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Strict mean-curvature barriers cannot have ordered boundary tangency

Statement

Work under the inherited ZFC convention. Let $(M, γ, N)$ be a smooth time-oriented Lorentzian four-manifold without boundary, of signature $(-, +, +, +)$ , with its Levi–Civita connection $\nabla$ . Let $Σ_{1}, Σ_{2}$ be smooth embedded spacelike hypersurfaces meeting at $q$ , with the same tangent hyperplane there. In the boundary case assume both are smooth embedded hypersurfaces up to a common smooth boundary surface near $q$ , with positive definite induced metric on their full tangent spaces at the boundary, and that their projected interiors select the same side of the common projected boundary in a common scalar-graph chart. Use future unit normals $n_{i}$ and $H_{i} = {tr}_{h_{i}} γ (\nabla n_{i}, \cdot) = {div}_{γ} n_{i}$ , with divergence computed using a local unit timelike extension; maximal means $H_{i} = 0$ .

At an interior contact, or at a boundary contact under these common-boundary conditions, suppose $Σ_{1}$ lies to the future of $Σ_{2}$ on their common local neighborhood. Precisely, in a common scalar-graph chart with future timelike time direction, the first graph height is at least the second on the common spatial neighborhood, a closed half-neighborhood at the boundary. Then $H_{1} (q) \geq H_{2} (q)$ . Consequently a strictly negative-mean-curvature spacelike hypersurface cannot touch a maximal one from the future; a strictly positive-mean-curvature hypersurface cannot touch it from the past.

For the $K$ -flow application, let $U \subseteq M$ be open, let $K$ be a smooth vector field on $U$ , and let $C \subseteq U$ be a smooth embedded three-dimensional collar with boundary the smooth surface $S$ , with $K ∣_{S} = 0$ . Let $ϕ : D \to U$ be the maximal flow of $K$ , with open domain $D \subseteq R \times U$ . Let $f, g : C \to R$ be smooth up to $S$ , with $(h (x), x) \in D$ for every $x \in C$ and $h \in {f, g}$ . Define $F_{h} (x) = ϕ (h (x), x)$ directly, and assume both maps are smooth embeddings up to $S$ with spacelike images on their full tangent spaces, including the boundary. Assume $K$ is future timelike on the exterior collar saturation ${ϕ (t, x) : x \in C ∖ S, (t, x) \in D}$ . These variable-time maps specify the boundary-compatible flow graphs; no transversality of $K$ at $S$ is imposed.

If $f \geq g$ on this collar and $f (p) = g (p)$ at $p \in S$ , both graph images pass through $p$ , have the same tangent hyperplane and inward tangent half-space, and the $f$ -graph lies to the future of the $g$ -graph on a common local half-neighborhood. Thus the comparison gives $H (F_{f} (C)) (p) \geq H (F_{g} (C)) (p)$ , and both strict-barrier exclusions apply. Equal boundary heights supply tangency without equality of $d f_{p}$ and $d g_{p}$ . The comparison uses only finite flow intervals near the common time at each contact; it requires no estimate uniform over unbounded times. For $f \leq g$ , interchange the two graphs.

Statement references

Second-order contact on a closed half-domain has nonnegative Hessian

Mean curvature at identical spacelike graph jets has positive Hessian symbol

Equal fixed-surface flow heights imply boundary tangency

Ordered timelike flow graphs lie on the corresponding geometric side

Spacelike sections, maximality and lapse–shift data

Normal graphs and the nonlinear mean-curvature map

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

The Axiom of Choice

Metric compatibility, torsion freedom and the metric-selected connection class

Local and global flows generated by a vector field

The fundamental theorem on flows

Facts & Assumptions

[F1]

A nonnegative $C^{2}$ graph difference on a closed smooth one-sided neighborhood, with zero value and first derivative at contact and a local $C^{2}$ extension, has positive semidefinite Hessian. The same holds at an interior minimum. Second-order contact on a closed half-domain has nonnegative Hessian

[F2]

Tangent spacelike hypersurfaces have common scalar-graph charts. When their boundary images and projected interior sides agree, their graphs have extensions across a common projected boundary and can be put on one half-neighborhood. At the shared first jet their mean-curvature difference is the contraction of the Hessian difference with a positive definite matrix. In Lorentzian normal coordinates with time vector the common future normal at contact, that matrix is the identity. Mean curvature at identical spacelike graph jets has positive Hessian symbol

[F3]

