Macroscopic scalar curvature and areas of cycles

2017/05/08 by Alpert, Hannah, Funano, Kei
#53C21 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.02923

Abstract

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× \mathbbS¹. Given an upper bound on the volumes of unit balls in the Riemannian universal cover (\widetildeΣ× \mathbbS¹,\widetildeg), we get a lower bound on the area of the ℤ₂--homology class [Σ× ∗] on Σ× \mathbbS¹, 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.

Related

• Urysohn width of hypersurfaces and positive macroscopic scalar curvature
• Average area ratio and normalized total scalar curvature of hyperbolic n-manifolds
• Macroscopic scalar curvature and local collapsing
• Volume and macroscopic scalar curvature
• Growth in the universal cover under large simplicial volume