Alphabeta MathArticle items · 19 — Cutoff maximum principle and uniform collar regularity
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 19–40.
1 · Prerequisites
- Stationary vacuum model and asymptotic normalization
- Regular non-degenerate stationary vacuum spacetime
- Mass-one convention and Harrison denominator
- Stationary maximal graphs and normalized graph heights
- Lemma 3.3: common spacelike reference collar
- Boundary angle bound and normal-graph cone
- Angle differential inequality near S
- Spacelike sections, maximality and lapse–shift data
- The Axiom of Choice
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Projection estimates for a Gaussian collar cutoff
- Weighted cutoff maximum gives a uniform normal angle
- Gaussian maximal graph equation on a compact spacelike slope cone
- Uniform elliptic extension and flattening before the boundary gradient estimate
- Uniform boundary Schauder bootstrap after a C1-Holder seed
- Leon Simon, Lectures on PDE · Lecture 18, Theorem 2 (C^1,beta estimate, boundary), printed p.217, assumptions (P), (*) and (**) on pp.216–217; PDF sheet111
- Sequence closedness and completeness without an interval-family premise
- Remark 1.3: lapse bounds and stationary product
- Openness via the weighted implicit-function branch and smooth boundary bootstrap
- A subset of ℝ is connected if and only if it is order-convex, that is, an interval
- Uniform end estimates indexed only by existing admissible slices
- Proposition 2.1: Einstein–Maxwell equations and regularity
- Metric space: d(x,y) = 0 iff x = y, symmetry, and the triangle inequality; pseudometric and ultrametric
- The absolute value makes ℝ a metric space: d(x,y) = |x-y| is a metric, its open balls are the intervals (x-r, x+r), and it is unbounded
- The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
- 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
- A point lies in the closure of A iff some sequence in A converges to it, and a set is closed iff it is sequentially closed
- Separated sets, disconnection, and connected subset of ℝ
- The closure equals the set together with its limit points, equals the set of points every neighbourhood of which meets it, and is the smallest closed superset; a set is closed iff it contains its limit points
- Rigidity of stationary black holes with small ADM angular momentum · Section 5.3, Lemma 5.1 and the following continuation paragraph
- Normal graphs and the nonlinear mean-curvature map
- Lemma 3.2: compact stationary-height bounds
- Existence and boost-coordinate description of Hawking-flow heights
- K-flow barriers with boundary tangency comparison and uniform height bound
- Causal exterior, chronological boundaries and Cauchy data
- Local causality and temporal functions on compact tubes
- Transformed lapse equation and bounds
- Rescaled annular estimates through four derivatives
- Positive-lapse stationary graphs give temporal level functions and closed restrictions
- Shifting a temporal function permits a controlled compact restriction
- Uniform interior graph geometry and a bounded finite chart cover
- Anchored lapse propagation and quantitative stationary coordinate transfer
- Controlled horizon cuts, compact restrictions and a uniform end anchor
- Local Hawking and axial fields, boost and pole
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- Energy-condition inputs for the transformed geometry
- Asymptotically flat positive-q ends and cylindrical limit
- Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence
- 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
- On a convex open set, a uniform bound ‖Df(z)v‖₂≤ M‖v‖₂ implies ‖f(y)-f(x)‖₂≤ M‖y-x‖₂
- Stationary Harrison one-form and metric
- Closed Ernst form and existence of normalized potential
- Positivity of the Harrison denominator and horizon bound
- Indexed compact heights and uniform smooth horizon collars
- Compact estimates for arbitrary sequences of individually existing slices
- Stationary section time and translated maximal sections
- Temporal functions are time functions, and time functions forbid closed causal curves
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Asymptotic maximal graph equation and coefficients
- Bounded boost times give a uniform Gaussian collar cone
- The fundamental theorem of calculus and integration by parts
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- Uniformly close integrable functions have integrals differing by at most the interval length times their uniform error
- Closedness, periods and smooth global choice of W0
- ADM section time and causal spatial speed
- Collar barrier against finite section-time boundary arrival
- Proper end radius and exclusion of finite-time spatial infinity
- Combining the collar and end barriers into interior compact confinement
- An interior compact causal trajectory has a finite-time endpoint
- Explicit compactly supported smooth cutoffs
- John M. Lee, Introduction to Riemannian Manifolds, second edition (2018) · Theorem 5.25 and normal-tube definitions, printed p. 133 (PDF p. 145); Corollary 6.39, p. 180 (PDF p. 192); Theorem 6.40, p. 181 (PDF p. 193); Example 6.44, p. 183 (PDF p. 195).
