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
- The Lax--Milgram theorem
- The Rellich--Kondrachov theorem for 1≤ p<n on bounded extension domains
- Higher-order boundary regularity for Dirichlet problems
- Boundary Schauder estimate for the Dirichlet problem
- Interior Schauder estimate for uniformly elliptic equations
- The Axiom of Choice
- Higher-order Sobolev embedding
- Smooth charts, atlases, and structures with boundary
- Riemannian volume density
- Riemannian volume is the radon measure of the riemannian density
- Measurable integration extends smooth density integration
- Smooth partitions of unity exist on manifolds with boundary
- Integer-order Sobolev spaces and their norms
- Integer-order Sobolev spaces are Banach
- Hilbert space
- Bounded Cᵏ domains admit integer-order Sobolev extension
- The Lᵖ trace operator on a bounded C¹ domain
- Zero weak gradient gives componentwise constants
- Nested-domain induction for interior elliptic derivatives
- Weak divergence-form equations are invariant under C² boundary charts
- The cutoff difference-quotient commutator estimate
- Cᵏ boundary flattening preserves local W^k,p
- Armin Schikorra, Partial Differential Equations I & II, version October 1, 2025 · §8.9, Theorem 8.31, printed p.153 (PDF page 153), with admissibility of smooth bounded domains in Theorem 8.30, printed p.152.
- David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2001 reprint of the 1998 edition · Chapter 6, §6.4, Theorems 6.17 and 6.19, printed pp.109–112, including the local-boundary-portion paragraph after Theorem 6.19; §6.2, Lemma 6.4, printed p.95, and Corollary 6.7, printed p.100, equation (6.38).
- Compact linear operator
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- The closure Hölder spaces are Banach spaces
- AC implies DC implies countable choice
- 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
- Smooth manifolds and their smooth charts
- Null expansions, screen quotients and focal data
- Null Raychaudhuri and focal-point bound
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- Finite smooth cutoffs relative to compact sets and boundary neighborhoods
- Weak first derivatives and Riemannian chart norm comparison
- Compactly supported chart mollification in uniform and first Sobolev norms
- Riemannian metric and riemannian manifold
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Riemannian distance on a connected manifold
- Extended riemannian distance on a disconnected manifold
- Riemannian speed and length
- Local comparison of a riemannian metric with the euclidean metric
- Interior and boundary of a manifold with boundary
- Formal adjoint of a real drift Laplacian on a closed 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
- Smooth maps are continuous
- Identity maps and composites of smooth maps are smooth
- Dual connection
- Product connection on tensor and hom bundles
- Orientation-free density integration and its properties
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
Compact-domain scalar Poisson solvability and fourth-order estimates
Lemma
Compact multiplication by a faster-decaying potential
Lemma
Zero null optics supplies the missing null Ricci condition
Lemma
Smooth approximation of compactly supported interior heights
Lemma