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 $(\widetilde{\Sigma \times {\mathbb{S}}^1},\tilde{g})$, 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.

arXiv comment14 pages, 0 figures; revised to match final version accepted by GAFA