Macroscopic scalar curvature and areas of cycles

Abstract

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