Geometric and Functional Analysis

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

Macroscopic scalar curvature and areas of cycles

  • Hannah AlpertEmail author
  • Kei Funano


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 


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

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