For the smooth collar, fixed-surface field, open maximal-flow domain and variable-time embeddings specified in the Statement, both maps fix $S$ . Equal boundary heights give equal differentials on the full collar tangent space and the same inward image half-space, without equality of height derivatives. Equal fixed-surface flow heights imply boundary tangency

[F4]

For these boundary-compatible spacelike embeddings, future timelikeness of $K$ on the exterior collar saturation makes ordered flow heights give future-side scalar-graph ordering near equal-height contact, including a common half-neighborhood at $S$ . Only finite intervals near the common time are used. Ordered timelike flow graphs lie on the corresponding geometric side

[F5]

Spacelikeness, the future unit normal, the induced metric and maximality have the stated hypersurface meanings, with $H = {div}_{γ} n$ for a unit timelike extension. Spacelike sections, maximality and lapse–shift data

[F6]

The graph mean-curvature convention is the trace of $(V, W) \mapsto γ (\nabla_{V} n, W)$ , equivalently the unit-extension divergence. The $K$ -maps in the Statement are specified directly as in [F3]; they do not require $K$ transverse to $C$ at $S$ . Normal graphs and the nonlinear mean-curvature map Equal fixed-surface flow heights imply boundary tangency

[F7]

The metric signature, time orientation and future timelike directions are the mathematical Lorentzian conventions. Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

[F8]

Every finite-dimensional real or complex inner product space has an orthonormal basis. This applies to the real three-dimensional tangent space with its positive definite induced metric $h$ at contact. Every finite-dimensional real or complex inner product space has an orthonormal basis

[A1]

The geometric suppliers retain their inherited Axiom of Choice convention; this local composition adds no choice assumption beyond ZFC. The Axiom of Choice

Proof

Given: The ambient, contact and ordering hypotheses in the Statement; for the $K$ -flow conclusion, its explicit collar, flow-domain, embedding and exterior-timelikeness hypotheses.

Goal: Prove the oriented mean-curvature comparison, its strict-barrier consequences, and its boundary-compatible $K$ -flow application.

1.1givenF2F5F6F7

Write the two hypersurfaces as scalar graphs $t = a (z)$ and $t = b (z)$ in the common future-timelike chart in the ordering hypothesis. In the boundary case their boundary images agree and their projected interiors select the same side by hypothesis; [F2] supplies $C^{2}$ extensions of the graph functions across the common projected boundary. Their common domain is thus a smooth one-sided neighborhood. Tangency gives $(a - b) (q) = 0$ and $D (a - b) (q) = 0$ , while the given future-side ordering gives $a - b \geq 0$ on this common domain.

2.1F1F2F8step 1.1algebra

Apply [F1] to $w = a - b$ , using its extension in the boundary case. Its Hessian $B = D^{2} w (q)$ is positive semidefinite. At the shared first jet, [F2] gives $H_{1} (q) - H_{2} (q) = α h^{i j} B_{i j}$ with $α = - γ (n, \partial_{t}) > 0$ and positive definite induced metric $h$ . In an $h$ -orthonormal basis $e_{1}, e_{2}, e_{3}$ , whose existence follows from [F8] for this real three-dimensional inner product space, this contraction is $α \sum_{i = 1}^{3} B (e_{i}, e_{i}) \geq 0$ . This proves the comparison on the actual one-sided domain without assuming a minimum on its exterior.

3.1F5step 2.1algebra

If the first surface is a future barrier with strictly negative mean curvature and the second is maximal, step 2.1 contradicts $H_{1} < 0 = H_{2}$ . If the barrier lies to the past, take the maximal surface as the first surface: step 2.1 gives $0 \geq H_{b a r r i e r}$ , contradicting its strict positive sign.

4.1givenF3F4step 1.1step 2.1step 3.1∎

In the $K$ -flow application all the hypotheses of [F3] and [F4] are explicitly assumed. For $p \in S$ with equal heights, [F3] gives $F_{f} (p) = F_{g} (p) = p$ , a common tangent hyperplane and inward tangent half-space; both images have boundary $S$ locally. Their projected interiors therefore choose the same side. Since $f \geq g$ and $K$ is future timelike on the exterior saturation, [F4] supplies their future-side ordering on a common scalar-graph half-neighborhood, using only a finite time interval near $f (p) = g (p)$ . Steps 1.1 and 2.1 give the claimed curvature inequality, and step 3.1 gives both barrier exclusions; interchanging $f, g$ gives the reversed-order version.

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