Macroscopic scalar curvature and areas of cycles

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.