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An Upper Bound for the Smallest Area of a Minimal Surface in Manifolds of Dimension Four

  • Nan WuEmail author
  • Zhifei Zhu
Article
  • 15 Downloads

Abstract

In this paper, we prove that if M is a closed 4-dimensional Riemannian manifold with trivial first homology group, Ricci curvature \(|Ric|\le 3\), diameter \({{\,\mathrm{diam}\,}}(M)\le D\), and volume \({{\,\mathrm{vol}\,}}(M)>v>0\), then the smallest area of a 2-dimensional minimal surface in M is bounded by F(vD), for some function F that only depends on v and D. In order to prove this result, we first establish upper bounds for the first homological filling function of M that are of independent interest. This part of our work is based on recent results of Cheeger and Naber about manifolds with Ricci curvature bounded from both sides.

Keywords

Minimal surface Ricci curvature Homological filling function 

Mathematics Subject Classification

52C23 

Notes

Acknowledgements

The authors are grateful to Alexander Nabutovsky and Regina Rotman for suggesting this problem and numerous helpful discussions. We also thank Vitali Kapovitch and Robert Haslhofer for useful discussions about the epsilon-regularity theorem. We thank Aaron Naber for answering several questions about his work [7] with Jeff Cheeger. We thank the anonymous referee for providing many useful comments.

References

  1. 1.
    Almgren, F.J.: The theory of varifolds. Mimeographed notes (1965)Google Scholar
  2. 2.
    Anderson, M.T., Cheeger, J.: Diffeomorphism finiteness for manifolds with ricci curvature and \(L^2\)-norm of curvature bounded. Geom. Funct. Anal. 1(3), 231–252 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Anderson, M.T.: Ricci curvature bounds and Einstein metrics on compact manifolds. J. Am. Math. Soc. 2(3), 455–490 (1989)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Anderson, M.T.: The \(L^2\) structure of moduli spaces of Einstein metrics on 4-manifolds. Geom. Funct. Anal. 2(1), 29–89 (1992)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cheeger, J., Colding, T.H.: Lower bounds on Ricci curvature and the almost rigidity of warped products. Ann. Math. 144(1), 189–237 (1996)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cheeger, J., Colding, T.H.: On the structure of spaces with Ricci curvature bounded below. I. J. Differ. Geom. 46(3), 406–480 (1997)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cheeger, J., Naber, A.: Regularity of Einstein manifolds and the codimension 4 conjecture. Ann. Math. 182(3), 1093–1165 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Colding, T.H.: Ricci curvature and volume convergence. Ann. Math. 145(3), 477–501 (1997)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Coornaert, M.: Topological Dimension and Dynamical Systems. Springer, New York (2015)zbMATHGoogle Scholar
  10. 10.
    Federer, H.: Geometric Measure Theory. Springer, New York (2014)zbMATHGoogle Scholar
  11. 11.
    Glynn-Parker, P., Liokumovich, Y.: Width, Ricci curvature, and minimal hypersurfaces. J. Differ. Geom. 105(1), 33–54 (2017)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Gromov, M.: Filling riemannian manifolds. J. Differ. Geom. 18(1), 1–147 (1983)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gromov, M.: Metric Structures for Riemannian and Non-riemannian Spaces. Springer, New York (2007)zbMATHGoogle Scholar
  14. 14.
    Lickorish, W.R.: A representation of orientable combinatorial 3-manifolds. Ann. Math. 76, 531–540 (1962)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Nabutovsky, A., Rotman, R.: Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem. J. Eur. Math. Soc. (JEMS) 5(3), 203–244 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Nabutovsky, A., Rotman, R.: Volume, diameter and the minimal mass of a stationary 1-cycle. Geom. Funct. Anal. 14(4), 748–790 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Nabutovsky, A., Rotman, R.: Curvature-free upper bounds for the smallest area of a minimal surface. Geom. Funct. Anal. 16(2), 453–475 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pitts, J.T.: Existence and Regularity of Minimal Surfaces on Riemannian Manifolds (MN-27). Princeton University Press, Princeton (2014)Google Scholar
  19. 19.
    Schoen, R., Simon, L.: Regularity of stable minimal hypersurfaces. Commun. Pure Appl. Math. 34(6), 741–797 (1981)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Schoen, R., Simon, L., Yau, S.T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134, 276–288 (1975)MathSciNetzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Math DepartmentDuke UniversityDurhamUSA
  2. 2.University of TorontoTorontoCanada

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