Temporal normal angle and compact positive-height portions

Work in ZFC on a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$. Let $g$ be its metric and $\mathrm{\Sigma }$ a smooth embedded spacelike maximal hypersurface, smooth up to its actual boundary $\mathrm{\partial }\mathrm{\Sigma }$, with future unit normal $n$. Maximal means its induced mean curvature vanishes.

Fix a compact ambient set $Q$, allowed to be empty, and let $\tau$ be a smooth real function on an open neighborhood of $\mathrm{\Sigma }\cup Q$. A compact positive-height portion is the entire set $P=\mathrm{\Sigma }\cap \{\tau \ge 0\}$, assumed compact and contained in $Q$; it is not a clipping of $\mathrm{\Sigma }$ by $Q$. Require the metric gradient of $\tau$ to be past-directed timelike on an open neighborhood of $P$. On that temporal neighborhood put $\alpha =\sqrt{-g^{-1}(d\tau ,d\tau )}>0$ and $V=-{\alpha }^{-1}{\mathrm{grad}}_g\tau$, where ${\mathrm{grad}}_g\tau$ is characterized by $g({\mathrm{grad}}_g\tau ,Y)=d\tau (Y)$ for every ambient tangent vector $Y$. Along the hypersurface in that neighborhood put $v=-g(n,V)$. Thus $V$ is future unit timelike and $v$ is the temporal normal angle.

Write $h=g{\mathrm{\mid }}_{T\mathrm{\Sigma }}$. For a real function $f$ smooth on an open subset of $\mathrm{\Sigma }$, smooth up to any actual boundary there, write ${\mathrm{\nabla }}_{\mathrm{\Sigma }}f={\mathrm{grad}}_{h}f$ and ${\mathrm{\Delta }}_{\mathrm{\Sigma }}f={\mathrm{div}}_h({\mathrm{\nabla }}_{\mathrm{\Sigma }}f)$ for the positive-trace Laplacian. If $f$ is an ambient function, these operators act on its restriction to $\mathrm{\Sigma }$. At actual boundary points use the smooth one-sided extensions of these local operators.

Fix an arbitrary real exponent $\mu >0$. For nonempty $Q$ set $T={\max }_{q\in Q}\max \{\tau (q),0\}$, and for empty $Q$ set $T=0$. Set $B={\sup }_{\mathrm{\partial }\mathrm{\Sigma }\cap \{\tau >0\}}{\tau }^{\mu }v$, with the supremum zero for an empty set. At positive height only the actual hypersurface boundary contributes a boundary term.

Spacelike sections, maximality and lapse–shift data

Causal exterior, chronological boundaries and Cauchy data

The Axiom of Choice

The inverse metric, musical isomorphisms and index raising

The Riemannian gradient is the metric dual of the differential

Riemannian divergence

Laplace–Beltrami operator as the trace of the Hessian

A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism

The normal and induced metric are those of the spacelike hypersurface interface; the temporal orientation is that of the causal interface. The inverse metric and metric dual of $d\tau$ have the musical-isomorphism convention. Dividing a past timelike gradient by its positive Lorentzian norm and reversing its sign gives a future unit timelike $V$. In a Lorentz-orthonormal frame with time axis $V$, the unit equation for the future $n$ gives $v=\sqrt{1+\mathrm{\mid }{n}^{\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{c}\mathrm{e}}{\mathrm{\mid }}^2}\ge 1$.

The Riemannian gradient and divergence suppliers define the two intrinsic operators. The Laplace–Beltrami supplier identifies their composition with the positive metric trace of the Hessian and supplies the stated one-sided boundary convention. The gradient's inverse-metric formula also extends smoothly in boundary half-charts, since $h$ stays positive definite and $f$ is smooth up to the boundary; its values there are determined by its values in the interior.

For nonempty $Q$, the continuous function $q\mapsto \max \{\tau (q),0\}$ attains a finite maximum by the compactness supplier's extreme-value assertion. If $Q$ is empty, so is $P$, and the explicit convention defines $T$. On nonempty compact $P$, smoothness up to the actual boundary makes $v$ continuous and bounded for each individual hypersurface, again by that extreme-value assertion. Since $\mu >0$, the weight ${\tau }^{\mu }$ is continuous on $P$, with value zero at zero height, and $0\le \tau \le T$ there. Thus the actual boundary supremum is finite, and its empty-set convention covers empty $P$ as well. No uniform bound along a sequence is assumed.