Proper end radius and exclusion of finite-time spatial infinity

In a boundary chart write $h=(h_{ij})$. Positive definiteness makes $\det (h_{ij})$ nonzero, so the cofactor formula shows that the inverse matrix $(h^{ij})$ is smooth up to the boundary. The identities $\mathrm{\mid }dR{\mathrm{\mid }}_h^2={\sum }_{i,j}{h}^{ij}({\mathrm{\partial }}_{i}R)({\mathrm{\partial }}_{j}R)$ and $\mathrm{\mid }X{\mathrm{\mid }}_h^2={\sum }_{i,j}{h}_{ij}{X}^i{X}^j$ therefore give continuous nonnegative norms, including at boundary points. The vector with components $Y^i={\sum }_j{h}^{ij}{\mathrm{\partial }}_{j}R$ satisfies $h(Y,X)=dR(X)$ and $\mathrm{\mid }Y{\mathrm{\mid }}_h=\mathrm{\mid }dR{\mathrm{\mid }}_h$, so Cauchy–Schwarz gives $\mathrm{\mid }X(R)\mathrm{\mid }\le \mathrm{\mid }dR{\mathrm{\mid }}_h\mathrm{\mid }X{\mathrm{\mid }}_h$. Thus $q=N\mathrm{\mid }dR{\mathrm{\mid }}_h+\mathrm{\mid }X(R)\mathrm{\mid }$ is continuous on $\bar{S}$.

If $K_0$ is nonempty, let $M_0$ be the maximum of $q$ on $K_0$; if $K_0$ is empty, set $M_0=0$, with no extreme-value assertion needed there. On $S\setminus K_0$, step 1.1 and the given estimate imply $q\le \mathrm{\mid }dR{\mathrm{\mid }}_h(N+\mathrm{\mid }X{\mathrm{\mid }}_h)\le C_0$. Hence $C_R=\max \{0,C_0,M_0\}$ bounds $q$ along every spatial curve in $S$. The section-time speed lemma gives $\mathrm{\mid }(R\circ x)\mathrm{\prime }\mathrm{\mid }\le C_R$ almost everywhere. On the compact interval between any two section times $v,v_0$ in the domain of the curve, F2 makes the scalar composition absolutely continuous; the fundamental theorem, with its choice hypotheses supplied by F1, gives $\mathrm{\mid }R(x(v))-R(x(v_0))\mathrm{\mid }\le C_R\mathrm{\mid }v-v_0\mathrm{\mid }$.

If $\mathrm{\mid }v-v_0\mathrm{\mid }\le T$, step 2.1 places the image in $\{R\le R(x(v_0))+C_{R}T\}$, which is compact in $\bar{S}$ by the supplied proper-radius hypothesis. This proves confinement in the manifold with boundary; it does not exclude boundary accumulation.

For the sufficient end realization, let $C$ be the larger compact core containing the transition region so that every point outside $C$ lies in a stated product end where $R=r$. For any real $m$, put $L=\max \{r_0,m\}$. The interval $[r_0,L]$ is compact, so each $[r_0,L]\times F_j$ is compact; its image under the end identification is compact by continuity. The union $P$ of those finitely many images with $C$ is compact. If $R(p)\le m$, either $p\in C$ or its end coordinate satisfies $r=R(p)\le m\le L$, so $p\in P$. Continuity of $R$ makes $\{R\le m\}$ closed in $\bar{S}$, hence closed in $P$ and compact. This includes the case $m<r_0$, where any points of the sublevel already lie in $C$. A bounded or nonproper reparameterization of an end coordinate alone supplies no such conclusion.