Alphabeta MathMaximal slices, causal cuts and stability
Assemble the analytic and geometric inputs for maximal-slice continuation, causal completeness, null-boundary cuts, area comparison and horizon flux rigidity. Use the common data and the preceding field inputs while retaining every local argument and imported-premise qualification.
1 · Prerequisites
- Finite smooth cutoffs relative to compact sets and boundary neighborhoods
- Smooth approximation of compactly supported interior heights
- Lipschitz height functions, cut measure and Jacobians
- Weak first derivatives and Riemannian chart norm comparison
- Smooth charts, atlases, and structures with boundary
- Riemannian metric and riemannian manifold
- Riemannian distance on a connected manifold
- Extended riemannian distance on a disconnected manifold
- Riemannian speed and length
- Riemannian metrics induce metrics on dual tensor and exterior bundles
- A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism
- Chain rule for smooth half-space maps
- The second fundamental theorem: if G is differentiable on [a,b] with G' = f and f is integrable, then ∫ₐᵇ f = G(b)-G(a)
- The Axiom of Choice
- Weighted exterior elliptic domains and admissible weights
- Exterior exhaustion bounds for normalized Schrödinger solutions
- Hopf positivity and uniform linear growth on a fixed compact collar
- Multiplicative interior Harnack for geometric Schrödinger solutions
- Finite Harnack chains on fixed compact connected interior sets
- Positive anchor values and the limits of uniform Harnack propagation
- Spacelike sections, maximality and lapse–shift data
- Causal exterior, chronological boundaries and Cauchy data
- Real powers for positive bases, with the zero-base positive-exponent convention
- Temporal normal angle and compact positive-height portions
- Temporal angle and tangential height gradient identity
- From hyperbolic angle to logarithmic angle inequalities
- Compact weighted maximum and the actual boundary term
- Differential exclusion of a large interior weighted angle maximum
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1 and (B.1), PDF/printed pp.35–36; proof (B.8)–(B.12), pp.38–39, especially (B.11)
- AC implies DC implies countable choice
- ADM section time and causal spatial speed
- Combining the collar and end barriers into interior compact confinement
- An interior compact causal trajectory has a finite-time endpoint
- The existence of a Cauchy set implies strong causality and compact causal diamonds
- Proper end radius and exclusion of finite-time spatial infinity
- Null expansions, screen quotients and focal data
- The entire target segment stays on the chronological boundary
- Endpoint orthogonality without an acausal target assumption
- Null focal points exclude later horismos generators
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- A compact chronological target supplies a causal endpoint
- A compact-target boundary point has a null geodesic segment
- Local null normal pairs and the positive screen quotient
- The Jacobi equation, geodesic deviation and the initial-value theory of Jacobi fields
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Metric compatibility, torsion freedom and the metric-selected connection class
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Every tangent vector is the velocity of a smooth curve
- Heine-Borel in ℝⁿ: with the Euclidean metric a subset of ℝⁿ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Boundary-constant-preserving smooth approximation
Lemma
Scalar metric Jacobian and weak-gradient identification
Lemma
Exterior Schrödinger maximum principles, Hopf growth and Harnack
Theorem
Temporal cutoff estimate for normals to maximal hypersurfaces
Lemma
Stationary ADM speed bounds and Cauchy sections
Lemma