Prescribed Steklov spectra and multiplicities on surfaces

Nifa, Anass

doi:10.5281/zenodo.22949635

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Prescribed Steklov spectra and multiplicities on surfaces

Anass Nifa4 September 2026

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Abstract

Let $0<a_1\le\cdots\le a_N$ and $H>a_N$ . On every compact connected orientable surface with $b\ge1$ boundary components and genus at least an explicit integer $\Gamma(N,b)$ , we construct a smooth metric whose first $N$ positive Steklov eigenvalues are $a_1,\ldots,a_N$ and whose next eigenvalue exceeds $H$ . Repetitions are allowed. For connected boundary, $\Gamma(N,1)=\lceil N(N-1)/4\rceil$ . This is the least genus permitted by the complete-graph band construction used in the proof; no optimality among all metrics is asserted. The metric may also have any sufficiently small prescribed boundary length, any positive prescribed area, and trivial isometry group. All these statements concern the ordinary, unweighted Steklov problem. The proof realises positive graph Laplacians by Dirichlet and boundary forms, estimates the complete finite spectral matrix, and applies degree after smoothing. For a family of invariant perforations of a closed hyperbolic surface, we identify the Steklov spectral subspace converging to the first positive Laplace eigenvalue with its Laplace eigenspace, equivariantly under the finite group used in the construction. The first Steklov multiplicity is at least the smallest dimension of a real irreducible constituent of that Laplace eigenspace; if the latter is irreducible, its multiplicity is preserved exactly. The surfaces of Fortier Bourque, Gruda-Mediavilla, Petri and Pineault yield hyperbolic examples of genera $10$ , $17$ and $37$ with first Steklov multiplicities $16$ , $21$ and at least $24$ , respectively. They disprove Jammes’s proposed equality between maximal first multiplicity and the relative chromatic number minus one.

MSC 2020

58J50: Spectral problems; spectral geometry; scattering theory on manifolds
35P05: General topics in linear spectral theory for PDEs
35J25: Boundary value problems for second-order elliptic equations
53A05: Surfaces in Euclidean and related spaces
05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)

Mathematics Subject Classification 2020

Record

Version DOI: 10.5281/zenodo.22949635
All versions: 10.5281/zenodo.22949634
Subjects: Steklov eigenvalue; spectral prescription; eigenvalue multiplicity; surface with boundary; graph Laplacian; homogenisation
Language: English

Cite this paper

Anass Nifa. Prescribed Steklov spectra and multiplicities on surfaces. Preprint, Zenodo, v1 (2026). doi:10.5281/zenodo.22949635

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References

As listed in the manuscript bibliography.

