# Macroscopic scalar curvature and areas of cycles

## TL;DR

This paper establishes a lower bound on the area of certain cycles in a Riemannian manifold based on bounds on the universal cover's volume, linking scalar curvature concepts with geometric topology.

## Contribution

It extends scalar curvature and volume bounds to derive area estimates for cycles in hyperbolic manifolds with product metrics, based on Guth's theorem.

## Key findings

- Lower bounds on cycle areas proportional to hyperbolic area
- Connection between universal cover volume bounds and cycle areas
- Extension of scalar curvature methods to product manifolds

## 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.