  For a sequence $\{(M_i, g_i, x_i)\}$ of pointed Riemannian manifolds with
boundary, the sequence $\{(M_i,\tilde g_i,x_i)\}$ is its conformal satellite if
the metric $\tilde g_i$ is conformal to $g_i$, that is, $\tilde
g_i=u^{\frac{4}{n-2}}_ig_i$. Assuming the manifolds $(M_i,g_i,x_i)$ have
uniformly bounded geometry, we show that both sequences have smoothly
Cheeger-Gromov convergent subsequences provided the conformal factors $u_i$ are
principal eigenfunctions of an appropriate elliptic operator. Part of our
result is a Cheeger-Gromov compactness for manifolds with boundary. We use
stable versions of classical elliptic estimates and inequalities found in the
recently established 'flatzoomer' method.
