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
- The Riemannian gradient is the metric dual of the differential
- David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2001 reprint of the 1998 edition · Chapter 6, section 6.4: Theorem 6.17, printed p.109; Theorem 6.19, printed p.111, and its local-boundary-portion conclusion on printed p.112.
- Compact Laplace-type operators: smooth kernel, closed range and Green inverse
- Interior Schauder estimate for uniformly elliptic equations
- Boundary Schauder estimate for the Dirichlet problem
- Compact Laplace-type bundle operators: regularity and shifted parametrices
- The Axiom of Choice
- AC implies DC implies countable choice
- Riemannian volume density
- Measurable integration extends smooth density integration
- Smooth partitions of unity exist on manifolds
- Coordinate formula for riemannian divergence
- The integral of the divergence of an integrable C1 field vanishes
- Fredholm and Banach continuation interfaces
- The closure Hölder spaces are Banach spaces
- Bounded inverse theorem
- Gilbarg and Trudinger, Elliptic Partial Differential Equations of Second Order (2001 reprint) · Chapter 6, section 6.4, Theorem 6.17, printed pp.109–110.
- Causal exterior, chronological boundaries and Cauchy data
- Null expansions, screen quotients and focal data
- In a globally hyperbolic manifold the causal relation is closed and truncated causal cones over compact sets are compact
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- 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
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- The inverse metric, musical isomorphisms and index raising
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
- Compact sets admit relatively compact open neighborhoods and finite chart pieces
- Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space
- Timelike, null, spacelike and causal vectors of a Lorentz vector space
- Time orientation, future and past timelike/null/causal vectors, and the closed cones
- The timelike set has exactly two convex components, described in orthonormal coordinates
- The Cauchy-Schwarz inequality for finite sums
- Smooth manifolds and their smooth charts
- Tensor transition laws define a smooth vector bundle
- Coordinate differentials form the dual cotangent basis
- Vector bundle charts and transition functions
- Cʳ and smooth maps between smooth manifolds
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Spacelike sections, maximality and lapse–shift data
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- Fundamental theorem of calculus for absolutely continuous functions
- C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation
- Local comparison of a riemannian metric with the euclidean metric
- Products of smooth manifolds have a canonical product smooth structure
- Pointwise norm and angle from a riemannian metric
- The action of a vector field on smooth functions
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Riemannian metric and riemannian manifold
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
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
A positive global supersolution with the exact exterior decay
Lemma
Compact extra components contribute index zero
Lemma
A compact chronological target supplies a causal endpoint
Lemma
A temporal covector has uniform compact negativity and orientation margins
Lemma