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

  • Nan WuEmail author
  • Zhifei Zhu


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.


Minimal surface Ricci curvature Homological filling function 

Mathematics Subject Classification




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.


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© Mathematica Josephina, Inc. 2019

Authors and Affiliations

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

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