Scalar Curvature Volume Comparison Theorems for Almost Rigid Sphere

TL;DR

This paper extends Bray's football theorem to higher dimensions, providing volume comparison results for manifolds with scalar curvature bounds under symmetry or Ricci curvature constraints.

Contribution

It generalizes Bray's volume comparison theorem to high-dimensional manifolds with additional symmetry or Ricci curvature bounds.

Findings

Established volume upper bounds in high dimensions

Extended the theorem under axis symmetry assumptions

Provided conditions involving Ricci curvature upper bounds

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.