Skip to main content
SpringerLink
Log in

Tubular Excision and Steklov Eigenvalues

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript

Abstract

Given a closed manifold M and a closed connected submanifold \(N\subset M\) of positive codimension, we study the Steklov spectrum of the domain \(\varOmega _\varepsilon \subset M\) obtained by removing the tubular neighborhood of size \(\varepsilon \) around N. All nonzero eigenvalues in the mid-frequency range tend to infinity at a rate which depends only on the codimension of N in M. Eigenvalues above the mid-frequency range are also described: they tend to infinity following an unbounded sequence of clusters. This construction is then applied to obtain manifolds with unbounded perimeter-normalized spectral gap and to show the necessity of using the injectivity radius in some known isoperimetric-type upper bounds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Al Sayed, A., Bogosel, B., Enrot, A., Nacry, F.: Maximization of the Steklov eigenvalues with a diameter constraint. SIAM J. Math. Anal. 53(1), 710–729 (2021)

    Article  MathSciNet  Google Scholar 

  2. Chiadò Piat, V., Nazarov, S.A.: Steklov spectral problems in a set with a thin toroidal hole. Partial Differ. Equ. Appl. Math. 1, 100007 (2020)

    Article  Google Scholar 

  3. Cianci, D., Girouard, A.: Large spectral gaps for Steklov eigenvalues under volume constraints and under localized conformal deformations. Ann. Glob. Anal. Geom. 54(4), 529–539 (2018)

    Article  MathSciNet  Google Scholar 

  4. Colbois, B., El Soufi, A., Girouard, A.: Isoperimetric control of the Steklov spectrum. J. Funct. Anal. 261(5), 1384–1399 (2011)

    Article  MathSciNet  Google Scholar 

  5. Colbois, B., El Soufi, A., Girouard, A.: Compact manifolds with fixed boundary and large Steklov eigenvalues. Proc. Am. Math. Soc. 147(9), 3813–3827 (2019)

    Article  MathSciNet  Google Scholar 

  6. Colbois, B., Girouard, A.: The spectral gap of graphs and Steklov eigenvalues on surfaces. Electron. Res. Announc. Math. Sci. 21, 19–27 (2014)

    MathSciNet  MATH  Google Scholar 

  7. Colbois, B., Girouard, A.: Metric upper bounds for Laplace and Steklov eigenvalues (2021). arXiv: 2108.03101

  8. Colbois, B., Girouard, A., Gittins, K.: Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. J. Geom. Anal. 29(2), 1811–1834 (2019)

    Article  MathSciNet  Google Scholar 

  9. Colbois, B., Girouard, A., Raveendran, B.: The Steklov spectrum and coarse discretizations of manifolds with boundary. Pure Appl. Math. Q. 14(2), 357–392 (2018)

    Article  MathSciNet  Google Scholar 

  10. Colbois, B., Gittins, K.: Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index. Differ. Geom. Appl. 78, 21 (2021). (Paper No. 101777)

    Article  MathSciNet  Google Scholar 

  11. Colbois, B., Verma, S.: Sharp Steklov upper bound for submanifolds of revolution. J. Geom. Anal. 31(11), 11214–11225 (2021)

    Article  MathSciNet  Google Scholar 

  12. Fraser, A., Schoen, R.: Shape optimization for the Steklov problem in higher dimensions. Adv. Math. 348, 146–162 (2019)

    Article  MathSciNet  Google Scholar 

  13. Girouard, A., Karpukhin, M., Lagacé, J.: Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems. Geom. Funct. Anal. 31(3), 513–561 (2021)

    Article  MathSciNet  Google Scholar 

  14. Girouard, A., Lagacé, J.: Large Steklov eigenvalues via homogenisation on manifolds. Invent. Math. 226(3), 1011–1056 (2021)

    Article  MathSciNet  Google Scholar 

  15. Gray, A.: Tubes. Progress in Mathematics, 2nd edn, vol 221. Birkhäuser Verlag, Basel (2004) (with a preface by Vicente Miquel)

  16. Hong, H.: Higher dimensional surgery and Steklov eigenvalues. J. Geom. Anal. 31(12), 11931–11951 (2021)

    Article  MathSciNet  Google Scholar 

  17. Kokarev, G.: Variational aspects of Laplace eigenvalues on Riemannian surfaces. Adv. Math. 258, 191–239 (2014)

    Article  MathSciNet  Google Scholar 

  18. NIST digital library of mathematical functions, Release 1.1.0 of 2020-12-15. http://dlmf.nist.gov/

  19. Weinstock, R.: Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3, 745–753 (1954)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Bruno Colbois and Jean Lagacé for reading an early version of the article and Léonard Tschanz and Bruno Colbois for helping with the multiplicity in the proof of Theorem 1.4. The author is supported by NSERC. This work is a part of the PhD thesis of the author under the supervision of Alexandre Girouard.

Author information

Authors and Affiliations

  1. Département de mathématiques et de statistique, Université Laval, Pavillon Alexandre-Vachon, Quebec, QC, G1V 0A6, Canada

    Jade Brisson

Authors
  1. Jade Brisson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jade Brisson.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brisson, J. Tubular Excision and Steklov Eigenvalues. J Geom Anal 32, 166 (2022). https://doi.org/10.1007/s12220-022-00905-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s12220-022-00905-3

Keywords

Mathematics Subject Classification

Search

Navigation