Scalar Curvature Volume Comparison Theorems for Almost Rigid Sphere

Abstract: Bray's football theorem (\cite{bray2009penrose}) is a weakening of Bishop theorem in dimension 3. It gives a sharp volume upper bound for a three dimensional manifold with scalar curvature larger than $n(n-1)$ and Ricci curvature larger than $\varepsilon$. This paper extends Bray's football theorem in high dimensions, assuming the manifold is axis symmetric or the Ricci curvature has an upper bound.