open ball $V_q$ in $Y_2$ about $q$ such that for any $q' \in V_q$ we have both $\sigma_{q'}(t)  \subset  {E}_1( Y_1 \times [R, \infty) )$ and $\dot\sigma_{q'}(t)  >  0$ for all $t \geq T$. For each such $q'$, let $p'$ be the unique element of $Y_1$ such that $\gamma_{p'}$ is asymptotic to $\sigma_{q'}$. By compactness, for any $q' \in \overline{V_q}$, $d(\sigma_{q'}(t), \gamma_{p'}(t))$ is uniformly bounded for $t \in [0, T]$. We need only check that a uniform bound holds for the tails of the geodesics emanating from $V_q$.

In order to estimate $d(\sigma_{q'}(t), \gamma_{p'}(t))$ for $t\geq T$, proceed as in the proof of Proposition \[prop:Ebaronto\]. By continuity of the exponential map and by shrinking $V_q$ if necessary all constants may be chosen independently of $q'$.

We may now prove the topological part of Theorem \[thm:main\]:

\[thmmarsh\] If $M$ is a complete, noncompact Riemannian manifold with an essential subset $K$ with pinched negative curvature as in (\[curvasmp\]), then $M$ admits a geometric compactification as a topological manifold with boundary.

Throughout the proof we use the notation for $\gamma$ and $\sigma$ as described preceding Lemma \[lemma:unifbdd\].

Suppose $K_1, K_2$ are two essential subsets of $M$. The propositions above imply that we get 