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
- Riemannian metric and riemannian manifold
- Laplace–Beltrami operator as the trace of the Hessian
- Riemannian gradient
- Coordinate formula for riemannian divergence
- Riemannian volume density
- Smooth partitions of unity exist on manifolds
- Integer-order Sobolev spaces and their norms
- The notation Hᵏ and the reserved zero-boundary symbol
- Positive-part truncation calculus and admissible cut-off weak tests
- Zero weak gradient gives componentwise constants
- The Axiom of Choice
- Weak derivative of a locally integrable function
- Lebesgue measurable sets, the family L(ℝⁿ), and the restricted set function λₙ
- Borel measurable and Lebesgue measurable functions on ℝⁿ
- Leon Simon, Lectures on Partial Differential Equations · Lecture 18, Theorem 1 and its displayed proof, printed p.211 (PDF sheet 108); coefficient hypotheses (i)–(iii), Lecture 17, printed pp.199–200 (PDF sheets 102–103), including the bounded-measurable q=∞ convention and essential-supremum remark on p.200; specialization R=1 and θ=1/2.
- 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
- Coordinate balls form a basis of a topological manifold
- Smooth manifolds and their smooth charts
- Topological manifolds without boundary: Hausdorff, second-countable, and locally Euclidean spaces
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
- 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
- Derivations at a point and the tangent space
- Coordinate derivations form a basis of the tangent space
- The differential of a smooth map
- Coordinate formula for the differential
- A type (r,s) tensor on a finite-dimensional vector space
- Cᵏ maps and multi-index derivative notation in Euclidean space
- 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
- Every Lorentz form admits an ordered basis with matrix diag(-1,1,1,1)
- The timelike set has exactly two convex components, described in orthonormal coordinates
- The Cauchy-Schwarz inequality for finite sums
- Lipschitz map, α-Hölder map for rational 0 < α ≤ 1, and contraction
- The nonnegative Lebesgue integral
- An open subset of a smooth manifold has a canonical restricted smooth structure
- The Axiom of Countable Choice (AC_ω)
- Assuming countable choice, L(ℝⁿ) is a sigma-algebra containing every elementary set and λₙ is a complete measure extending elementary volume
- Assuming countable choice, every Borel subset of ℝⁿ is Lebesgue measurable
- A continuous map has Borel preimages of Borel sets
- Sequential suprema, infima, limsup, liminf, and pointwise limits of measurable functions are measurable
- A map into ℝⁿ is measurable exactly when its coordinates are measurable
- Composition with a Borel measurable outer map preserves measurability
- Measure-null sets and almost-everywhere statements relative to a measure
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A nonnegative integral over a null set vanishes
- A C¹ map is locally Lipschitz on compact coordinate subsets
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- Manifold charts, coordinate domains, and coordinate functions
- Continuity of a map of topological spaces at a point and globally
- For every fixed finite size at least one, the determinant of a real square matrix is a polynomial in its matrix entries
- A finite square real matrix is invertible if and only if its determinant is nonzero
- Sums, products, absolute values, finite maxima and minima, and quotients of continuous real-valued maps on a topological space are continuous where defined
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 comparison with a nonnegative potential
Theorem
Imported unit-ball Harnack inequality with bounded potential
Lemma
Compact sets admit relatively compact open neighborhoods and finite chart pieces
Lemma
Compact timelike-length bounds restrict to open subdomains
Lemma