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Stationary maximal graphs have an explicit uniformly spacelike flux

Statement

Let $g$ be a smooth time-oriented Lorentzian metric of signature $(-, +, +, +)$ in a stationary chart $(t, x)$ , so its coefficients are independent of $t$ . For a smooth graph $t = u (x)$ , put $Q (x, p) = - g^{00} + 2 g^{0 i} p_{i} - g^{i j} p_{i} p_{j}$ , with spatial indices summed from $1$ to $3$ , and assume $Q (x, D u) > 0$ . Then $n_{0} = - g^{- 1} d (t - u) / \sqrt{Q (x, D u)}$ is a unit timelike normal. For any future timelike vector field $T$ along the graph, $n_{0}$ is future-directed exactly when $d (t - u) (T) > 0$ . Under this sign condition, the future mean curvature is $H = J^{- 1} \partial_{x^{i}} P^{i} (x, D u)$ , where $J = \sqrt{- \det g}$ and $P^{i} (x, p) = \frac{J (g^{i j} p_{j} - g^{i 0})}{\sqrt{Q (x, p)}} .$ The derivative in this formula is the total derivative of the composed flux. With the opposite time orientation the future normal and $H$ both change sign, so maximality is still equivalent to $\partial_{x^{i}} P^{i} (x, D u) = 0$ . In particular the displayed $n_{0}$ is future-directed whenever $\partial_{t}$ is future timelike.

Fix $0 < α < 1$ . Suppose the chart is a Cartesian end with $r = ∣ x ∣$ and $g_{00} + 1, g_{i j} - δ_{i j} = O_{5} (r^{- 1})$ , $g_{0 i} = O_{5} (r^{- 2})$ . On the fixed annulus $A = {y \in R^{3} : 1 < ∣ y ∣ < 2}$ , let $G_{R} (y) = g (R y)$ denote the array of metric coefficients evaluated at $R y$ , set $J_{R} (y) = J (R y)$ , and define $Q_{R}, P_{R}$ by the same coefficient formulas using $G_{R}$ . There are $R_{*}$ and $p_{0} > 0$ such that, for $R \geq R_{*}$ and $∣ p ∣ \leq p_{0}$ , $Q_{R} \geq κ > 0$ , $(G_{R}^{i j}) \geq κ I$ , and $G_{R}, G_{R}^{- 1}, J_{R}, J_{R}^{- 1}$ have common bounds. The matrices $\partial_{p} P_{R}$ are symmetric and satisfy $λ I \leq \partial_{p} P_{R} \leq Λ I$ , and $P_{R}$ has a common joint $C^{4, α}$ bound on $A \times {∣ p ∣ \leq p_{0}}$ . For every fixed finite number of additional slope derivatives, those derivatives of $P_{R}$ have common joint $C^{4, α}$ bounds as well. With derivatives in $y$ taken at fixed $p$ , the zero-slope divergence $s_{R} (y, 0) = \sum_{i} \partial_{y_{i}} P_{R}^{i} (y, 0)$ obeys $∥ s_{R} (\cdot, 0) ∥_{C^{3, α} (A)} \leq C R^{- 2} .$ All constants are independent of $R$ ; the last coefficient constants may depend on the number of slope derivatives. These are explicit flux bounds. For $w (y) = u (R y) / R$ , $D_{y} w = D u (R y)$ and the maximal equation rescales to ${div}_{y} P_{R} (y, D_{y} w) = 0$ . Using these flux bounds in a subsequent solution regularity estimate additionally requires a smooth solution with $∣ D w ∣ \leq p_{0}$ and any uniform height bound required by that estimate, such as $∥ w ∥_{\infty} \leq M_{0}$ ; metric decay alone supplies no solution or height bound.

