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Complete boundary initial-data and vanishing momentum interface

Statement

Under the regular stationary-vacuum hypotheses, the section $\overline{C} = C \cup S$ is a complete smooth three-dimensional initial-data manifold with nonempty compact boundary and one Euclidean end. It satisfies $h_{i j} = δ_{i j} + O_{2} (r^{- 1})$ , $k = O_{1} (r^{- 3})$ , $μ = J = 0$ , and ${tr}_{h} k = O (r^{- 3})$ . These imply local $C^{2, α} \times C^{1, α}$ regularity for any $0 < α < 1$ , the usual asymptotic-flat initial-data bounds with any $1 / 2 < q \leq 1$ , and vanishing ADM linear momentum. These are initial-data properties independent of maximality.

Use the induced tensors $h = g ∣_{T \overline{C}}$ and $k (U, V) = - g (\nabla_{U} n, V)$ with the future unit normal $n$ . Local Hölder regularity is componentwise on relatively compact smaller coordinate balls in the interior and on smaller boundary half-balls, with the derivatives extending continuously to the flat boundary; it asserts no uniform Hölder norm over the entire noncompact manifold.

Facts & Assumptions

[F1]

Completeness includes the smooth boundary and there is exactly one asymptotically Euclidean end. The exterior Cauchy component has a complete asymptotically Euclidean closure

[F2]

The vacuum constraint densities vanish and are norm integrable, including with the supplied boundary convention. Vacuum exterior-section constraints imply the dominant energy condition

[F3]

The induced end metric and extrinsic curvature obey the displayed differentiated estimates. Extrinsic curvature decay on the original stationary end Differentiated asymptotic bounds and parameter uniformity

[F4]

The regular stationary-vacuum hypotheses specify the smooth ambient metric, the spacelike section $Σ_{0}$ , its end chart and $C = Σ_{0} \cap E$ . Regular non-degenerate stationary vacuum spacetime

[F5]

The induced metric $h$ , the future unit normal $n$ , and $k (U, V) = - g (\nabla_{U} n, V)$ are smooth on the original spacelike section; the boundary convention includes the smooth induced metric at $S$ . Spacelike sections, maximality and lapse–shift data

[F6]

The local $C^{m, α}$ classes use Hölder seminorms of the order- $m$ derivatives on relatively compact sets; the boundary-extension class also requires continuous extension of derivatives to the boundary. Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains

[F7]

The ADM momentum uses the outward $h$ -unit sphere normal and induced area measure; changing the overall second-fundamental-form sign reverses momentum but preserves its vanishing. Asymptotic mass and axial angular-momentum conventions

[F8]

On a convex open subset of Euclidean space, a totally differentiable map whose total derivative has operator norm at most $L$ satisfies $∣ f (x) - f (y) ∣ \leq L ∣ x - y ∣$ . On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂

Proof

Given: The regular stationary-vacuum hypotheses and the supplied interfaces [F1]–[F8].

Goal: Establish the initial-data, local regularity, decay and zero-momentum conclusions without maximality.

1.1F1F2F3F4F5F6F8algebra

The geometric, completeness and boundary assertions follow from [F1] under the hypotheses of [F4]. By [F5], $h$ and $k$ are restrictions of smooth tensors on $Σ_{0}$ , also across $S$ . In a smaller ball or flattened boundary half-ball whose closure lies in a smooth chart, component derivatives through order three for $h$ and order two for $k$ are bounded. Apply [F8] to each second component derivative of $h$ and each first component derivative of $k$ on the open ball, or on the convex open interior of the half-ball. The bounded next derivatives give a finite common operator-norm bound $L$ , so the differences are at most $L ∣ x - y ∣$ . On a boundary half-ball, continuity of these derivatives extends the same inequality to pairs on the flat boundary by taking interior limits. For $0 < α < 1$ this is at most $L D^{1 - α} ∣ x - y ∣^{α}$ , where $D$ is the finite diameter. Smoothness gives the continuous boundary extensions, proving the local classes of [F6]. The end estimates of [F3] imply $h - δ = O_{2} (r^{- q})$ and $k = O_{1} (r^{- 1 - q})$ for every $1 / 2 < q \leq 1$ , and contraction with $h^{- 1} = δ + O (r^{- 1})$ gives ${tr}_{h} k = O (r^{- 3})$ . The zero densities are integrable by [F2].

2.1F3F7algebra∎

In the ADM momentum flux $P_{i} = (8 π)^{- 1} \lim_{r \to \infty} \int_{S_{r}} (k_{i j} - ({tr}_{h} k) h_{i j}) ν_{h}^{j} d A_{h}$ from [F7], the tensor factor is $O (r^{- 3})$ by [F3]. The metric bound $h = δ + O (r^{- 1})$ makes the Cartesian components of $ν_{h}$ bounded and the sphere area $O (r^{2})$ . Thus the absolute value of each flux is $O (r^{- 1})$ and tends to zero. The convention in [F7] uses $K = - k$ , which reverses that flux and still yields zero.

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