- Single-potential construction of the Harrison one-form family
- The existence of a Cauchy set implies strong causality and compact causal diamonds
- Comparison spheres and their stationary cylinders
- Null expansions, screen quotients and focal data
- Proposition 4.1: positive-expansion spheres of limiting area 4π
- Normalized outward null normal at large coordinate spheres
- Compact-target null generators and the required endpoint conditions
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison
- Negative stability eigenvalue produces outer trapped graphs
- Stationary transfer of timelike curves
- Preservation of bifurcate horizons and surface gravity
- Invertibility, Lorentz signature and stationary norm
- A bifurcate future horizon has an oriented defining pair near its section
- Small outward normal graphs have controlled inward null initial jets
- The inward null family reaches the future local horizon in short affine time
- An exterior-contained null segment to a distant target cannot start inward
- The surviving source-normal ray has the outgoing affine normalization
- Compactness gives an earliest translated-cylinder causal contact
- First contact lies on the chronological and causal future boundary
- Time reversal supplies a source-normal generator without interior source focal points
- The earliest contact generator is orthogonal to the distant target cut
- The stationary Killing energy is constant along every affine null segment
- An asymptotically flat stationary end bounds null speed by Killing energy
- Compact normalized source data give a target-independent energy bound
- Distant targets force arbitrarily long source-normalized affine segments
- A sufficiently long focal-free boundary segment contradicts negative outgoing expansion
- Killing two-form and normalized Ernst notation
- Eventual chronological access to translated comparison spheres
- 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
- If f ≤ g on [a,b] and both are integrable then ∫ₐᵇ f ≤ ∫ₐᵇ g; and m(b-a) ≤ ∫ₐᵇ f ≤ M(b-a)
- 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
- 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
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Local comparison of a riemannian metric with the euclidean metric
- ℝ and ℝⁿ for n ≥ 1 with the Euclidean metric are complete, componentwise from the Cauchy criterion in ℝ
- The metric volume form, current flux through 3-faces and finite-chain charges
- A maximal stationary exterior has future access to its far tail
- Compact remainder, causal cut and generator endpoint map
- Eventual access is uniform on the trimmed compact inside core
- A finite temporal target cannot be reached from a sufficiently small collar
- Intersecting a chronological past boundary with a spacelike section
- A closed subset of a compact metric space is compact
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- The image of a compact metric space under a continuous map is compact, and so is the image of any compact subset
- The riemannian distance topology is the manifold topology
- Horizon expansion variation and Einstein–Maxwell curvature
- Principal eigenfunctions and MOTS spectral stability
- Admissible domains for outward normal graphs
- Smooth functions and tensor fields extend locally across the boundary
- Smooth locally defined functions can be glued by a partition of unity
- Collar neighborhood theorem
- Compatible tubular charts realize a prescribed normal identification
- Choice-free smooth inverse function theorem in Euclidean space
- Smooth partitions of unity exist on manifolds with boundary
- 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)
- Invariant Ernst data and smooth K-invariant gauge
- Harrison electromagnetic form
- Horizon null frame, flux densities and stability operator notation
- Lipschitz height functions, cut measure and Jacobians
- Boundary membership excludes interior focal points from the target
- Temporal normalization makes the null geodesic flow compact over compact bases
- Compact causal diamonds close the selected-generator endpoint domain
- The closed selected domain has smooth endpoint extensions
- The selected endpoints cover the compact cut and remain on its past boundary
- Distinct boundary generators cannot cross strictly before a spacelike cut
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- For n≥1, every Euclidean closed ball and every Euclidean sphere of positive radius is compact
- Metric compatibility, torsion freedom and the metric-selected connection class
- Uniform transformed component estimates
- Uniform denominator and one-form estimates
- The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Theorem 2.22 and its proof, PDF pp. 35–36 of 202; ambient regularity convention, PDF p. 3
- Surface measure, flux densities and null normal gauge
- Riemannian divergence theorem
- Global Gauss-Bonnet for closed oriented surfaces
- Cauchy-Schwarz inequality for L²
- Proposition 2.1: electric and magnetic fluxes
- Positive stationary mass from the bifurcate vacuum boundary
- Truncated causal-boundary height for smooth approximation
