Fix $n \in \mathbb{N}$, $K \in \mathbb{R}$, $D > 0$, and $v > 0$. Let $\mathcal{M}(n,K,D,v)$ be the class of isometry classes of closed connected $n$-dimensional Riemannian manifolds $(M,g)$ such that
Then $\mathcal{M}(n,K,D,v)$ is precompact in the Gromov-Hausdorff topology. Equivalently, every sequence $((M_j,g_j))_{j=1}^{\infty}$ in $\mathcal{M}(n,K,D,v)$ has a subsequence whose associated compact metric spaces $(M_j,d_{g_j})$ converge in the Gromov-Hausdorff topology to a compact [metric space](/page/Metric%20Space).