Scalar curvature volume comparison theorems for almost rigid spheres

Abstract:

Bray’s football theorem gives a sharp volume upper bound for a three dimensional manifold with scalar curvature no less than $n(n-1)$ and Ricci curvature at least $\varepsilon _0 \bar {g}$. This paper extends Bray’s football theorem in higher dimensions, assuming the manifold is axisymmetric or the Ricci curvature has a uniform upper bound. Effectively, we show that if the Ricci curvature of an $n$-manifold is close to that of a round n-sphere, a lower bound on scalar curvature gives an upper bound on the total volume.