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 × S1. Given an upper bound on the volumes of unit balls in the Riemannian universal cover (Σ × S1~ , g~) , we get a lower bound on the area of the Z2–homology class [ Σ × * ] on Σ × S1, 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.
ASJC Scopus subject areas
- Geometry and Topology