Morse theory for minimal surfaces in manifolds

Article
  • 16 Downloads

Abstract

A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.

Keywords

Minimal surfaces Harmonic maps Abstract critical point theory Morse theory 

Mathematics Subject Classification

49Q05 58E20 58E05 37B30 

Notes

Acknowledgements

This research was supported by Hannam University in 2016.

References

  1. 1.
    Chang, K.C.: Infinite Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Basel (1993)CrossRefMATHGoogle Scholar
  2. 2.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Springer, Berlin (1989)CrossRefMATHGoogle Scholar
  3. 3.
    Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations. Princeton University Press, Princeton (1983)MATHGoogle Scholar
  5. 5.
    Gromov, M.L., Rohlin, V.A.: Imbeddings and immersion in Riemannian geometry. Russ. Math. Surv. 25, 1–57 (1970)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Heinz, E., Hildebrandt, S.: Some remarks on minimal surfaces in Riemannian manifolds. Commun. Pure Appl. Math. XXIII, 371–377 (1970)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hildebrand, S., Kaul, H., Widman, K.O.: An existence theorem for harmonic mappings of Riemannian manifolds. Acta Math. 138, 1–16 (1977)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Hamilton, R.: Harmonic Maps of Manifolds with Boundary, LNM 471. Springer, Berlin (1975)CrossRefGoogle Scholar
  9. 9.
    Hohrein, J.: Existence of unstable minimal surfaces of higher genus in manifolds of nonpositive curvature. Dissertation (1994)Google Scholar
  10. 10.
    Jäger, W., Kaul, H.: Uniqueness and stability of harmonic maps and their Jacobi field. Manuscr. Math. 28, 269–291 (1979)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Jost, J.: Riemannian Geometry and Geometric Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  12. 12.
    Kim, H.: A variational approach to the regularity of the minimal surfaces of annulus type in Riemannian manifolds. Differ. Geom. Appl. 25, 466–484 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kim, H.: Unstable minimal surfaces of annulus type in manifolds. Adv. Geom. 3, 401–436 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Kim, H.: The second derivative of the energy functional. Honam Math. J. 34(2), 191–198 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kim, H.: A note on the Jacobi fields on manifolds. Honam Math. J. 38(2), 385–391 (2016)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lemaire, M.: Boundary value problems for harmonic and minimal maps of surfaces into manifolds. Ann. Sc. Sup Pisa 4(9), 91–103 (1982)MathSciNetMATHGoogle Scholar
  17. 17.
    Schulz, F.: Regularity Theory for Quasilinear Elliptic System and Monge–Ampére Equations in Two Dimensions, LMN 1445. Springer, Berlin (1990)CrossRefGoogle Scholar
  18. 18.
    Struwe, M.: Plateau’s Problem and the Calculus of Variations. Princeton University Press, Princeton (1998)MATHGoogle Scholar
  19. 19.
    Struwe, M.: A critical point theory for minimal surfaces spanning a wire in \({\mathbb{R}}^k\). J. Reine Angew. Math. 349, 1–23 (1984)MathSciNetMATHGoogle Scholar
  20. 20.
    Struwe, M.: A Morse theory for annulus type minimal surfaces. J. Reine u. Angew. Math. 386, 1–27 (1986)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHannam UniversityDaejeonRepublic of Korea

Personalised recommendations