\Lambda \leq b . \end{cases} \nonumber \\
\lambda' = \begin{cases}  \lambda  & \text{ if } \lambda < a, \\
         \frac{a}{2} & \text{ if } \lambda \geq a . \end{cases}\end{aligned}$$ **Metric estimate:** $$\begin{aligned}
\label{eqn:metric}
L_3 \; \cosh^2( a( r+ L_1 ) ) \; \delta_{\beta \nu} \leq g_{\beta \nu}(y, r) \leq L_4 \; \sinh^2( b( r+ L_2 ) ) \; \delta_{\beta \nu}, \end{aligned}$$ where $\displaystyle L_3 = \left( \frac{\omega}{\cosh^2{ a L_1 } } \right)$, $\displaystyle L_4 = \left( \frac{\Omega}{\sinh^2{ b L_2 } } \right)$, and $\Omega, \omega$ are described above.

\[lemma:metriccomp\]

Essential Subsets {#sec:tconvex}
=================

In this section we provide a sufficient condition for the submanifold $K \subset M$ to be an essential subset. We assume that $K$ is 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. We discuss a condition that ensures that $E\colon Y \times [0, \infty) {\rightarrow}\overline{{M \backslash K}}$ is a diffeomorphism. This property allows us to relax the hypothesis that $M$ be simply connected in the Anderson-Schoen result; essential subsets replace the requirement that the map $exp_p\colon T_p M {\rightarrow}M$ be a diffeomorphism which is ensured by the 