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
- The Riemannian gradient is the metric dual of the differential
- Laplace–Beltrami operator as the trace of the Hessian
- Riemannian volume density
- Gradient hessian and divergence connection formulas
- Riemannian divergence theorem
- Smooth partitions of unity exist on manifolds with boundary
- Orientation-free density integration and its properties
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- The Axiom of Choice
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
- The boundary tangent space is the boundary-tangent hyperplane
- Pullback of a riemannian metric is riemannian exactly for immersions
- Positive smooth densities give Radon volume
- Intrinsic density measure and its chart restriction
- Measurable integration extends smooth density integration
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- Positive-codimension immersed submanifolds are null
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Diffeomorphisms and local diffeomorphisms of manifolds
- Riemannian divergence
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Embedded smooth submanifolds with boundary
- Immersions and embeddings for manifolds with boundary
- A smooth differential k-form
- Local coordinate expression for a differential form
- Riemannian volume form on an oriented manifold
- The Lebesgue integral is linear on L¹(μ)
- The modulus of an integral is bounded by the integral of the modulus
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Smooth manifolds and their smooth charts
- Oriented smooth manifolds and oriented charts
- Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)
- Coordinate differentials form the dual cotangent basis
- Wedge monomials in a dual basis form a basis
- 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
- A subset of ℝⁿ with the product topology is compact exactly when it is closed and bounded, the product topology being the Euclidean metric topology
- The Borel sigma-algebra of a topological space
- A continuous map has Borel preimages of Borel sets
- Integrable real and complex functions, and their integrals
- The integral of a nonnegative simple function
- The nonnegative integral agrees with the simple integral on simple functions
- Integrating against a density agrees with integrating the product
- The Axiom of Countable Choice (AC_ω)
- Pointwise Borel nonnegative densities
- Finite and countable subadditivity of measures
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- The notation Hᵏ and the reserved zero-boundary symbol
- Local weak solutions of a divergence-form operator
- Xavier Fernández-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE · Theorem 3.17, printed p. 93, PDF p. 99
- Uniformly elliptic nondivergence-form operators and their frozen coefficients
- Interior Schauder estimate for uniformly elliptic equations
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
Divergence comparison in a smooth compact boundary collar
Lemma
Compact flux saturation and pointwise density rigidity
Lemma
Gradient Hölder control by differentiated flux and one-dimensional lifting
Lemma