The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 1344–1386 | Cite as

Immersions with Bounded Second Fundamental Form

Article

Abstract

We first consider immersions on compact manifolds with uniform Lp-bounds on the second fundamental form and uniformly bounded volume. We show compactness in arbitrary dimension and codimension, generalizing a classical result of J. Langer. In the second part, this result is used to deduce a localized version, being more convenient for many applications, such as convergence proofs for geometric flows.

Keywords

Immersions Compactness Second fundamental form Bounded curvature Compact and noncompact manifolds 

Mathematics Subject Classification

53C42 53C23 53B25 

Notes

Acknowledgement

I would like to thank my advisor Ernst Kuwert for his support. Moreover, I would like to thank Manuel Breuning for proofreading my dissertation [6], where the results of this paper were established first.

References

  1. 1.
    Alt, H.W.: Lineare Funktionalanalysis, 4th edn. Springer, Berlin (2002) MATHGoogle Scholar
  2. 2.
    Baker, C.: The mean curvature flow of submanifolds of high codimension. Ph.D. thesis (2010) Google Scholar
  3. 3.
    Baker, C.: A partial classification of Type 1 singularities of the mean curvature flow in high codimension. Preprint (2011). arXiv:1104.4592
  4. 4.
    Bauer, M., Kuwert, E.: Existence of Minimizing Willmore Surfaces of Prescribed Genus. Int. Math. Res. Not. 10, 553–576 (2003) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Breuning, P.: Compactness of immersions with local Lipschitz representation. Ann. Inst. Henri Poincaré 29, 545–572 (2012) CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Breuning, P.: Immersions with local Lipschitz representation. Ph.D. thesis, Freiburg (2011) Google Scholar
  7. 7.
    Bröcker, T., Jänich, K.: Introduction to differential topology. Cambridge University Press, Cambridge (1982) MATHGoogle Scholar
  8. 8.
    Cheeger, J.: Finiteness theorems for Riemannian manifolds. Am. J. Math. 92, 61–74 (1970) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Cooper, A.A.: A compactness theorem for the second fundamental form. Preprint (2011). arXiv:1006.5697v4
  10. 10.
    Corlette, K.: Immersions with bounded curvature. Geom. Dedic. 33, 153–161 (1990) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Delladio, S.: On Hypersurfaces in Rn+1 with Integral Bounds on Curvature. J. Geom. Anal. 11, 17–41 (2000) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (1992) MATHGoogle Scholar
  13. 13.
    Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces, second printing with corrections. Birkhäuser, Boston (2001) Google Scholar
  14. 14.
    Hirsch, M.W.: Differential Topology. Graduate Texts in Mathematics, vol. 33. Springer, New York (1976) MATHGoogle Scholar
  15. 15.
    Huisken, G.: Asymptotic behavior for singularities of the mean curvature flow. J. Differ. Geom. 31, 285–299 (1990) MATHMathSciNetGoogle Scholar
  16. 16.
    Hutchinson, J.E.: Second Fundamental Form for Varifolds and the Existence of Surfaces Minimising Curvature. Indiana Univ. Math. J. 35(1), 45–71 (1986) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Kuwert, E., Schätzle, R.: The Willmore flow with small initial energy. J. Differ. Geom. 57, 409–441 (2001) MATHGoogle Scholar
  18. 18.
    Langer, J.: A Compactness Theorem for Surfaces with L p-Bounded Second Fundamental Form. Math. Ann. 270, 223–234 (1985) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Link, F.: Gradient Flow for the Willmore Functional in Riemannian Manifolds. Ph.D. thesis, Freiburg (2013). arXiv:1308.6055
  20. 20.
    Mondino, A.: Existence of integral m-varifolds minimizing ∫|A|p and ∫|H|p, p>m, in Riemannian manifolds. Preprint (2010). arXiv:1010.4514v1
  21. 21.
    Ndiaye, C.B., Schätzle, R.: A convergence theorem for immersions with L 2-bounded second fundamental form. Rend. Semin. Mat. Univ. Padova 127, 235–247 (2012). doi:10.4171/RSMUP/127-12 CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Schlichting, A.: Mittlerer Krümmungsfluss fast konvexer Lipschitzflächen. Diploma thesis, Freiburg (2009) Google Scholar
  23. 23.
    Simon, L.: Lectures on Geometric measure theory. Proc. of the Centre for Math. Analysis, vol. 3. Australian National University, Canberra (1983) MATHGoogle Scholar
  24. 24.
    Smith, G.: An Arzela-Ascoli theorem for immersed submanifolds. Ann. Fac. Sci. Toulouse 16(4), 817–866 (2007) CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Struwe, M.: Variational Methods, 4th edn. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34. Springer, Berlin (2008) MATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2014

Authors and Affiliations

  1. 1.Fakultät für Mathematik des Karlsruher Institut für TechnologieInstitut für AnalysisKarlsruheGermany

Personalised recommendations