A Compactness Theorem for Riemannian Manifolds with Boundary and Applications

In this paper, we prove $W^{1,p}$ ($p>n$) and $C^{0,\alpha}$ ($0 < \alpha < 1$) precompactness for classes of Riemannian $n$-manifolds with boundary satisfying uniform $L^{\infty}$
bounds on curvature, mean curvature, diameter, and the $(n-1)$-volume of the boundary. In particular, we identify a class of convex manifolds and a class of uniformly mean-convex manifolds that
exhibit this compactness without any a priori bounds on the injectivity radii. In these cases, we show the boundary data excludes interior volume collapse.
If injectivity radius bounds are assumed, the sectional curvature bounds can be replaced with bounds on the Ricci tensor. A central feature of these results is that we only assume a
pointwise bound on the mean curvature, which constrains the regularity of the convergence.
We obtain two geometric stability theorems as applications of this compactness result, based
on rigidity results of Cohn-Vossen/Pogorelov and Hopf. The first theorem applies to $3$-manifolds with pointwise curvature close to $0$ whose boundaries are Gromov-Hausdorff close to a
fixed metric on $S^2$ with positive curvature. We show that
such manifolds are $C^{0, \alpha}$ close to the solid region enclosed by a Weyl embedding of the fixed metric into $\mathbb R^3$.
The second theorem demonstrates that if a $3$-manifold with
$\chi(\partial M) = 2$ has Ricci curvature close to $0$ and boundary mean curvature close to $1$, then $M$ is $C^{0, \alpha}$ close to the Euclidean unit ball in $\mathbb R^3$. In these theorems, the geometric stability is within a precompact class identified above, so that the notion of ``close'' in the hypotheses depends on the constants determining that class.