[1]
S. Audet-Beaumont, Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity, Canad. Math. Bull. 69 (2026), no. 2, 572–586. DOI: 10.4153/S0008439525101288.
DOI
[2]
D. Bucur, A. Henrot and M. Michetti, Asymptotic behaviour of the Steklov spectrum on dumbbell domains, Comm. Partial Differential Equations 46 (2021), no. 2, 362–393. DOI: 10.1080/03605302.2020.1840587.
DOI
[3]
D. Bucur and M. Nahon, Stability and instability issues of the Weinstock inequality, Trans. Amer. Math. Soc. 374 (2021), no. 3, 2201–2223. DOI: 10.1090/tran/8302.
DOI
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DOI
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DOI
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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.
[7]
Y. Colin de Verdière, Sur une hypothèse de transversalité d'Arnol'd, Comment. Math. Helv. 63 (1988), no. 2, 184–193.
[8]
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.
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K. Devriendt, R. Lambiotte and P. Van Mieghem, Constructing Laplacian matrices with Soules vectors: inverse eigenvalue problem and applications, arXiv:1909.11282v1 (2019), Proposition 7.1, p. 16 of the PDF (Proposition 13 in the HTML rendering).
arXiv v1
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DOI
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H. M. Farkas and I. Kra, Riemann Surfaces, second ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992.
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M. Fortier Bourque, É. Gruda-Mediavilla, B. Petri and M. Pineault, Three counterexamples to a conjecture of Colin de Verdière on multiplicity, arXiv:2312.03504v2 (2026), Theorem 1.1 and Sections 5.1–5.4. The first version, arXiv:2312.03504v1 (2023), is entitled Two counterexamples to a conjecture of Colin de Verdière on multiplicity.
arXiv v2 · arXiv v1
[13]
A. Girouard, A. Henrot and J. Lagacé, From Steklov to Neumann via homogenisation, Arch. Ration. Mech. Anal. 239 (2021), no. 2, 981–1023. DOI: 10.1007/s00205-020-01588-2.
DOI
[14]
A. Girouard, M. Karpukhin and J. Lagacé, Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems, Geom. Funct. Anal. 31 (2021), no. 3, 513–561. DOI: 10.1007/s00039-021-00573-5. Numbered statements refer to arXiv:2004.10784v2.
DOI · arXiv v2
[15]
A. Girouard and J. Lagacé, Large Steklov eigenvalues via homogenisation on manifolds, Invent. Math. 226 (2021), 1011–1056. DOI: 10.1007/s00222-021-01058-w.
DOI
[16]
A. Girouard, L. Parnovski, I. Polterovich and D. A. Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 379–389. DOI: 10.1017/S030500411400036X.
DOI
[17]
A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, J. Spectr. Theory 7 (2017), no. 2, 321–359. DOI: 10.4171/JST/164.
DOI
[18]
X. He, Prescription of finite Dirichlet eigenvalues and area on surface with boundary, J. Geom. Phys. 198 (2024), article 105100. DOI: 10.1016/j.geomphys.2024.105100.
DOI
[19]
X. He and Z. Wang, Prescription of the Robin spectrum, arXiv:2503.01451v4 (2025).
arXiv v4
[20]
P. Jammes, Prescription du spectre de Steklov dans une classe conforme, Anal. PDE 7 (2014), no. 3, 529–550. DOI: 10.2140/apde.2014.7.529.
DOI
[21]
P. Jammes, Multiplicité du spectre de Steklov sur les surfaces et nombre chromatique, Pacific J. Math. 282 (2016), no. 1, 145–171. DOI: 10.2140/pjm.2016.282.145.
DOI
[22]
Y. Jin and Z. Wang, The Steklov determinant and compactness of isospectral planar domains, arXiv:2608.22330v1 (2026).
arXiv v1
[23]
M. Karpukhin and J. Lagacé, Flexibility of Steklov eigenvalues via boundary homogenisation, Ann. Math. Qué. 48 (2024), no. 1, 175–186. DOI: 10.1007/s40316-022-00207-8.
DOI
[24]
M. Karpukhin, G. Kokarev and I. Polterovich, Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 6, 2481–2502. DOI: 10.5802/aif.2918.
DOI
[25]
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
[26]
H. Matthiesen and R. Petrides, Free boundary minimal surfaces of any topological type in Euclidean balls via shape optimization, arXiv:2004.06051v2 (2020).
arXiv v2
[27]
S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. (2) 40 (1939), no. 2, 400–416.
[28]
A. Nifa, Local spectral sections and topology of eigenspaces, preprint (2026). DOI: 10.5281/zenodo.22859622.
DOI
[29]
G. Ringel and J. W. T. Youngs, Solution of the Heawood map-coloring problem, Proc. Natl. Acad. Sci. USA 60 (1968), no. 2, 438–445. DOI: 10.1073/pnas.60.2.438.
DOI
[30]
G. W. Soules, Constructing symmetric nonnegative matrices, Linear Multilinear Algebra 13 (1983), no. 3, 241–251. DOI: 10.1080/03081088308817523.
DOI
[31]
N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory Ser. B 26 (1979), no. 2, 217–225. DOI: 10.1016/0095-8956(79)90058-3.
DOI
[32]
X. Zhan and J.-X. Zhou, A transfer principle for Steklov eigenvalue estimates of graphs, arXiv:2608.17264v1 (2026).
arXiv v1

[reference-1] [1]
S. Audet-Beaumont, Constructing surfaces with first Steklov eigenvalue of arbitrarily large multiplicity, Canad. Math. Bull. 69 (2026), no. 2, 572–586. DOI: 10.4153/S0008439525101288.
DOI

[reference-2] [2]
D. Bucur, A. Henrot and M. Michetti, Asymptotic behaviour of the Steklov spectrum on dumbbell domains, Comm. Partial Differential Equations 46 (2021), no. 2, 362–393. DOI: 10.1080/03605302.2020.1840587.
DOI

[reference-3] [3]
D. Bucur and M. Nahon, Stability and instability issues of the Weinstock inequality, Trans. Amer. Math. Soc. 374 (2021), no. 3, 2201–2223. DOI: 10.1090/tran/8302.
DOI

[reference-4] [4]
B. Colbois and A. Girouard, The spectral gap of graphs and Steklov eigenvalues on surfaces, Electron. Res. Announc. Math. Sci. 21 (2014), 19–27. DOI: 10.3934/era.2014.21.19.
DOI

[reference-5] [5]
Y. Colin de Verdière, Sur la multiplicité de la première valeur propre non nulle du laplacien, Comment. Math. Helv. 61 (1986), no. 2, 254–270. DOI: 10.1007/BF02621914.
DOI

[reference-6] [6]
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-7] [7]
Y. Colin de Verdière, Sur une hypothèse de transversalité d'Arnol'd, Comment. Math. Helv. 63 (1988), no. 2, 184–193.

[reference-8] [8]
K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985.

[reference-9] [9]
K. Devriendt, R. Lambiotte and P. Van Mieghem, Constructing Laplacian matrices with Soules vectors: inverse eigenvalue problem and applications, arXiv:1909.11282v1 (2019), Proposition 7.1, p. 16 of the PDF (Proposition 13 in the HTML rendering).
arXiv v1

[reference-10] [10]
R. A. Duke, The genus, regional number, and Betti number of a graph, Canad. J. Math. 18 (1966), 817–822. DOI: 10.4153/CJM-1966-081-6.
DOI

[reference-11] [11]
H. M. Farkas and I. Kra, Riemann Surfaces, second ed., Graduate Texts in Mathematics, vol. 71, Springer-Verlag, New York, 1992.

