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Radial barrier, absorption and height decay

PDF p. 14: §3.2.1; (3.6)–(3.7); PDF p. 15: §3.2.1; (3.6)–(3.7)

Statement

Fix $δ \in (0, 1)$ and $μ \in (0, 1)$ . Assume the candidate family, normalized Cartesian ends and uniform estimates of Asymptotic maximal graph equation and coefficients and Uniform asymptotic slope bound. In particular, each normalized graph height ${\tilde{u}}_{q}$ tends to zero as $r_{q} = ∣ x_{q} ∣ \to \infty$ , uniformly in direction for that fixed $q$ ; the sections are assumed to exist. Write $L_{q} v = a_{q}^{i j} \partial_{i j} v + b_{q}^{j} \partial_{j} v$ , using the coefficients supplied by the graph equation, evaluated on the given graph.

There are $R_{δ, μ}$ and $c_{δ, μ} > 0$ , independent of $q \in [δ, 1]$ , such that $L_{q} (r_{q}^{- μ}) \leq - c_{δ, μ} r_{q}^{- μ - 2}$ for $r_{q} \geq R_{δ, μ}$ . Fix any $r_{*} \geq R_{δ, μ}$ and put $M_{q} (r_{*}) = \max_{r_{q} = r_{*}} ∣ {\tilde{u}}_{q} ∣$ . There is a constant $K_{δ, μ}$ independent of $q$ and $r_{*}$ such that, for $r_{q} \geq r_{*}$ , $∣ {\tilde{u}}_{q} (x_{q}) ∣ \leq (M_{q} (r_{*}) + K_{δ, μ} / r_{*}) (r_{*} / r_{q})^{μ}$ . If $C_{δ}$ is a uniform constant in $∣ \partial_{x_{q}} {\tilde{u}}_{q} ∣ \leq C_{δ} r_{q}^{- 1 / 2}$ , then $M_{q} (r_{*}) \leq 4^{- μ} (M_{q} (r_{*}) + K_{δ, μ} / r_{*}) + 2 C_{δ} \sqrt{r_{*}}$ . Consequently, for a fixed sufficiently large $r_{*}$ , $∣ {\tilde{u}}_{q} ∣ \leq C_{δ, μ} r_{q}^{- μ}$ on $r_{q} \geq r_{*}$ , with $C_{δ, μ}$ independent of $q$ .

Facts & Assumptions

[F1]

On a common end the graph equation is $L_{q} {\tilde{u}}_{q} = f_{q}$ , with $a_{q}^{i j} = δ^{i j} + O (r_{q}^{- 1})$ , $b_{q}^{j} = O (r_{q}^{- 2})$ and $∣ f_{q} ∣ \leq C r_{q}^{- 3}$ uniformly in $q$ . The principal coefficient is uniformly positive definite on a sufficiently distant common end. Asymptotic maximal graph equation and coefficients

[F2]

On a common end the normalized slope satisfies $∣ \partial_{x_{q}} {\tilde{u}}_{q} ∣ \leq C_{δ} r_{q}^{- 1 / 2}$ uniformly in $q$ . Uniform asymptotic slope bound

[F3]

The normalized end uses $r_{q} = ∣ x_{q} ∣$ , and the graph-height normalization requires convergence to zero uniformly in angle for each fixed parameter. Stationary maximal graphs and normalized graph heights

Proof

Given: The family and hypotheses in the Statement.

Goal: Establish the radial barrier, comparison, absorption inequality and uniform height decay.

1.1F1F3algebra

In three Cartesian spatial dimensions, direct differentiation gives $Δ (r^{- μ}) = - μ (1 - μ) r^{- μ - 2}$ , while $∣ \partial_{i} r^{- μ} ∣ \leq C_{μ} r^{- μ - 1}$ and $∣ \partial_{i j} r^{- μ} ∣ \leq C_{μ} r^{- μ - 2}$ . The coefficient errors in [F1] therefore contribute at most $C_{δ, μ} r^{- μ - 3}$ . Since $μ (1 - μ) > 0$ , increasing the common radius gives the claimed strictly negative bound, uniformly in $q$ .

2.1F1F3step 1.1algebra

Fix $r_{*}$ beyond that radius and choose $K_{δ, μ} > C / c_{δ, μ}$ , with $C$ from [F1]. Set $B = (M_{q} (r_{*}) + K_{δ, μ} / r_{*}) (r_{*} / r)^{μ}$ . Then $L_{q} (\pm {\tilde{u}}_{q} - B) > 0$ for $r \geq r_{*}$ : indeed $- L_{q} B$ is strictly greater than $C r^{- 3}$ because $r \geq r_{*}$ and $μ < 1$ . At the inner sphere $\pm {\tilde{u}}_{q} - B \leq 0$ . For each fixed $q$ , [F3] and $B \to 0$ imply that on a sufficiently distant outer sphere this difference is at most any prescribed $ε > 0$ . On the closed intervening annulus it cannot attain a positive interior maximum exceeding $ε$ : at such a maximum the gradient vanishes and the Hessian is negative semidefinite, so positive definiteness of $a_{q}$ gives $L_{q} (\pm {\tilde{u}}_{q} - B) \leq 0$ , a contradiction. Letting the outer radius tend to infinity and then $ε ↓ 0$ proves the comparison bound.

3.1F2step 2.1algebra

Along each radial segment from radius $r_{*}$ to $4 r_{*}$ , [F2] bounds the height change by $C_{δ} \int_{r_{*}}^{4 r_{*}} s^{- 1 / 2} d s = 2 C_{δ} \sqrt{r_{*}}$ . Taking the maximum over directions and applying step 2.1 at radius $4 r_{*}$ yields the stated absorption inequality.

4.1step 2.1step 3.1algebra∎

Since $1 - 4^{- μ} > 0$ , the inequality in step 3.1 gives a bound on $M_{q} (r_{*})$ independent of $q$ once the common $r_{*}$ is fixed. Substitution into step 2.1 gives the asserted $C_{δ, μ} r^{- μ}$ bound.

Source notes

The article reference attributes the original estimates. The proof does not use that reference as a premise; the comparison argument is given on bounded annuli and uses each individual graph's end limit, not a uniform rate of convergence in $q$ .

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Used by · 2

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Sources

The uniqueness of smooth stationary black holes · PDF pages 14–15, §3.2.1, equations (3.5)–(3.7)