Statement references

Normal graphs and the nonlinear mean-curvature map

Ellipticity and zero-slope forcing of the stationary maximal flux

One extra pointwise derivative supplies scaled Hölder control

Differentiated asymptotic bounds and parameter uniformity

Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

The derivative of the metric density and the chart formula for the divergence of a vector field

Matrix inversion preserves Cᵏ regularity where the determinant is nonzero

The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

Sum, product, quotient, and chain rules from difference quotients

Facts & Assumptions

[F1]

Mean curvature is the pullback of the divergence of the future unit normal, with the stated Lorentzian and graph conventions. Normal graphs and the nonlinear mean-curvature map

[F2]

For the explicit stationary flux, uniform positive bounds on $Q_{R}$ and $(G_{R}^{i j})$ , together with bounded metric, inverse and density, give symmetric uniform ellipticity. A common background $C^{4, α}$ bound gives a common joint $C^{4, α}$ flux bound. Its separate zero-slope conclusion is only $C^{2, α}$ ; the stronger estimate here is proved below from the additional differentiated decay. Ellipticity and zero-slope forcing of the stationary maximal flux

[F3]

A differentiated bound $f = O_{m + 1} (r^{- b})$ gives uniform $C^{m, α}$ bounds for $R^{b} f (R \cdot)$ on a fixed annulus, also for finite component arrays. One extra pointwise derivative supplies scaled Hölder control

[F4]

The notation $O_{5} (r^{- b})$ uses Cartesian derivatives through order five and a common eventual exterior radius. Differentiated asymptotic bounds and parameter uniformity

[F5]

Use ordinary componentwise Euclidean Hölder norms; joint flux norms differentiate in both $y$ and $p$ , with $R$ a parameter. Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

[F6]

In these Lorentzian coordinates the metric density is $J = \sqrt{- \det g}$ and ${div}_{g} V = J^{- 1} \partial_{μ} (J V^{μ})$ for smooth vector fields. This supplier has domain mathematics. The derivative of the metric density and the chart formula for the divergence of a vector field

[F7]

Matrix inversion preserves finite differentiability on the invertible locus; its proof uses the cofactor-over-determinant formula. Quantitative bounds below are obtained locally by differentiating the inverse identity. Matrix inversion preserves Cᵏ regularity where the determinant is nonzero

[F8]

The total derivative of a composite is the composite of its derivatives. Scalar sums, products and quotients with nonzero denominator have their usual derivative rules, iterated on coordinate lines. The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a) Sum, product, quotient, and chain rules from difference quotients

Proof

Given: The hypotheses in the Statement, using the decay assumptions only for the end estimates.

Goal: Prove the oriented normal formula, the maximal flux equation and all stated uniform rescaled flux bounds.

1.1givenF1algebra

The covector $a = d (t - u)$ annihilates the graph tangents $\partial_{x^{i}} + u_{i} \partial_{t}$ and satisfies $g^{- 1} (a, a) = - Q (x, D u) < 0$ . Thus its kernel is spacelike and $n_{0} = - g^{- 1} a / \sqrt{Q}$ is a unit timelike normal. The equality $g (n_{0}, T) = - a (T) / \sqrt{Q}$ gives precisely the claimed future sign condition. If $T = \partial_{t}$ is future timelike, then $a (T) = 1$ . Reversing the normal reverses its divergence and hence the signed mean curvature.

1.2givenF4F7F8algebra

The decay hypotheses put $g$ uniformly close to $η = diag (- 1, 1, 1, 1)$ sufficiently far out, with determinant bounded away from zero and $g_{00}$ bounded away from zero. The cofactor formula bounds $g^{- 1}$ there. Differentiating $g^{- 1} g = I$ gives $\partial_{k} g^{- 1} = - g^{- 1} (\partial_{k} g) g^{- 1}$ ; iteration with the product rule through order five gives $g^{- 1} - η = O_{5} (r^{- 1})$ . The mixed entry satisfies the exact identity $g^{i 0} = - g^{i j} g_{j 0} / g_{00}$ , obtained from $g^{i μ} g_{μ 0} = 0$ . Each differentiated product therefore has at least one mixed covariant factor of order $r^{- 2}$ , and the other factors and their derivatives have the inverse bounds just established. Consequently $g^{i 0} = O_{5} (r^{- 2})$ . The determinant is a finite sum of products of metric entries; the identities $\sqrt{v + h} - \sqrt{v} = h / (\sqrt{v + h} + \sqrt{v})$ and $(\sqrt{v})^{'} = 1 / (2 \sqrt{v})$ for $v > 0$ , followed by repeated product and quotient differentiation, similarly show $J - 1 = O_{5} (r^{- 1})$ and $h - 1 = O_{5} (r^{- 1})$ for $h = J / \sqrt{- g^{00}}$ . All denominators are bounded away from zero on this end.

