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
- Gradient Hölder control by differentiated flux and one-dimensional lifting
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Interior Schauder estimate for uniformly elliptic equations
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Leibniz's rule on a compact rectangle: an interior parameter derivative with a continuous extension may be passed through a Riemann integral
- Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error
- The mean value theorem, as the case g(x) = x of Cauchy's: for f continuous on [a,b] with a < b and differentiable on (a,b) there is c ∈ (a,b) with f(b) - f(a) = f'(c)(b-a)
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- Xavier Fernández-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE · Theorem 2.20 and Corollary 2.21, printed pp. 46–47, PDF pp. 52–53
- Weighted exterior elliptic domains and admissible weights
- Riemannian gradient
- Coordinate formula for riemannian divergence
- Laplace–Beltrami operator as the trace of the Hessian
- The fundamental theorem of calculus and integration by parts
- A continuous function on a closed rectangle has repeated Riemann integrals in every coordinate order, all equal to its multiple integral
- Explicit compactly supported smooth cutoffs
- Riemannian metric and riemannian manifold
- Divergence on a bounded C1 Euclidean domain
- Borel measurable and Lebesgue measurable functions on ℝⁿ
- Integer-order Sobolev spaces and their norms
- Imported unit-ball Harnack inequality with bounded potential
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
Uniform coefficient regularity before the final graph estimate
Lemma
Coordinate and weak forms of the geometric Schrödinger equation
Lemma