Total scalar curvature under a curvature operator lower bound

Abstract

Let $(M^n,g)$ be a complete, simply connected Riemannian manifold without boundary, of dimension $n\ge 3$, with curvature operator at least that of the unit sphere. We prove that $${\int }_M\mathrm{scal}(x)d{\mathrm{Vol}}_x\le n(n-1){\omega }_n,$$ where ${\omega }_n$ is the volume of the unit $n$-sphere. Equality holds if and only if $(M,g)$ is isometric to the unit round sphere. In fact, we obtain a stronger bound containing $\mathrm{Vol}(M,g)$. In even dimensions, the proof follows from the Chern-Gauss-Bonnet formula. In odd dimensions, we apply the corresponding boundary formula to Deruelle's Ricci expander filling.

11 pages; comments welcome