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 ) n(n-1) and Ricci curvature at least ε 0 g ¯ \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 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.