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Logarithmic angle inequality through the full range v at least one

Statement

Let $U$ be a smooth future unit timelike field on a Lorentzian four-region with a uniform bound $∣ D U ∣_{e} + ∣ D^{2} U ∣_{e} + ∣ Ric ∣_{e} \leq K$ , where $e = g + 2 U^{♭} \otimes U^{♭}$ . For every smooth spacelike maximal hypersurface in the region, let $v = - g (n, U)$ . There is a constant $C$ depending only on the background bound and dimension such that $Δ_{Σ} \log v \geq \frac{1}{4} ∣ \nabla_{Σ} \log v ∣^{2} - C v^{2}$ at every point, including the entire range $1 \leq v \leq 2$ . For the compact-family assertion, let $P$ be a nonempty compact metric parameter space and let $(g_{p}, U_{p})$ be smooth Lorentzian metrics and future unit timelike observer fields on one common open ambient domain. Assume continuity in the $C^{2}$ compact-open topology: in each fixed chart the components and all spatial derivatives through order two vary continuously with $p$ , uniformly on every compact subset of the chart. Fix a common compact ambient set $L$ covered by finitely many nonempty compact coordinate sets $L_{j}$ contained in those open chart domains. Then on $L$ the constant $C$ can be chosen independently of $p$ and of the candidate hypersurface. In particular this applies to jointly smooth data on a neighborhood of a compact parameter set and the common compact ambient set. If a common background bound $K$ is explicitly supplied on the whole region for all family members, the constant is uniform on that region without requiring compactness of the parameter space. No hypersurface boundary, compact positive-height portion, bound on the candidate second fundamental form or temporal cutoff is needed for this pointwise conclusion.

Facts & Assumptions

[F1]

Maximality makes the symmetric three-dimensional form $k$ trace free. Spacelike sections, maximality and lapse–shift data

[F2]

The trace-free spectral/gradient inequality is Trace-free three-dimensional coercivity for the logarithmic gradient.

[F3]

The uniform all-range scalar estimates, including $\nabla_{Σ} v = k Z + Q$ , $∣ Z ∣^{2} = v^{2} - 1$ , $∣ Q ∣ \leq K_{1} v^{2}$ and the compact-family scope, are Uniform scalar angle inequalities from local background tensors.

[F4]

For a future unit observer and future unit hypersurface normal, $e$ is positive definite and $v = - g (n, U) \geq 1$ ; use the induced $e$ tensor operator norms. Boost-frame bounds for a smooth observer and arbitrary spacelike normals

[F5]

With $h = g ∣_{T Σ}$ , the intrinsic gradient is defined by $h (\nabla_{Σ} f, X) = d f (X)$ for tangent $X$ . The Riemannian gradient is the metric dual of the differential

[F6]

The intrinsic Laplacian uses the positive-divergence convention $Δ_{Σ} f = {tr}_{h} {Hess}_{h} f = {div}_{h} \nabla_{Σ} f$ , where ${Hess}_{h} f (X, Y) = h (\nabla_{X}^{h} \nabla_{Σ} f, Y)$ . Laplace–Beltrami operator as the trace of the Hessian

Proof

Given: The smooth observer, background bound and arbitrary maximal hypersurface. Goal: Prove the displayed inequality for all $v \geq 1$ .

1.1F1F4F5givenalgebra

By [F1], the induced metric $h$ is Riemannian and $k$ is symmetric and trace free; identify $k$ with its $h$ -raised endomorphism. By [F4], $e$ is positive definite and $v \geq 1$ . Hence $\log v$ is smooth. For tangent $X$ , [F5] gives $h (\nabla_{Σ} \log v, X) = d (\log v) (X) = v^{- 1} d v (X) = h (v^{- 1} \nabla_{Σ} v, X)$ , so $v \nabla_{Σ} \log v = \nabla_{Σ} v$ .

2.1F2F3step 1.1choosealgebra

Fix $ε = 1 / 11$ and put $G = ∣ \nabla_{Σ} \log v ∣_{h}^{2}$ . By [F3] and $v \geq 1$ , $Δ_{Σ} v / v \geq \frac{10}{11} ∣ k ∣_{h}^{2} - C_{1} v^{2}$ . Step 1.1 and the vector identity and bounds in [F3] satisfy the hypotheses of [F2] with $A = \nabla_{Σ} \log v$ , so $G \leq \frac{8}{11} ∣ k ∣_{h}^{2} + C_{2} v^{2}$ . The second inequality rearranges to $\frac{10}{11} ∣ k ∣_{h}^{2} \geq \frac{5}{4} G - \frac{5}{4} C_{2} v^{2}$ , with constants depending only on the background bound and dimension.

2.2F5F6step 1.1algebra

Using step 1.1 and the connection product rule in the Hessian formula of [F6], for tangent $X, Y$ one has ${Hess}_{h} (\log v) (X, Y) = h (\nabla_{X}^{h} (v^{- 1} \nabla_{Σ} v), Y) = v^{- 1} {Hess}_{h} v (X, Y) - v^{- 2} d v (X) d v (Y)$ . By [F5], tracing $d v \otimes d v$ gives $∣ \nabla_{Σ} v ∣_{h}^{2}$ . Taking the trace as prescribed in [F6] therefore yields $Δ_{Σ} \log v = Δ_{Σ} v / v - G$ .

3.1F3step 1.1step 2.1step 2.2algebra∎

Insert step 2.1 into step 2.2 to obtain $Δ_{Σ} \log v \geq \frac{1}{4} G - (C_{1} + \frac{5}{4} C_{2}) v^{2}$ . No division by $\sqrt{v^{2} - 1}$ occurred, so step 1.1 makes this calculation valid also at $v = 1$ and throughout $1 \leq v \leq 2$ . Under the stated compact-family hypotheses, [F3] gives a single finite background bound $K_{L}$ on $L$ for all $p \in P$ . The same calculation therefore gives one $C$ on $L$ , independently of $p$ and the candidate hypersurface. With an explicitly supplied common background bound $K$ on the whole region, the calculation instead gives one $C$ there. The algebra uses fixed $ε$ and numerical coefficients only. No hypersurface boundary, compact positive-height portion, candidate second-fundamental-form bound or temporal cutoff was used.

Source notes

The cited Appendix B uses normal-angle differentiation, Codazzi and trace-free three-dimensional algebra, then restricts to $v > 2$ for its arcosh argument. This is methodological attribution only; the all-range logarithmic conclusion above is proved from the declared local suppliers and operator definitions.

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External dependencies

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1 hypotheses, printed pp.35–36; proof (B.8)–(B.11), printed pp.38–39 (PDF pages 35–36 and 38–39).

Methodological attribution: normal-angle differentiation and Codazzi yield a scalar Laplacian bound; trace-free three-dimensional algebra retains a positive gradient term in the subsequent arcosh calculation.

Hypotheses: The cited lemma uses C2-convergent Lorentz metrics near a compact set, a fixed uniformly temporal function and compact nonnegative-height hypersurface portions. Its arcosh calculation assumes v>2.

Reading limits: Consulted the online PDF text of these hypotheses and (B.8)–(B.11) in this repair; no rendered images were read. This citation supplies background attribution only. The all-range logarithmic inequality is proved locally from the declared geometric and scalar suppliers and operator definitions. No global weighted estimate or compact-positive-portion hypothesis is imported.

Sources

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1 hypotheses, printed pp.35–36; proof (B.8)–(B.11), printed pp.38–39 (PDF pages 35–36 and 38–39).