[reference-12] [12]
M. Fortier Bourque, É. Gruda-Mediavilla, B. Petri and M. Pineault, Three counterexamples to a conjecture of Colin de Verdière on multiplicity, arXiv:2312.03504v2 (2026), Theorem 1.1 and Sections 5.1–5.4. The first version, arXiv:2312.03504v1 (2023), is entitled Two counterexamples to a conjecture of Colin de Verdière on multiplicity.
arXiv v2 · arXiv v1

[reference-13] [13]
A. Girouard, A. Henrot and J. Lagacé, From Steklov to Neumann via homogenisation, Arch. Ration. Mech. Anal. 239 (2021), no. 2, 981–1023. DOI: 10.1007/s00205-020-01588-2.
DOI

[reference-14] [14]
A. Girouard, M. Karpukhin and J. Lagacé, Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems, Geom. Funct. Anal. 31 (2021), no. 3, 513–561. DOI: 10.1007/s00039-021-00573-5. Numbered statements refer to arXiv:2004.10784v2.
DOI · arXiv v2

[reference-15] [15]
A. Girouard and J. Lagacé, Large Steklov eigenvalues via homogenisation on manifolds, Invent. Math. 226 (2021), 1011–1056. DOI: 10.1007/s00222-021-01058-w.
DOI

[reference-16] [16]
A. Girouard, L. Parnovski, I. Polterovich and D. A. Sher, The Steklov spectrum of surfaces: asymptotics and invariants, Math. Proc. Cambridge Philos. Soc. 157 (2014), no. 3, 379–389. DOI: 10.1017/S030500411400036X.
DOI

[reference-17] [17]
A. Girouard and I. Polterovich, Spectral geometry of the Steklov problem, J. Spectr. Theory 7 (2017), no. 2, 321–359. DOI: 10.4171/JST/164.
DOI

[reference-18] [18]
X. He, Prescription of finite Dirichlet eigenvalues and area on surface with boundary, J. Geom. Phys. 198 (2024), article 105100. DOI: 10.1016/j.geomphys.2024.105100.
DOI

[reference-19] [19]
X. He and Z. Wang, Prescription of the Robin spectrum, arXiv:2503.01451v4 (2025).
arXiv v4

[reference-20] [20]
P. Jammes, Prescription du spectre de Steklov dans une classe conforme, Anal. PDE 7 (2014), no. 3, 529–550. DOI: 10.2140/apde.2014.7.529.
DOI

[reference-21] [21]
P. Jammes, Multiplicité du spectre de Steklov sur les surfaces et nombre chromatique, Pacific J. Math. 282 (2016), no. 1, 145–171. DOI: 10.2140/pjm.2016.282.145.
DOI

[reference-22] [22]
Y. Jin and Z. Wang, The Steklov determinant and compactness of isospectral planar domains, arXiv:2608.22330v1 (2026).
arXiv v1

[reference-23] [23]
M. Karpukhin and J. Lagacé, Flexibility of Steklov eigenvalues via boundary homogenisation, Ann. Math. Qué. 48 (2024), no. 1, 175–186. DOI: 10.1007/s40316-022-00207-8.
DOI

[reference-24] [24]
M. Karpukhin, G. Kokarev and I. Polterovich, Multiplicity bounds for Steklov eigenvalues on Riemannian surfaces, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 6, 2481–2502. DOI: 10.5802/aif.2918.
DOI

[reference-25] [25]
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.

[reference-26] [26]
H. Matthiesen and R. Petrides, Free boundary minimal surfaces of any topological type in Euclidean balls via shape optimization, arXiv:2004.06051v2 (2020).
arXiv v2

[reference-27] [27]
S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. (2) 40 (1939), no. 2, 400–416.

[reference-28] [28]
A. Nifa, Local spectral sections and topology of eigenspaces, preprint (2026). DOI: 10.5281/zenodo.22859622.
DOI

[reference-29] [29]
G. Ringel and J. W. T. Youngs, Solution of the Heawood map-coloring problem, Proc. Natl. Acad. Sci. USA 60 (1968), no. 2, 438–445. DOI: 10.1073/pnas.60.2.438.
DOI

[reference-30] [30]
G. W. Soules, Constructing symmetric nonnegative matrices, Linear Multilinear Algebra 13 (1983), no. 3, 241–251. DOI: 10.1080/03081088308817523.
DOI

[reference-31] [31]
N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory Ser. B 26 (1979), no. 2, 217–225. DOI: 10.1016/0095-8956(79)90058-3.
DOI

[reference-32] [32]
X. Zhan and J.-X. Zhou, A transfer principle for Steklov eigenvalue estimates of graphs, arXiv:2608.17264v1 (2026).
arXiv v1