Bonnet-Myers type theorems for $Q$-curvature on four-manifolds

Abstract

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. First, if the $Q$-curvature $Q_g\ge 6k^2$ and scalar curvature $R_g\ge -12k$ for some positive constant $k$, then $(M^4,g)$ is either Einstein with $Ri{c}_g=-3kg$ or compact with $R_g\ge 12k$. As a corollary, the fundamental group ${\pi }_1(M^4)$ satisfies $|{\pi }_1(M^4)|\le 16{\pi }^2/({\int }_{M^4}{Q}_{g}d{\mu }_g),$ under the additional assumption $R_g>-12k$. Second, if the scalar curvature $R_g>0$ and $Q_g\ge \theta R_g$ for a positive constant $\theta$, then $M^4$ is compact and the diameter of $(M^4,g)$ is at most $4\pi /\sqrt{15\theta }$.

29 pages. New version