# Gromov's compactness theorem for manifolds with boundary

The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature then there exists a subsequence which converges in the Gromov-Hausdorff sense to a compact metric space.

My question is whether there are similar results when $\{M_i^n\}$ are compact manifolds with boundary. Say one may require that the diameters are uniformly bounded, the Ricci (or even sectional) curvature of $M_i$'s and the second fundamental form of $\partial M_i$'s are uniformly bounded from below. Does it follow that there exists a subsequence converging in the Gromov-Hausdorff sense?

ADDED: In Perales' survey http://arxiv.org/abs/1310.0850 mentioned in a comment by Belegradek there is Wong's Thm 3.6 which is very close to what I am looking for. It says: let $\{M_i^n\}$ be a sequence of compact Riemannian manifolds with boundary such that $Ricci(M_i)\geq r$, $Diam(M_i)\leq D$, $\lambda_1\leq II\leq \lambda_2$ for all $i$, where $II$ is the second fundamental form. Then there is a Gromov-Hausdorff convergent subsequence.

I am wondering if the upper bound on $II$ can be removed. The motivating situation I have in mind is the Blaschke selection theorem saying that a bounded sequence of convex sets in $\mathbb{R}^n$, or sphere $\mathbb{S}^n$, or hyperbolic space $\mathbb{H}^n$, has a Hausdorff convergent subsequence. In these cases $Ricci(M_i)=const$ , $II\geq const$, but there is no upper bound on $II$.

• See recent paper by Perales and Sormani, "Sequences of open Riemannian manifolds with boundary" arxiv.org/abs/1301.3961, and references therein, and the survey arxiv.org/abs/1310.0850. – Igor Belegradek Jul 28 '15 at 12:15
• Thanks for the useful reference! Influenced by it, I have made my question slightly more precise. – MKO Jul 28 '15 at 13:55