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
- Smooth partition of unity on a manifold with boundary
- Topological manifolds with boundary
- Smooth charts, atlases, and structures with boundary
- Smooth functions on relatively open half-space sets
- A smooth bump between concentric Euclidean balls
- 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
- 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
- For a metric space with its metric topology, compactness in the topological sense is compactness in the metric sense, and the two notions of compact subset coincide
- 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
- 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
- Sums, scalar multiples, products and quotients: (f+g)'(c) = f'(c) + g'(c), (α f)'(c) = α f'(c), (fg)'(c) = f'(c)g(c) + f(c)g'(c), and (f/g)'(c) = (f'(c)g(c) - f(c)g'(c))/g(c)² when g(c) ≠ 0
- Chain rule for smooth half-space maps
- Interior and boundary of a manifold with boundary
- The interior is an open smooth n-manifold
- Riemannian metric and riemannian manifold
- Lipschitz graph area and calculus under Countable Choice
- Integer-order Sobolev spaces and their norms
- Weak derivative of a locally integrable function
- Riemannian volume density
- The Axiom of Countable Choice (AC_ω)
- Uniqueness of a weak derivative as an almost-everywhere class
- A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions
- Riemannian metrics induce metrics on dual tensor and exterior bundles
- The essential supremum of a measurable function with respect to a measure
- The mollifier family generated by a unit-mass smooth bump
- Interior mollification commutes with weak derivatives
- ‖τ_h f - f‖ₚ → 0 in Lᵖ(ℝⁿ) as h → 0, for 1 ≤ p < ∞
- Convolution with a mollifier is smooth, and derivatives pass under the integral sign
- The Axiom of Choice
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- Convolution of two functions on ℝⁿ
- Minkowski's integral inequality
- The modulus of an integral is bounded by the integral of the modulus
- A linear map T of ℝⁿ sends Lebesgue measurable sets to Lebesgue measurable sets, with λₙ(T[E])=|det T| λₙ(E) when T is invertible and T[E] Lebesgue null when it is not
- Lebesgue outer measure, Lebesgue measurability and Lebesgue measure are unchanged by translation
- Borel measurable and Lebesgue measurable functions on ℝⁿ
- The Lᵖ norm descends to the quotient and makes Lᵖ a normed space for 1 ≤ p ≤ ∞
- The support of a function on ℝⁿ and its compactly supported Riemann integral
- The spaces C_c(ℝⁿ) and C_c^∞(ℝⁿ)
- Fundamental theorem of riemannian geometry
- Riemannian divergence theorem
- Riemannian gradient
- Laplace–Beltrami operator as the trace of the Hessian
- Density bundle and smooth density fields
- The riemannian volume density is coordinate independent
- Integral of a compactly supported smooth density
- Orientation-free density integration and its properties
- Smooth partitions of unity exist on manifolds with boundary
- Coordinate formula for riemannian divergence
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
Finite smooth cutoffs relative to compact sets and boundary neighborhoods
Lemma
Weak first derivatives and Riemannian chart norm comparison
Lemma
Compactly supported chart mollification in uniform and first Sobolev norms
Lemma