2.1F1F6step 1.1algebra

Extend $u$ independently of $t$ and use the same formula for $n_{0}$ where $Q > 0$ . This is a smooth unit extension, independent of $t$ , with $n_{0}^{i} = (g^{i j} u_{j} - g^{i 0}) / \sqrt{Q}$ . Stationarity and the density formula give ${div}_{g} n_{0} = J^{- 1} \partial_{x^{i}} (J n_{0}^{i}) = J^{- 1} \partial_{x^{i}} P^{i} (x, D u)$ . Under the future sign condition this is $H$ by F1; otherwise it is $- H$ . Both cases give the same zero-mean-curvature equation.

2.2givenF2F3F5F8step 1.2algebra

Apply F3 with $m = 4$ to the metric errors and to the inverse, $J$ and $h$ estimates of step 1.2. It gives common $C^{4, α} (A)$ bounds on $G_{R}$ , its inverse and $h_{R} (y) = h (R y)$ , with $∥ d_{R} ∥_{C^{4, α} (A)} \leq C R^{- 2}$ for $d_{R}^{i} (y) = g^{i 0} (R y)$ . The same applications for $0 \leq m < 4$ control the lower-derivative Hölder seminorms as needed in products. For example choose $p_{0} = 1 / 4$ . Uniform closeness to $η$ implies a common positive lower bound for $Q_{R}$ and $(G_{R}^{i j})$ on this slope ball, and common bounds for $G_{R}, G_{R}^{- 1}, J_{R}, J_{R}^{- 1}$ . F2 then gives the claimed ellipticity and joint $C^{4, α}$ bound for $P_{R}$ . To retain any fixed number of extra slope derivatives, differentiate the displayed flux formula in $p$ : every term is a finite product of inverse entries, density, slope monomials and negative half-integer powers of $Q_{R}$ . Its derivatives through four further orders in $(y, p)$ involve background derivatives only through order four. The common positive denominator bounds control all smooth scalar factors and their derivatives. For their Hölder estimates use $[f g]_{α} \leq ∥ f ∥_{\infty} [g]_{α} + ∥ g ∥_{\infty} [f]_{α}$ and, the identities $a^{- 1} - b^{- 1} = (b - a) / (a b)$ and $\sqrt{a} - \sqrt{b} = (a - b) / (\sqrt{a} + \sqrt{b})$ , iterated over finite products, bound increments of all the scalar powers when their arguments stay in a fixed positive bounded interval. Combining these identities with the componentwise bounds proves the asserted extra-slope-derivative bounds, with constants allowed to depend on their finite number.

3.1F3F5F8step 2.1step 2.2algebra∎

At zero slope $P_{R}^{i} (y, 0) = - h_{R} (y) d_{R}^{i} (y)$ . The product rule and Hölder product inequality of step 2.2 give $∥ P_{R} (\cdot, 0) ∥_{C^{4, α} (A)} \leq C ∥ h_{R} ∥_{C^{4, α} (A)} ∥ d_{R} ∥_{C^{4, α} (A)} \leq C R^{- 2}$ , using the lower-derivative bounds noted there. One $y$ derivative therefore yields the asserted $C^{3, α}$ zero-slope estimate, rather than only the weaker zero-slope conclusion of F2. Finally $D_{y} w = D u (R y)$ and ${div}_{y} P_{R} (y, D w) = R (\partial_{x^{i}} P^{i} (x, D u)) ∣_{x = R y}$ , so step 2.1 proves the rescaled maximal equation. The coefficient bounds were obtained without a solution estimate. Smoothness and a slope bound must hold for a graph to which one applies such an estimate, and a common bound for $u (R y) / R$ must be supplied whenever the estimate requires a height bound.

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