hypothesis of simple connectedness used in the Anderson-Schoen paper. In addition, the result here proves the regularity of the *entire* compactification ${M^{*}}= M \cup {M(\infty)}$. The Anderson-Schoen theorem only proves the regularity for the boundary at infinity ${M(\infty)}$.

The outline of this paper is as follows. In Section \[sec:notation\] we outline our notation and explain our comparison theorems. In Section \[sec:tconvex\] we provide a condition for an essential subset. In Section \[sec:tracey\] we describe the compactification of $M$ as a topological manifold, and then in Section \[sec:compreg\] we set up the necessary estimates to show the compactified manifold has a well defined $C^{a/b}$ structure.

We would like to thank our advisor Jack Lee for suggesting this problem to us and for his constant guidance. We would also like to thank Robin Graham for suggestions on an early draft of this paper.

Notation and basic estimates {#sec:notation}
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In this section we outline our notation and provide the estimates that will be used in the subsequent comparison geometry. Throughout this paper $M$ denotes a complete, connected, noncompact Riemannian $(n+1)$-manifold with metric $g$. The letter $K$ will always denote an essential subset and $Y:=\partial K$ will denote the smooth hypersurface boundary. Throughout 