- The entire target segment stays on the chronological boundary
- Null optical comparison with affine and screen conventions
- The parametrized implicit function theorem with Cᵏ regularity
- Uniform coordinate-sphere area and null expansion
- A compact normal-flow collar has uniformly small area distortion
- Uniform localization of translated past-boundary cuts
- The truncated past height is Lipschitz and has exactly the pulled-back cuts as levels
- Every negative-time cut has a closed selected domain and smooth endpoint extensions
- Level-set area transfer retains generator multiplicity
- For a topological space the following agree: no separation exists, the only clopen subsets are ∅ and X, and every continuous map to the two-point discrete space is constant
- For n≥2, the sphere Sⁿ⁻¹ is path-connected and connected
- Every path-connected space is connected, and every path component lies inside a component
- Coarea smoothing with fixed boundary values and area control
- Divergence comparison in a smooth compact boundary collar
- A compact smooth boundary has a uniform inward normal exponential collar
- The inward normal coordinate is intrinsic distance from the boundary
- Distance Laplacian and normal-collar volume density are uniformly bounded
- The thin-collar flux error is uniform over all smooth comparison regions
- Isometric target areas remain fixed while collar errors vanish
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Riemannian volume form on an oriented manifold
- Dominated convergence
- Riemannian volume is the radon measure of the riemannian density
- Compact flux saturation and pointwise density rigidity
- Lorentzian exterior calculus and Hodge star
- Complex self-dual forms and metric-dependent contraction
- Normal boost action and invariant two-forms at a fixed surface
- A common positive Harrison domain across the compact sphere
- Rank-one tangent changes, signature and Hodge transfer
- Periodic isometries and invariant normal averaging
- Constant homothety preserves the original vacuum and maximal section
- The mass factor in the recovered Ernst identity
- Exact asymptotic and orbit package for the IK criterion
- Exact bifurcate-sphere package for the IK criterion
- Ernst sublevels and the Mars–Simon tensor interface
- Ionescu–Klainerman local Kerr criterion with full interface
- Kerr metric, parameters, exterior and extremality
- AIK global Kerr identification under the full GR/SBS/PK contract
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
Physics theoremDependency level 19
Cutoff maximum principle and uniform collar regularity
Physics theoremDependency level 20
Open–closed continuation for the set of individually admissible parameters
Physics theoremDependency level 20
Lemma 3.4: uniform graph regularity at the bifurcation sphere
Physics theoremDependency level 21
Uniform compact graph and positive-lapse estimates
Physics theoremDependency level 21
Theorem 3.1: complete maximal slices for positive q
Physics theoremDependency level 22
ADM form and stable causality of positive-q metrics
Physics theoremDependency level 22
Closedness and smooth limiting maximal sections
Physics theoremDependency level 23
Causal speed and finite-section-time escape estimates
Physics theoremDependency level 24
Lemma 4.1: Cauchy sections and global hyperbolicity
Physics theoremDependency level 25
Lemma 4.2: null generators of the causal past boundary
Physics theoremDependency level 25
First cylinder contact, outward source branch and the focusing contradiction
Physics theoremDependency level 26
Lemma 4.3: localization of the causal cut near S
Physics theoremDependency level 26
Nonnegative principal eigenvalue of the horizon stability operator
Physics theoremDependency level 27
Closed generator set and nonintersection before the cut
Physics theoremDependency level 27
Stable-horizon flux-energy and area-charge inequalities
Physics theoremDependency level 28
Theorem 4.2: charged horizon area lower bound
Physics theoremDependency level 28
Raychaudhuri expansion and generator Jacobians
Physics theoremDependency level 29
Lipschitz causal height and level-set Jacobian bound
Physics theoremDependency level 30
Coarea, smooth approximation and regular area-controlled levels
Physics theoremDependency level 31
Lemma 4.4: smooth compact comparison region
Physics theoremDependency level 32
Divergence comparison between S and the translated causal cut
Physics theoremDependency level 33
Theorem 4.1: limiting horizon area upper bound
Physics theoremDependency level 34
Corollary 4.1: area and pointwise limiting flux rigidity
Physics theoremDependency level 35
Boost invariance and limiting Maxwell normal form
Physics theoremDependency level 36
Inverse Harrison relation and Killing-scalar identity
Physics theoremDependency level 37
Pole identities and the Ernst scalar inequality reduction
Physics theoremDependency level 38
Axial averaging and positive boost at the pole
Physics theoremDependency level 39
Proposition 5.1: recovery of Ionescu–Klainerman conditions
Physics theoremDependency level 40