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Rescaled annular estimates through four derivatives

PDF p. 15: §3.2.1, Schauder display after (3.7)

Statement

Fix $δ \in (0, 1)$ . Assume the candidate family and normalized stationary Cartesian ends of Asymptotic maximal graph equation and coefficients, including the common differentiated metric estimates and slope estimate stated there. The sections are assumed to exist and are smooth, spacelike and maximal. In the coordinates $(t_{q}, x_{q})$ write $r_{q} = ∣ x_{q} ∣$ and $u_{q} = {\tilde{u}}_{q}$ for the normalized graph height defined in Stationary maximal graphs and normalized graph heights. Constants and threshold radii below may depend on $δ$ but are independent of $q \in [δ, 1]$ .

There are $R_{δ}$ and $C_{δ}$ such that, for all $R \geq R_{δ}$ and $q \in [δ, 1]$ , $\sum_{j = 0}^{4} R^{j} \sup_{R < r_{q} < 2 R} ∣ D_{x_{q}}^{j} u_{q} ∣ \leq C_{δ} (\sup_{R / 2 < r_{q} < 4 R} ∣ u_{q} ∣ + R^{- 1}) .$ Here $D^{j}$ denotes the finite array of Cartesian partial derivatives of total order $j$ , with $D^{0} u_{q} = u_{q}$ . The $R^{- 1}$ term is the rescaled contribution of the $O (r_{q}^{- 3})$ zero-slope forcing. Consequently, for $0 \leq j \leq 4$ , $∣ D_{x_{q}}^{j} u_{q} ∣ \leq C_{δ, j} r_{q}^{- 3 / 4 - j}$ on a common sufficiently distant end.

Facts & Assumptions

[F1]

The stationary maximal graph equation is $\partial_{i} A_{q}^{i} (x, D u_{q}) = 0$ , where $A_{q}^{i} (x, p) = \sqrt{- \det g_{q}} (g_{q}^{i j} p_{j} - g_{q}^{i 0}) / \sqrt{- g_{q}^{00} + 2 g_{q}^{0 j} p_{j} - g_{q}^{j k} p_{j} p_{k}}$ ; all metric components are in normalized coordinates. Asymptotic maximal graph equation and coefficients

[F2]

Uniformly for $q \in [δ, 1]$ , $g_{00} = - 1 + O_{5} (r_{q}^{- 1})$ , $g_{0 i} = O_{5} (r_{q}^{- 2})$ , and $g_{i j} = δ_{i j} + O_{5} (r_{q}^{- 1})$ . In $O_{5} (r^{- b})$ , each spatial derivative of order $k \leq 5$ is bounded by a common constant times $r^{- b - k}$ . Asymptotically flat positive-q ends and cylindrical limit Differentiated asymptotic bounds and parameter uniformity

[F3]

On a common end, $∣ D u_{q} ∣ \leq C_{δ} r_{q}^{- 1 / 2}$ . Uniform asymptotic slope bound

[F4]

For every $μ \in (0, 1)$ the normalized heights satisfy $∣ u_{q} ∣ \leq C_{δ, μ} r_{q}^{- μ}$ uniformly on a common end. Radial barrier, absorption and height decay

[F5]

The annular bootstrap applies to a height-independent divergence flux with a common ellipticity interval and joint $C^{4, α}$ bounds on a neighborhood of a common slope ball, a common normalized height bound, and zero-slope divergence bounded by $K / R^{2}$ in $C^{2, α}$ ; it then gives the displayed four-derivative estimate. Uniform annular Schauder bootstrap for spacelike graph equations

[F6]

Hölder norms use the Euclidean distance on the fixed rescaled annulus. Work in ZFC, including the choice assumptions of the analytic supplier. Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains The Axiom of Choice

Proof

Given: The candidate family, normalized ends and hypotheses in the Statement.

Goal: Prove the uniform annular estimate and the stated derivative decay.

1.1givenF1F3F4algebra

Put $A = {1 / 2 < ∣ y ∣ < 4}$ , $v_{q, R} (y) = u_{q} (R y)$ , $w_{q, R} = v_{q, R} / R$ , and $P_{q, R} (y, p) = A_{q} (R y, p)$ . Then $D_{y} w_{q, R} = D_{x} u_{q} (R y)$ and the equation in [F1] becomes ${div}_{y} P_{q, R} (y, D_{y} w_{q, R}) = 0$ . By [F3], after increasing the common radius, the realized slopes lie in $∣ p ∣ \leq 1 / 4$ ; by [F4], the normalized heights $w_{q, R}$ are uniformly bounded on $A$ .

2.1F1F2F6step 1.1algebra

On a fixed larger slope ball, for example $∣ p ∣ \leq 1 / 3$ , [F2] makes the denominator in [F1] uniformly positive for sufficiently large $R$ . Matrix inversion and differentiation of the displayed flux show that $P_{q, R}$ and its joint derivatives through order four are uniformly bounded there; the fifth spatial derivative bounds in [F2], after scaling, also give the required uniform $α$ -Hölder control for any fixed $0 < α < 1$ . The slope derivative $P_{q, R, p}$ is symmetric and uniformly positive definite with a common upper bound: it converges uniformly on the slope ball to the slope derivative of $p / \sqrt{1 - ∣ p ∣^{2}}$ . These bounds hold on a slope neighborhood of $∣ p ∣ \leq 1 / 4$ .

2.2F1F2F6step 1.1algebra

Let $s_{q, R} (y, p) = \sum_{i} \partial_{y_{i}} P_{q, R}^{i} (y, p)$ , differentiating at fixed $p$ . At $p = 0$ , [F1] and [F2] give $A_{q} (x, 0) = O_{5} (r_{q}^{- 2})$ . Thus, for spatial derivatives through order three, $D_{y}^{k} s_{q, R} (y, 0) = O (R^{- 2})$ on $A$ , uniformly in $q$ and $R$ . In particular its $C^{2, α} (A)$ norm is at most $K_{δ} R^{- 2}$ : the third derivative bound controls the Hölder seminorm of its second derivatives on the fixed annulus. The zero-slope divergence in the original equation is $O (r_{q}^{- 3})$ , and scaling it by the two powers of $R$ used to pass from $u_{q}$ to $v_{q, R}$ accounts for the $R^{- 1}$ forcing in the final estimate.

3.1F4F5step 1.1step 2.1step 2.2algebra∎

Steps 1.1, 2.1 and 2.2 verify all structural hypotheses of [F5], including initial quasilinear regularity through that supplier, on the fixed annulus with constants independent of $q$ and $R$ . Applying [F5] yields the displayed estimate. For $μ = 3 / 4$ , [F4] bounds the larger-annulus height supremum by $C_{δ} R^{- 3 / 4}$ ; since $R^{- 1} \leq R^{- 3 / 4}$ for $R \geq 1$ , division by $R^{j}$ gives $∣ D^{j} u_{q} ∣ \leq C_{δ, j} R^{- 3 / 4 - j}$ on $R < r_{q} < 2 R$ . Choosing $R$ comparable to any sufficiently large $r_{q}$ proves the pointwise decay.

Source notes

The article reference attributes the original estimate. The argument uses the declared graph equation, end estimates, barrier result and mathematical bootstrap as premises; it does not use the article's claim as a premise.

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Sources

The uniqueness of smooth stationary black holes · PDF p. 15, §3.2.1, Schauder display after (3.7)