Skip to main content
SpringerLink
Log in

p-Fundamental tone estimates of submanifolds with bounded mean curvature

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

In this paper, we estimate the p-fundamental tone of submanifolds in a Cartan–Hadamard manifold. First, we obtain lower bounds for the p-fundamental tone of geodesic balls and submanifolds with bounded mean curvature. Moreover, we provide the p-fundamental tone estimates of minimal submanifolds with certain conditions on the norm of the second fundamental form. Finally, we study transversely oriented codimension-one \(C^2\)-foliations of open subsets \(\Omega \) of Riemannian manifolds M and obtain lower bound estimates for the infimum of the mean curvature of the leaves in terms of the p-fundamental tone of \(\Omega \).

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. Allard, W.K.: On the first variation of a varifold. Ann. Math. (2) 95, 417–491 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, M.T.: The Compactification of Aminimal Submanifold in Euclidean Space by the Gauss Map. IEHS (1984) (preprint)

  3. Barbosa, J.L.M., Bessa, G.P., Montenegro, J.F.: On Bernstein–Heinz–Chern–Flanders inequalities. Math. Proc. Camb. Philos. Soc. 144(2), 457–464 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Belloni, M., Kawohl, B.: A direct uniqueness proof for equations involving the \(p\)-Laplace operator. Manuscr. Math. 109(2), 229–231 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernstein, S.: Sur la généralisation du problème de Dirichlet. Math. Ann. 69(1), 82–136 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bernstein, S.: Sur les surfaces définies au moyen de leur courbure moyenne ou totale. Ann. Sci. École Norm. Sup. 27, 233–256 (1910)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bessa, G.P., Montenegro, J.F.: Eigenvalue estimates for submanifolds with locally bounded mean curvature. Ann. Glob. Anal. Geom. 24(3), 279–290 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cheeger, J., Ebin, D.G.: Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York (1975)

  9. Chen, Q.: On the volume growth and the topology of complete minimal submanifolds of a Euclidean space. J. Math. Sci. Univ. Tokyo 2(3), 657–669 (1995)

    MathSciNet  MATH  Google Scholar 

  10. Chern, S.S.: On the curvature of a piece of hypersurface in Euclidean space. Abh. Math. Sem. Univ. Hamburg 29, 77–91 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  11. Cheung, L.F., Leung, P.F.: Eigenvalue estimates for submanifolds with bounded mean curvature in the hyperbolic space. Math. Z. 236(3), 525–530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dung, N. T., Seo, K.: \(p\)-harmonic functions and connectedness at infinity of complete submanifolds in a Riemannian manifold. Ann. Mat. Pura Appl. (4) (to appear). doi:10.1007/s10231-016-0625-0

  13. Efimov, N.V.: Investigation of a complete surface of negative curvature. Dokl. Akad. Nauk SSSR: N.S. 93, 393–395 (1953)

    MathSciNet  Google Scholar 

  14. Flanders, H.: Remark on mean curvature. J. Lond. Math. Soc. 41, 364–366 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fontenele, F., Silva, S.L.: Sharp estimates for size of balls in the complement of a hypersurface. Geom. Dedicata 115, 163–179 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gimeno, V.: On the fundamental tone of minimal submanifolds with controlled extrinsic curvature. Potential Anal. 40(3), 267–278 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gimeno, V., Palmer, V.: Volume growth, number of ends, and the topology of a complete submanifold. J. Geom. Anal. 24(3), 1346–1367 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gimeno, V., Palmer, V.: Extrinsic isoperimetry and compactification of minimal surfaces in Euclidean and hyperbolic spaces. Isr. J. Math. 194(2), 539–553 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Haymann, W.: Some bounds for principal frequency. Appl. Anal. 7(3), 247–254 (1977/78)

  20. Heinz, E.: Über Flächen mit eindeutiger projektion auf eine ebene, deren krümmungen durch ungleichungen eingschränkt sind. Math. Ann. 129, 451–454 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jorge, L., Koutrofiotis, D.: An estimate for the curvature of bounded submanifolds. Am. J. Math. 103(4), 711–725 (1981)

    Article  MathSciNet  Google Scholar 

  22. Lima, B.P., Montenegro, J.F.B., Santos, N.L.: Eigenvalue estimates for the p-Laplace operator on manifolds. Nonlinear Anal. 72, 771–781 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lindqvist, P.: On the equation \(\text{ div }(|\nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2}u=0\). Proc. Am. Math. Soc. 109(1), 157–164 (1990)

    MATH  Google Scholar 

  24. Makai, E.: A lower estimation of the principal frequencies of simply connected membranes. Acta Math. Acad. Sci. Hung. 16, 319–323 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  25. McKean, H.P.: An upper bound to the spectrum of \(\Delta \) on a manifold of negative curvature. J. Differ. Geom. 4, 359–366 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  26. Osserman, R.: A note on Hayman’s theorem on the bass note of a drum. Comment. Math. Helv. 52(4), 545–555 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  27. Palmer, V.: Isoperimetric inequalities for extrinsic balls in minimal submanifolds and their applications. J. Lond. Math. Soc. (2) 60(2), 607–616 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Salavessa, I.M.C.: Graphs with parallel mean curvature. Proc. Am. Math. Soc. 107(2), 449–458 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. Schoen, R., Yau, S.-T.: Lectures on differential geometry. In: Conference Proceedings and Lecture Notes in Geometry and Topology. I. International Press, Cambridge (1994)

  30. Seo, K.: Isoperimetric inequalities for submanifolds with bounded mean curvature. Monatsh. Math. 166(3–4), 525–542 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  31. Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Vol .3. Australian National University, Centre for Mathematical Analysis, Canberra (1983)

  32. Takeuchi, H.: On the first eigenvalue of the \(p\)-Laplacian in a Riemannian manifold. Tokyo J. Math. 21(1), 135–140 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Veron, L.: Some existence and uniqueness results for solution of some quasilinear elliptic equations on compact Riemannian manifolds. Differential equations and its applications (Budapest). In: Colloq. Math. Soc. János Bolyai, Vol. 62. North-Holland, Amsterdam, pp. 317–352 (1991)

Download references

Acknowledgements

The authors would like to thank the referees for their valuable comments on this manuscript. The authors also would like to thank Professor Pacelli Bessa for helpful suggestion during the elaboration of this paper and Professor Abdênago Barros for his careful reading and review on this manuscript and for fruitful conversations about the results. The first author was supported by CNPq/Brazil. The second author was supported by the National Research Foundation of Korea (NRF-2016R1C1B2009778).

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal Do Piauí - UFPI, Parnaíba, PI, Brazil

    Israel Evangelista

  2. Department of Mathematics, Sookmyung Women’s University, Cheongpa-ro 47-gil 100, Yongsan-ku, Seoul, 04310, Korea

    Keomkyo Seo

Authors
  1. Israel Evangelista
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Keomkyo Seo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Keomkyo Seo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Evangelista, I., Seo, K. p-Fundamental tone estimates of submanifolds with bounded mean curvature. Ann Glob Anal Geom 52, 269–287 (2017). https://doi.org/10.1007/s10455-017-9557-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-017-9557-1

Keywords

Mathematics Subject Classification

Search

Navigation