Skip to main content

Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds


We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds of bounded sectional curvature endowed with a weighted measure.

This is a preview of subscription content, access via your institution.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.


  1. Alencar, H., Do Carmo, M.P., Rosenberg, H.: On the first eigenvalue of Linearized operator of the r-th mean curvature of a hypersurface. Ann. Glob. Anal. Geom. 11, 387–395 (1993)

    Article  MathSciNet  Google Scholar 

  2. Alias, L.J., Malacarne, J.M.: On the first eigenvalue of the linearized operator of the higher order mean curvature for closed hypersurfaces in space forms. Illinois J. Math. 48(1), 219–240 (2004)

    Article  MathSciNet  Google Scholar 

  3. Auchmuty, G.: Steklov Eigenproblems and the Representation of Solutions of Elliptic Boundary Value Problems. Numer. Funct. Anal. Optim. 25(3–4), 321–348 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Batista, M., Cavalcante, M.P., Pyo, J.: Some isomperimetric inequalities and eigenvalue estimates in weighted manifolds. J. Math. Anal. Appl. 419(1), 617–626 (2014)

    Article  MathSciNet  Google Scholar 

  5. Dambrine, M., Kateb, D., Lamboley, J.: An extremal eigenvalue problem for the Wentzell-Laplace operator. Ann. I. H. Poincaré 33(2), 409–450 (2016)

    Article  MathSciNet  Google Scholar 

  6. Domingo-Juan, M.C., Miquel, V.: Reilly’s type inequality for the Laplacian associated to a density related with shrinkers for MCF. arXiv:1503.01332

  7. El Soufi, A., Ilias, S.: Une inégalité de type Reilly pour les sous-variétés de l’espace hyperbolique. Comment. Math. Helv. 67(2), 167–181 (1992)

    Article  MathSciNet  Google Scholar 

  8. Grosjean, J.F.: Extrinsic upper bounds for the first eigenvalue of elliptic operators. Hokkaido Math. J. 33, 319–339 (2004)

    Article  MathSciNet  Google Scholar 

  9. Grosjean, J.F.: Upper bounds for the first eigenvalue of the Laplacian on compact manifolds. Pac. J. Math. 206(1), 93–111 (2002)

    Article  Google Scholar 

  10. Heintze, E.: Extrinsic upper bound for \(\lambda _1\). Math. Ann. 280, 389–402 (1988)

    Article  MathSciNet  Google Scholar 

  11. Ilias, S., Makhoul, O.: A Reilly inequality for the first Steklov eigenvalue. Differ. Geom. Appl. 29(5), 699–708 (2011)

    Article  MathSciNet  Google Scholar 

  12. Reilly, R.C.: On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv. 52, 525–533 (1977)

    Article  MathSciNet  Google Scholar 

  13. Roth, J.: General Reilly-type inequalities for submanifolds of weighted Euclidean spaces. Colloq. Math. 144(1), 127–136 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Roth, J.: Reilly-type inequalities for Paneitz and Steklov eigenvalues. preprint hal-01539128

  15. Wang, R.: Reilly inequalities of elliptic operators on closed submanifolds. Bull. Aust. Math. Soc. 80, 335–346 (2009)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Abhitosh Upadhyay.

Additional information

Publisher's Note

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

The first author is supported by Fapesp, Grant 2019/23370-4 and the third author gratefully acknowledges the financial support from the Indian Institute of Technology Goa through Start-up Grant 2021/SG/AU/043.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Manfio, F., Roth, J. & Upadhyay, A. Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds. Ann Glob Anal Geom 62, 489–505 (2022).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Submanifolds
  • Reilly-type upper bounds
  • Eigenvalues estimates
  • Divergence-type operators
  • Steklov problems

Mathematics Subject Classification

  • 53C24
  • 53C42
  • 58J50