairwise-isometric family of metrics near a scaled round metric on the four-sphere has fixed exterior geometry, positive Ricci curvature, and a nontrivial rank-three bundle for the first three positive eigenvalues, counted with multiplicity. For the fixed round two-sphere, localized potential sections are smooth at zero and realize every smooth real vector bundle of positive finite rank over a compact smooth manifold in a suitable finite spectral cluster.

MSC 2020

58J50: Spectral problems; spectral geometry; scattering theory on manifolds
58C40: Spectral theory; eigenvalue problems on manifolds
35P05: General topics in linear spectral theory for PDEs
35S05: Pseudodifferential operators as generalizations of partial differential operators
57R22: Topology of vector bundles and fiber bundles

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22859622
All versions: 10.5281/zenodo.22859621
Subjects: Laplace operator; Schrödinger operator; eigenvalue multiplicity; weak Arnold hypothesis; local section; spectral bundle
Language: English

Cite this paper

Anass Nifa. Local spectral sections and topology of eigenspaces. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22859622

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References

As listed in the manuscript bibliography.

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[2]
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[3]
A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 6, 635–690.
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G. Berkolaiko and I. Zelenko, Morse inequalities for ordered eigenvalues of generic self-adjoint families, Invent. Math. 238 (2024), 283–330.
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T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976; corrected reprint, 1995.
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J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helv. 71 (1996), 213–228.
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arXiv v1
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[reference-1] [1]
A. A. Agrachev, Topology of quadratic maps and Hessians of smooth maps, J. Soviet Math. 49 (1990), no. 3, 990–1013.

[reference-2] [2]
A. A. Agrachev, On the space of symmetric operators with multiple ground states, Funct. Anal. Appl. 45 (2011), no. 4, 241–251.

[reference-3] [3]
A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 6, 635–690.

[reference-4] [4]
V. I. Arnol’d, Modes and quasimodes, Funct. Anal. Appl. 6 (1972), 94–101.

[reference-5] [5]
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. (9) 36 (1957), 235–249.

[reference-6] [6]
S. Bando and H. Urakawa, Generic properties of the eigenvalue of the Laplacian for compact Riemannian manifolds, Tohoku Math. J. (2) 35 (1983), no. 2, 155–172.

[reference-7] [7]
G. Berkolaiko and I. Zelenko, Morse inequalities for ordered eigenvalues of generic self-adjoint families, Invent. Math. 238 (2024), 283–330.

[reference-8] [8]
G. Besson, Propriétés génériques des fonctions propres et multiplicité, Comment. Math. Helv. 64 (1989), no. 4, 542–588.

[reference-9] [9]
D. D. Bleecker and L. C. Wilson, Splitting the spectrum of a Riemannian manifold, SIAM J. Math. Anal. 11 (1980), no. 5, 813–818.

[reference-10] [10]
Y. Colin de Verdière, Construction de laplaciens dont une partie finie du spectre est donnée, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 4, 599–615.

[reference-11] [11]
Y. Colin de Verdière, Sur une hypothèse de transversalité d’Arnold, Comment. Math. Helv. 63 (1988), 184–193.

[reference-12] [12]
S. Dyatlov, Lecture notes for 18.155: distributions, elliptic regularity, and applications to PDEs, MIT lecture notes, 10 December 2022, https://math.mit.edu/~dyatlov/18.155/155-notes.pdf.

[reference-13] [13]
J. N. V. Gomes and M. A. M. Marrocos, On eigenvalue generic properties of the Laplace–Neumann operator, J. Geom. Phys. 135 (2019), 21–31.

[reference-14] [14]
J. Greilhuber and W. Kepplinger, On Arnold’s transversality conjecture for the Laplace–Beltrami operator, arXiv:2312.16939v4 (2026).
arXiv v4

[reference-15] [15]
G. Grubb, Properties of normal boundary problems for elliptic even-order systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 1 (1974), nos. 1–2, 1–61.

[reference-16] [16]
G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr. 289 (2016), no. 7, 831–844.

[reference-17] [17]
X. He and Z. Wang, Riemannian metrics with prescribed volume and finite parts of Dirichlet spectrum, Calc. Var. Partial Differential Equations 63 (2024), no. 3, Paper No. 78, 24 pp.

[reference-18] [18]
L. Hörmander, The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Grundlehren der mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin, 1985.

[reference-19] [19]
T. Kato, Perturbation Theory for Linear Operators, second edition, Springer-Verlag, Berlin, 1976; corrected reprint, 1995.

[reference-20] [20]
J. Lohkamp, Discontinuity of geometric expansions, Comment. Math. Helv. 71 (1996), 213–228.

[reference-21] [21]
A. M. Micheletti and A. Pistoia, Multiple eigenvalues of the Laplace–Beltrami operator and deformation of the Riemannian metric, Discrete Contin. Dynam. Systems 4 (1998), no. 4, 709–720.

[reference-22] [22]
J. W. Milnor and J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, Princeton, 1974.

[reference-23] [23]
N. Nowaczyk, Existence of Dirac eigenvalues of higher multiplicity, Math. Z. 284 (2016), 285–307; arXiv:1504.00961v1, used for statement numbering.
arXiv v1

[reference-24] [24]
M. Teytel, How rare are multiple eigenvalues?, Comm. Pure Appl. Math. 52 (1999), no. 8, 917–934.

[reference-25] [25]
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059–1078.