Lipschitz height functions, cut measure and Jacobians

On a smooth Riemannian three-region $(\mathrm{\Sigma },k)$ a locally Lipschitz height $u$ has almost-everywhere chart derivatives, equal to weak derivatives; its metric gradient is defined almost everywhere by $k(\mathrm{\nabla }u,\cdot )=du$. Cut area means normalized two-dimensional Hausdorff measure ${\mathcal{H}}_k^2$. The initial domain for an endpoint construction is a Borel subset $A$ of a smooth Riemannian surface $(C,h)$, not the possibly nonsmooth image $D$. For $F:A\to \mathrm{\Sigma }$ with countably many local $C^1$ extensions, $J(F)$ means the nonnegative Gram Jacobian $J_{2}F$, and $N(F,A;y)$ counts the initial points arriving at $y$. The area formula is ${\int }_{A}J(F)d{\mu }_C={\int }_{F(A)}N(F,A;y)d{\mathcal{H}}_k^2(y)$; therefore surjectivity $F(A)=D$ gives $\mathrm{\mid }D\mathrm{\mid }\le {\int }_{A}J(F)d{\mu }_C$. Coincident endpoints do not invalidate this inequality. The identity $\mathrm{\mid }D\mathrm{\mid }={\int }_{A}J(F)d{\mu }_C$ holds if $N(F,A;y)=1$ for ${\mathcal{H}}_k^2$-almost every $y\in D$. If ${\int }_{A}J(F)d{\mu }_C<\mathrm{\infty }$, this multiplicity-one condition is also necessary for equality. Equality with infinite common value need not imply multiplicity one. All measures use the same area normalization and the specified fixed metrics; $J(F)\le 1$ is an additional geometric estimate, not part of the definition of Jacobian.

Apply [F1] in coordinate charts to the height and use the smooth inverse metric to obtain its measurable weak gradient. The area and Jacobian conventions are exactly [F2], and the formula and inequality are [F3]. Multiplicity one almost everywhere suffices for equality by that formula. For necessity in the finite case, put $I={\int }_{A}J(F)d{\mu }_C<\mathrm{\infty }$ and suppose $\mathrm{\mid }D\mathrm{\mid }=I$. The measurable set $E=\{y\in D:N(F,A;y)\ge 2\}$ satisfies $I={\int }_{D}N(F,A;y)d{\mathcal{H}}_k^2(y)\ge \mathrm{\mid }D\mathrm{\mid }+{\mathcal{H}}_k^2(E)$. Subtracting the finite value $\mathrm{\mid }D\mathrm{\mid }=I$ gives ${\mathcal{H}}_k^2(E)=0$. This subtraction is unavailable for infinite common value. Indeed, take $A=C={\mathbb{R}}^2\bigsqcup {\mathbb{R}}^2$ with its Euclidean component metrics and map each component isometrically onto the same plane $D$ in Euclidean $\mathrm{\Sigma }={\mathbb{R}}^3$. This map is smooth on both components, $J_{2}F=1$, $N=2$ everywhere on $D$, and both $\mathrm{\mid }D\mathrm{\mid }$ and $I$ are infinite.

For an endpoint family, [F3] supplies local extensions when the supplied $C^1$ family meets a slice $s=0$ transversely at its unique hit. For null generators of a smooth Lorentzian metric meeting a spacelike slice $\mathrm{\Sigma }=s^{-1}(0)$, use a smooth defining function $s$ with $0$ a regular value, as required by [F3]. At a hit $p$, the regular-level coordinates of [F5] give $T_p\mathrm{\Sigma }=\ker d{s}_p$. If the nonzero null tangent $\dot{\gamma }$ had $d{s}_p(\dot{\gamma })=0$, it would belong to $T_p\mathrm{\Sigma }$. Positive definiteness on that space would give $g_p(\dot{\gamma },\dot{\gamma })>0$, contradicting $g_p(\dot{\gamma },\dot{\gamma })=0$. Thus $d{s}_p(\dot{\gamma })\mathrm{\ne }0$, using precisely the conventions in [F4].

If the family is parametrized by a smooth chosen initial null normal and has smooth geodesic dependence on its open flow domain, it is such a supplied family; these are geometric inputs to be established by an application, not consequences of a Lipschitz height alone. When the defining function $s$ is also a time function, [F4] makes it strictly increasing along each future-oriented generator, and strictly decreasing with the opposite orientation. Either strict monotonicity makes a prescribed time-level hit unique.

A closed $A\subset C$ is Borel by [F2]. Thus an application must verify the smooth initial surface and chosen generator branch, open flow domain, unique transverse hits at a regular level, Borel domain and image coverage. It does not need to prove that $D$ is a smooth or Lipschitz surface. Focal degeneracy at endpoints is allowed as rank loss of the derivative. No inference of the required endpoint-map regularity is made merely from the statement that $D$ is a cut.