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Coarea, smooth approximation and regular area-controlled levels

PDF p. 26: Proof of Lemma 4.4, coarea and Sard displays

Statement

Work in ZFC under the regular mass-one stationary vacuum Harrison construction, with original maximal exterior section, $T ∣_{S} \equiv̸ 0$ , fixed sufficiently small positive $q$ , supplied smooth maximal section $Σ = Σ_{q, 0}$ , and sufficiently late comparison sphere $C = C_{t}$ , as in the localized construction. Let $R = {0 \leq ρ \leq η}$ be its supplied compact smooth collar, with boundary components $S$ and $S_{η}$ . Use the induced Riemannian metric $h_{0}$ on $Σ$ , the truncated height $u$ , and the zero-time endpoint data $A \subset C$ , $F : A \to D$ of the near-horizon construction, with the maximal generator family supplied by the endpoint-map result.

For every $ϵ > 0$ , one can choose a sufficiently small height parameter $δ > 0$ and a smooth function $u_{i}$ on $R$ , from a sequence converging to $u$ in $C^{0} \cap W^{1, 1}$ and preserving its constants $- 2 δ$ and $2 δ$ on fixed neighborhoods of $S$ and $S_{η}$ , respectively, and a regular value $b \in (- δ, - δ / 2)$ such that $D_{ϵ} = u_{i}^{- 1} (b)$ is smooth, closed, possibly disconnected, separates the two boundary components, and satisfies

$∣ D_{ϵ} ∣_{h_{0}} \leq \int_{A} J_{2} F d μ_{C} + ϵ \leq ∣ A ∣ + ϵ \leq ∣ C ∣ + ϵ .$

The compact smooth sublevel region $Ω_{ϵ} = {u_{i} \leq b} \subset R$ has boundary $S ⊔ D_{ϵ}$ . Here $∣ A ∣$ is its induced surface measure in $C$ , and $J_{2} F$ is the nonnegative Gram two-Jacobian in the fixed metrics; neither injectivity of $F$ nor multiplicity one is assumed.

Statement references

Lipschitz causal height and level-set Jacobian bound

Raychaudhuri expansion and generator Jacobians

Coarea smoothing with fixed boundary values and area control

Lipschitz height functions, cut measure and Jacobians

Truncated causal-boundary height for smooth approximation

Stationary section time and translated maximal sections

Regular non-degenerate stationary vacuum spacetime

Mass-one convention and Harrison denominator

Stationary vacuum model and asymptotic normalization

The Axiom of Choice

Compact remainder, causal cut and generator endpoint map

Closed generator set and nonintersection before the cut

Facts & Assumptions

[F1]

The model, regularity and mass-one normalization are those of Stationary vacuum model and asymptotic normalization, Regular non-degenerate stationary vacuum spacetime and Mass-one convention and Harrison denominator. Work in ZFC, including the Axiom of Choice. The Axiom of Choice

[F2]

The section time and induced metric use Stationary section time and translated maximal sections. The collar distance, zero-time cut $D$ , initial domain $A$ and endpoint map $F$ use Compact remainder, causal cut and generator endpoint map, with the maximal generator family of Closed generator set and nonintersection before the cut. The truncated height and translated cuts use Truncated causal-boundary height for smooth approximation. The endpoint theorem supplies closed, hence compact and Borel, $A \subset C$ and local smooth extensions of $F$ into the spacelike section, so its Gram Jacobian is measurable in the conventions of Lipschitz height functions, cut measure and Jacobians.

[F3]

For the fixed target and collar, Lipschitz causal height and level-set Jacobian bound supplies the boundary constants, Lipschitz height and, for every $z \in (- δ, - δ / 2)$ , the bound $H_{h_{0}}^{2} ({u = z}) \leq (1 + o_{δ} (1)) \int_{A} J_{2} F d μ_{C}$ , with $o_{δ} (1) \to 0$ as $δ ↓ 0$ .

[F4]

The sufficiently late target fixed in [F3] satisfies $\min_{C} v > 0$ , as ensured in Proof 2.1 of Lipschitz causal height and level-set Jacobian bound. Thus the zero-time conclusion of Raychaudhuri expansion and generator Jacobians applies on the entire domain $A$ : $0 \leq J_{2} F (c) \leq 1$ for every $c \in A$ , without requiring an exhaustion by the negative-cut domains. Measurability is supplied by [F2], not by the Jacobian definition.

[F5]

Relative smooth approximation, coarea, interval enlargement and regular-level selection apply to a compact smooth Riemannian three-region with two smooth boundary components, boundary-constant Lipschitz height and an almost-everywhere finite level-area bound. They give a regular value in the prescribed open interval, an interior level with area within any positive error of that bound, and the stated sublevel boundary. Coarea smoothing with fixed boundary values and area control

Proof

Given: The construction and $ϵ > 0$ in the Statement.

Goal: Construct $u_{i}$ , $b$ , $D_{ϵ}$ and $Ω_{ϵ}$ with the stated properties.

1.1givenF2F3F4algebra

Put $K = \int_{A} J_{2} F d μ_{C}$ . By [F2] and [F4], this integral is defined and $0 \leq K \leq ∣ A ∣ \leq ∣ C ∣ < \infty$ . With the target and collar fixed, choose $δ$ sufficiently small that the bound in [F3] holds and its nonnegative distortion allowance $r_{δ} = \max {o_{δ} (1), 0}$ satisfies $r_{δ} K < ϵ / 2$ . The resulting height is Lipschitz and has the two stated boundary constants. Thus every level in $I = (- δ, - δ / 2)$ has area at most $(1 + r_{δ}) K$ .

2.1step 1.1F2F5

Apply [F5] to $R$ , $u$ , $I$ and the bound $(1 + r_{δ}) K$ , with smoothing error $ϵ / 2$ . The endpoint constants lie on opposite sides of $I$ . The supplier gives a single smooth approximating sequence preserving the constants on fixed boundary neighborhoods and, for some index $i$ , a regular value $b \in I$ whose level separates the boundary components and has area at most $(1 + r_{δ}) K + ϵ / 2 < K + ϵ$ . It also gives the compact smooth sublevel region with boundary $S ⊔ D_{ϵ}$ .

3.1step 1.1step 2.1F5∎

Substitute the definition of $K$ and use $K \leq ∣ A ∣ \leq ∣ C ∣$ from step 1.1. This proves the displayed chain and the remaining conclusions. The coarea and Sard selection in [F5] does not require the original Lipschitz levels to be smooth or the endpoint map to be injective.

Source notes

The article citation supplies attribution only. The geometric level estimate retains the small flow-distortion factor; the proof absorbs it before applying the independent smoothing supplier. The parameter $δ$ used here is the height parameter, not a lower bound on $q$ .

Depends on

Used by · 2

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Sources

The uniqueness of smooth stationary black holes · PDF pp. 25–26, Lemma 4.4 and its proof