Temporal cutoff estimate for normals to maximal hypersurfaces

Let $g_j$ be smooth time-oriented Lorentz metrics of signature $(-,+,+,+)$ on a common smooth four-manifold, converging in $C^2$ to a Lorentz metric near a fixed compact set $Q$. Use a common time orientation and a fixed finite coordinate cover near $Q$, with uniformly bounded metric $C^2$ and inverse-metric $C^0$ norms. Let $\tau$ be a fixed smooth function, with past-directed timelike $g_j$-gradient and ${\alpha }_j^2=-g_j^{-1}(d\tau ,d\tau )\ge a_0^2>0$ on $Q\cap \{\tau \ge 0\}$. Let ${\mathrm{\Sigma }}_j$ be smooth spacelike maximal hypersurfaces, smooth up to their actual boundaries, with future unit normals $n_j$, such that the entire portions ${\mathrm{\Sigma }}_j\cap \{\tau \ge 0\}$ are compact and contained in $Q$. Set $V_j=-{\alpha }_j^{-1}{\mathrm{grad}}_{g_j}\tau$, $v_j=-g_j(n_j,V_j)$ and ${\tau }_{\max }={\max }_Q\max (\tau ,0)$ when $Q$ is nonempty, and ${\tau }_{\max }=0$ when $Q=\mathrm{\varnothing }$. There is $\mu >4$, depending only on $a_0$, the uniform metric and inverse bounds and $\mathrm{\parallel }\tau {\mathrm{\parallel }}_{C^3(Q)}$ in that cover, such that ${\sup }_{{\mathrm{\Sigma }}_j\cap \{\tau >0\}}{\tau }^{\mu }{v}_j\le 2\max \{2{\tau }_{\max }^{},{\sup }_{\mathrm{\partial }{\mathrm{\Sigma }}_j\cap \{\tau >0\}}{\tau }^{\mu }{v}_j\}$. A supremum over an empty set is zero. In particular the boundary supremum is retained unless it is empty or separately bounded. No uniform bound on the normals is assumed in advance.

Import (B.11) of Radu, Appendix B: with the geometric hypotheses of the Statement and ${\eta }_j=\mathrm{arcosh}{v}_j$, there is a uniform $C\ge 1$ such that on $\{v_j>2\}$, ${\mathrm{\Delta }}_{{\mathrm{\Sigma }}_j}{\eta }_j\ge \coth {\eta }_j(\frac{1}{4}\mathrm{\mid }{\mathrm{\nabla }}_{{\mathrm{\Sigma }}_j}{\eta }_j{\mathrm{\mid }}^2-C{v}_j^2)$ and $\mathrm{\mid }{\mathrm{\Delta }}_{{\mathrm{\Sigma }}_j}\tau \mathrm{\mid }\le C{v}_j^2$. Its constant depends only on $a_0$, the stated uniform coordinate metric/inverse bounds and the $C^3$ norm of $\tau$. This is an external geometric differential estimate, not a local reconstruction. The source derives it from (B.8)–(B.10) using Gauss–Codazzi, maximality and the trace-free eigenvalue bound in three tangent dimensions; the present proof imports that step with all its hypotheses.

The scalar argument follows the maximum-principle method of Radu, Appendix B, (B.11)–(B.12), using the explicitly proved conversion to log v and retaining the source’s conservative stated constant. The independent source is not the article under study. No graph existence, temporal-function existence, boundary-angle estimate or tensor-identity reconstruction is asserted. The compact sets must contain the entire nonnegative-height portions; replacing them by intersections with an arbitrary compact Q would introduce extra boundary terms.