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Stationary graph translation preserves source asymptotics

Statement

Let $k$ be a nonnegative integer, $R_{0} > 0$ , $E = {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ , and $u \in C^{k + 1} (E; R)$ . In a stationary end chart $V \subseteq R \times E$ , suppose $Γ_{u} = {(u (x), x) : x \in E} \subseteq V$ is a hypersurface graph. Write $D \subseteq R \times M$ for the domain of the stationary flow $ϕ$ , with coordinate law $ϕ_{a} (t, x) = (t + a, x)$ whenever the flow is defined and both points lie in $V$ . Fix $s \in R$ such that $(s, p) \in D$ and $ϕ_{s} (p) \in V$ for every $p \in Γ_{u}$ . Then $ϕ_{s} (Γ_{u})$ is the graph $t = u (x) + s$ over the same $E$ .

If $0 < α \leq 1$ and $∣ u ∣ + \sum_{j = 1}^{k + 1} r^{j} ∣ \partial^{j} u ∣ \leq C w_{α} (r)$ on $E$ , where $C \geq 0$ is finite, $w_{α} (r) = (1 + r)^{1 - α}$ for $α < 1$ and $w_{1} (r) = 1 + \log (1 + r)$ , then the fixed translate obeys the same bound with $C + ∣ s ∣$ in place of $C$ . Here $\partial^{j} u$ denotes the finite array of order- $j$ Cartesian derivatives, with its Euclidean norm. For $1 \leq j \leq k + 1$ and $0 < λ \leq 1$ , any additionally imposed finite weighted Hölder bound on $\partial^{j} u$ is unchanged: on every annulus $A_{R} = {R < ∣ x ∣ < 2 R} \subseteq E$ , the seminorm $[v]_{λ; A_{R}} = \sup_{x \neq y \in A_{R}} ∣ v (x) - v (y) ∣ / ∣ x - y ∣^{λ}$ is identical for $v = \partial^{j} (u + s)$ and $v = \partial^{j} u$ . Finiteness of the relevant Hölder seminorms is an additional hypothesis, not a consequence of the stated $C^{k + 1}$ regularity. These source-type bounds do not force $u \to 0$ or even convergence to a finite constant.

Facts & Assumptions

[F1]

Chart derivative bounds and weighted Hölder estimates have their fixed-coordinate meaning. The formula in the Statement also specifies the endpoint exponent $λ = 1$ . Differentiated asymptotic bounds and parameter uniformity

[F2]

A local flow has a domain $D$ ; the translation law and the conditions that the chosen $s$ is defined on all of $Γ_{u}$ and its image remains in $V$ are explicit hypotheses here, not consequences of the flow definition. Local and global flows generated by a vector field

[F3]

A hypersurface is an embedded submanifold of codimension one; $Γ_{u}$ is assumed to be such a hypersurface in the chart. Codimension and hypersurfaces

[F4]

If $f : [a, b] \to R^{m}$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , $a < b$ , and $∣ f^{'} (t) ∣ \leq M$ there for a finite $M \geq 0$ , then $∣ f (b) - f (a) ∣ \leq M (b - a)$ . The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)

[F5]

For every $ε > 0$ , $\log t / t^{ε} \to 0$ as $t \to + \infty$ . The logarithm grows more slowly than every positive real power

Proof

Given: The chart, graph, flow-domain conditions, regularity and estimates in the Statement.

Goal: Establish graph translation, preservation of the estimates and the absence of forced finite limiting height.

1.1F1F2

For every $x \in E$ , the domain and containment hypotheses permit use of the coordinate law at $(u (x), x)$ , giving $ϕ_{s} (u (x), x) = (u (x) + s, x)$ . As $x$ ranges over $E$ , these are exactly the points of the translated graph. All derivatives of positive order through $k + 1$ remain the same, and $∣ u + s ∣ \leq ∣ u ∣ + ∣ s ∣$ . Since $w_{α} \geq 1$ , replacing $C$ by $C + ∣ s ∣$ proves the claimed bound, while derivative Hölder differences, their seminorms and any fixed radius weights are identical.

2.1F1F4F5step 1.1∎

On $E = {r > 1}$ , the smooth function $u (x) = \log r$ is unbounded, but $r^{j} ∣ \partial^{j} u ∣$ is bounded for every fixed $j \geq 1$ : the first derivatives are homogeneous of degree $- 1$ , further differentiation lowers that degree, and their angular factors are smooth on the unit sphere. Also $∣ u ∣ \leq w_{1} (r) \leq C_{α} w_{α} (r)$ for $α < 1$ : apply [F5] with $ε = 1 - α > 0$ and $t = 1 + r$ . Choose $T \geq 2$ so that $\log t \leq t^{ε}$ for $t \geq T$ . There $1 + \log t \leq 2 t^{ε}$ ; for $2 < t \leq T$ , $1 + \log t \leq 1 + \log T \leq (1 + \log T) t^{ε}$ . Thus $C_{α} = \max {2, 1 + \log T}$ suffices. For completeness, if $v = \partial^{j} u$ and $x, y \in A_{R}$ with $∣ x - y ∣ \leq R / 2$ , the connecting segment stays at radius at least $R / 2$ . For distinct $x, y$ , the finite derivative array $f (t) = v (x + t (y - x))$ , $0 \leq t \leq 1$ , is smooth and has $∣ f^{'} (t) ∣ \leq C_{j} R^{- j - 1} ∣ x - y ∣$ by the next-derivative bound. Applying [F4] on $[0, 1]$ gives $∣ v (x) - v (y) ∣ \leq C_{j} R^{- j - 1} ∣ x - y ∣$ ; for $x = y$ the inequality is immediate. If $∣ x - y ∣ > R / 2$ , the individual derivative bounds give $∣ v (x) - v (y) ∣ \leq C_{j} R^{- j}$ . Consequently $[\partial^{j} u]_{λ; A_{R}} \leq C_{j, λ} R^{- j - λ}$ for $0 < λ \leq 1$ and $R \geq 2$ . This supplies the corresponding weighted Hölder control, including that at the highest derivative order. Thus the displayed growth class does not imply zero height or a finite limiting height; this is a statement about the estimate, not a spacetime counterexample.

Depends on

Used by · 1

Stationary maximal hypersurface existence with a horizon boundary

Dependency tree · 2 levels

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