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
- Formal adjoint of a real drift Laplacian on a closed manifold
- The Axiom of Choice
- Lars Andersson, Marc Mars and Walter Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §4, equation (10), standing hypotheses and Lemma 4.1(i)–(ii), printed p.9
- General spacelike two-surfaces, null gauges and fluxes
- The exponential domain is open and the exponential map is smooth
- The Axiom of Countable Choice (AC_ω)
- The smooth inverse function theorem on manifolds
- The exponential map scales geodesic time
- Domain and exponential map of a connection
- Smooth embeddings
- 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
- Annular scaling and fixed interior covers
- Ellipticity and zero-slope forcing of the stationary maximal flux
- Gradient Hölder control by differentiated flux and one-dimensional lifting
- Uniform coefficient regularity before the final graph estimate
- Linear fourth derivative bound with controlled realized coefficients
- Differentiated asymptotic bounds and parameter uniformity
- Normal graphs and the nonlinear mean-curvature map
- 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)
- Fundamental theorem of calculus for Banach-valued continuous curves
- The notation Hᵏ and the reserved zero-boundary symbol
- Local weak solutions of a divergence-form operator
- 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)
- Divergence on a bounded C1 Euclidean domain
- Xavier Fernández-Real and Xavier Ros-Oton, Regularity Theory for Elliptic PDE · Theorem 3.17, printed p. 93, PDF page 99 (zero-based page 98 in the currently retrieved 236-page PDF)
- Riemannian metric and riemannian manifold
- Laplace–Beltrami operator as the trace of the Hessian
- Coordinate and weak forms of the geometric Schrödinger equation
- Compact-domain comparison with a nonnegative potential
- 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
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Smooth charts, atlases, and structures with boundary
- Embedded smooth submanifolds with boundary
- A compactly supported Riemann integrand admits the global change-of-variables formula from a diffeomorphism near the relevant compact preimage
- Every continuous function on a closed nondegenerate rectangle in ℝᵐ is Riemann integrable
- Weighted exterior elliptic domains and admissible weights
- Multiplicative interior Harnack for geometric Schrödinger solutions
- 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
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
Theorem
Imported positive eigenpairs for a smooth closed elliptic operator
Lemma
Normal deformation families and an outgoing null gauge
Lemma
Uniform annular Schauder bootstrap for spacelike graph equations
Lemma
A variable-coefficient annular barrier and inward boundary derivative
Lemma
Rescaling a geometric Schrödinger equation without treating its potential as forcing
Lemma