Geometric and Functional Analysis

, Volume 27, Issue 4, pp 727–743 | Cite as

Macroscopic scalar curvature and areas of cycles

Article

Abstract

In this paper we prove the following. Let \({\Sigma}\) be an n–dimensional closed hyperbolic manifold and let g be a Riemannian metric on \({\Sigma \times \mathbb{S}^1}\). Given an upper bound on the volumes of unit balls in the Riemannian universal cover \({(\widetilde{\Sigma\times \mathbb{S}^1},\widetilde{g})}\), we get a lower bound on the area of the \({\mathbb{Z}_2}\)–homology class \({[\Sigma \times \ast]}\) on \({\Sigma \times \mathbb{S}^1}\), proportional to the hyperbolic area of \({\Sigma}\). The theorem is based on a theorem of Guth and is analogous to a theorem of Kronheimer and Mrowka involving scalar curvature.

Keywords and phrases

Macroscopic scalar curvature minimal surface hyperbolic manifold 

Mathematics Subject Classification

53C21 53C23 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. CM11.
    T. H. Colding and W. P. Minicozzi, A Course in Minimal Surfaces. Graduate Studies in Mathematics, 121. American Mathematical Society, Providence, RI (2011).Google Scholar
  2. GG74.
    M.A. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Graduate texts in mathematics: 14. Springer, New York (1974).Google Scholar
  3. GHL04.
    S. Gallot, D. Hulin, and J. Lafontaine, Riemannian Geometry. Third Edition. Universitext, Springer, Berlin (2004).Google Scholar
  4. Gro82.
    Gromov M.: Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 19(56), 5–99 (1982)MathSciNetMATHGoogle Scholar
  5. Gut10a.
    L. Guth, Metaphors in Systolic Geometry. In: Proceedings of the International Congress of Mathematicians, vol. II. Hindustan Book Agency, New Delhi (2010), pp. 745–768.Google Scholar
  6. Gut10b.
    Guth L.: Systolic inequalities and minimal hypersurfaces. Geom. Funct. Anal. 19(6), 1688–1692 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. Gut11.
    Guth L.: Volumes of balls in large riemannian manifolds. Ann. of Math. 173(1), 51–76 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. Has88.
    Hass J.: Surfaces minimizing area in their homology class and group actions on 3-manifolds. Math. Z. 199(4), 501–509 (1988)MathSciNetCrossRefMATHGoogle Scholar
  9. KM97.
    Kronheimer P. B., Mrowka T. S.: Scalar curvature and the thurston norm. Math. Res. Lett. 4(6), 931–937 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. Sch89.
    R. M. Schoen, Variational Theory for the Total Scalar Curvature Functional for Riemannian Metrics and Related Topics. Lecture Notes in Math., vol. 1365, Springer, Berlin (1989).Google Scholar
  11. Sim83.
    L. Simon, Lectures on Geometric Measure Theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983).Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Ohio State UniversityColumbusUSA
  2. 2.Tohoku UniversitySendaiJapan

Personalised recommendations