On the proof of Bray's conjecture

Abstract

Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\ge 3$. There exists a positive constant ${\epsilon }_n<1$ such that, if Ricci curvature ${\mathrm{Ric}}_g\ge {\epsilon }_n(n-1)g$ and the scalar curvature $R_g\ge n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.

This is our submitted version to a journal. This version includes the rigidity in the equality case and an application section