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
- Weighted exterior elliptic domains and admissible weights
- Differentiated asymptotic bounds and parameter uniformity
- Compact-domain scalar Poisson solvability and fourth-order estimates
- A positive global supersolution with the exact exterior decay
- Interior Schauder estimate for uniformly elliptic equations
- Boundary Schauder estimate for the Dirichlet problem
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- The closure Hölder spaces are Banach spaces
- Causal exterior, chronological boundaries and Cauchy data
- Null expansions, screen quotients and focal data
- A compact chronological target supplies a causal endpoint
- The Axiom of Choice
- Future null geodesic completeness and incompleteness
- Metric compatibility, torsion freedom and the metric-selected connection class
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Section 4.5.4, Theorem 4.123 (Avez–Seifert) and its proof, printed/PDF p. 148 of 202; section 2.1, geodesic convention and equations (2.1)–(2.3), printed/PDF pp. 25–26.
- The inverse metric, musical isomorphisms and index raising
- Compact sets admit relatively compact open neighborhoods and finite chart pieces
- A temporal covector has uniform compact negativity and orientation margins
- Inversion is uniformly stable on compact chart families
- Smooth manifolds and their smooth charts
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- ADM section time and causal spatial speed
- Fundamental theorem of calculus for absolutely continuous functions
- Gronwall's integral inequality with variable and constant coefficients
- Interior and boundary of a manifold with boundary
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- AC implies DC implies countable choice
- Smooth charts, atlases, and structures with boundary
- Smooth maps between manifolds with boundary
- Riemannian metric and riemannian manifold
- 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
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A product of finitely many compact spaces is compact in the product topology
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- The riemannian distance topology is the manifold topology
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- Local comparison of a riemannian metric with the euclidean metric
- Temporal normal angle and compact positive-height portions
- Gradient hessian and divergence connection formulas
- Laplace–Beltrami operator as the trace of the Hessian
- Sum, product, quotient, and chain rules from difference quotients
- Addition formulas, identities, parity, and derivatives of the hyperbolic functions
- Logarithm formulas for inverse sinh, inverse cosh, and inverse tanh on their natural domains
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- Continuity and derivatives of positive-base real powers
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
Scalar exterior inverse from exhaustion and uniform annular estimates
Lemma
A compact-target boundary point has a null geodesic segment
Lemma
A compact temporal covector persists under metric perturbation
Lemma
Collar barrier against finite section-time boundary arrival
Lemma
Proper end radius and exclusion of finite-time spatial infinity
Lemma
An interior compact causal trajectory has a finite-time endpoint
Lemma
Temporal angle and tangential height gradient identity
Lemma