  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.
