$\sigma$ and $\tau$ are *asymptotic* if $d_M( \sigma(t), \tau(t))$ remains bounded as $t {\rightarrow}+\infty$. Here $d_M$ is the distance function on $M$ induced by the metric $g$. We define ${M(\infty)}$ to be the set of all equivalence classes of this relation; this is the *geometric boundary at infinity*. \[page:asymp\]

In 1985, Michael Anderson and Richard Schoen proved that given a Cartan-Hadamard manifold $M$ with pinched sectional curvatures like $$\begin{aligned}
-\infty < -b^2 \leq \sec( M ) \leq -a^2 < 0, \end{aligned}$$ where $a$ and $b$ are positive constants, then geometric boundary at infinity has a $C^{a/b}$ structure [@Anderson-Schoen]. Motivated by this result we investigate to what extent the simply connected hypothesis may be relaxed when compactifying the manifold and what resulting regularity may be obtained for the compactified manifold with boundary. In particular, let $M$ be a complete, noncompact Riemannian $(n+1)$-manifold. Define an *essential subset* \[page:es\] $K$ of $M$ to be a compact $(n+1)$-dimensional Riemannian submanifold with boundary, such that $Y := \partial K$ is a smooth hypersurface that is convex with respect to the outward pointing unit normal and such that $\exp\colon N^{+}Y {\rightarrow}\overline{{M \backslash K}}$ is a diffeomorphism, where $N^{+}Y \approx Y \times [0, \infty)$ is the outward 