Alphabeta MathArticle items · 13 — Weighted cutoff maximum gives a uniform normal angle
Presents 29 items from the article and its immediate prerequisites, covering dependency levels 4–5.
1 · Prerequisites
- Smooth collars of a manifold boundary
- Projection estimates for a Gaussian collar cutoff
- Logarithmic angle inequality through the full range v at least one
- Riemannian metric and riemannian manifold
- The Riemannian gradient is the metric dual of the differential
- Laplace–Beltrami operator as the trace of the Hessian
- Gradient hessian and divergence connection formulas
- 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
- The Hessian is negative semidefinite at an interior local maximum
- Causal exterior, chronological boundaries and Cauchy data
- Null expansions, screen quotients and focal data
- A compact chronological target supplies a causal endpoint
- A compact-target boundary point has a null geodesic segment
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- The Axiom of Choice
- Spacelike sections, maximality and lapse–shift data
- Stationary maximal graphs and normalized graph heights
- Smooth manifolds and their smooth charts
- Products of smooth manifolds have a canonical product smooth structure
- Causal curves, chronological and causal reachability, and inextendibility
- Temporal functions are time functions, and time functions forbid closed causal curves
- AC implies DC implies countable choice
- Embedded smooth submanifolds with boundary
- Immersions and embeddings for manifolds with boundary
- Interior and boundary of a manifold with boundary
- For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and f(Ā) ⊆ f(A)
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A product of finitely many compact spaces is compact in the product topology
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Timelike, null, spacelike and causal vectors of a Lorentz vector space
- Cʳ and smooth maps between smooth manifolds
- The differential of a smooth map
- The inverse metric, musical isomorphisms and index raising
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Stationary Lorentzian data and tensor conventions
- A smooth Cauchy hypersurface gives a connected timelike-flow product
- The fundamental theorem on flows
- Time-t flow maps are diffeomorphisms between open domains
- The smooth inverse function theorem on manifolds
- Diffeomorphisms and local diffeomorphisms of manifolds
- The maximal equation upgrades pointwise order-four graph decay to weighted order five
- One extra pointwise derivative supplies scaled Hölder control
- Stationary maximal graphs have an explicit uniformly spacelike flux
- Weighted Hölder norms on a three-dimensional exterior
- Sum, product, quotient, and chain rules from difference quotients
- 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
- Local and global flows generated by a vector field
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- The parametrized implicit function theorem with Cᵏ regularity
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- A smooth spacelike reference section is a bounded rapidity graph
- Pushforwards and pullbacks of vector fields by a diffeomorphism
- The pullback of a covariant tensor field
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Normal graphs and the nonlinear mean-curvature map
- A variable-time flow through a fixed boundary has the boosted unit normal
- Bounded boost times give a uniform Gaussian collar cone
- Smooth maps between manifolds with boundary
- The chain rule for differentials of smooth maps
- In any metric space compactness implies countable compactness and limit point compactness, and each of countable compactness and limit point compactness implies sequential compactness; every implication here is proved without a choice principle
- The selected exterior is connected and causally convex
- Closure of the exterior Cauchy section meets only the double horizon
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- Stationary end access and future orientation need no maximal slice
- Regular non-degenerate stationary vacuum spacetime
- The flowout theorem
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal
- The causal domain of dependence of a partial Cauchy hypersurface is an open causally convex manifold on which the hypersurface is Cauchy
- Temporal normalization makes the null geodesic flow compact over compact bases
- Compact causal diamonds close the selected-generator endpoint domain
- Transverse hitting maps have local differentiable extensions
- Lipschitz height functions, cut measure and Jacobians
- The Euclidean implicit function theorem with derivative formula
- A regular level set is an embedded submanifold
- Complex self-dual forms and metric-dependent contraction
- Geometric electrovacuum equations and charge convention
- Stationary Harrison one-form and metric
- Harrison electromagnetic form
- Stationary flow-box data for a nonzero Killing field
- Bifurcate Killing horizons, rotation and normalization
- Regular bifurcate null horizon data before rigidity
- Smooth event-boundary branches are achronal null geodesic sheets
- A compact spacelike surface has a normal exponential tube
- Smooth transverse horizon branches are the null axes of the normal tube
- The normal future and past quadrants are the local causal sets of the sphere
- Mass-one convention and Harrison denominator
- Single-potential construction of the Harrison one-form family
- Integration on an oriented embedded submanifold
- Lorentzian exterior calculus and Hodge star
- Finite chart localization of compactly supported top forms, including 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
- Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ℝᵐ
- 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
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- The mass-one Killing form has electric seed flux minus four pi
- Harrison spatial two-form differs from the seed by an integrable tail
- A specified future ingoing Kerr extension and its time orientation
- The outer Kerr null hypersurface and its future generator
- Outgoing Kerr null curves and the selected future null infinity
- Absolute continuity on a compact interval
- A Lipschitz function after an absolutely continuous function is absolutely continuous
- The product of two absolutely continuous functions is absolutely continuous
- Fundamental theorem of calculus for absolutely continuous functions
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- The exponential function is smooth and (exp)'=exp
- 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)
- Integrable real and complex functions, and their integrals
- Time orientation, future and past timelike/null/causal vectors, and the closed cones
- Classification of future and past causal vectors by a timelike field
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- Complete ordered field (least-upper-bound property)
- Epsilon characterisation of the supremum
- Continuity of f : A → ℝ at a point of A and on A: the ε-δ condition, its agreement with lim_x → c f(x) = f(c) at a limit point, and continuity at an isolated point
- Saul A. Teukolsky, The Kerr Metric (arXiv:1410.2130v2) · §2, equations (2.1), (2.5)–(2.9), PDF pp. 2–3
- Asymptotic mass and axial angular-momentum conventions
- The Boyer–Lindquist spatial end and differentiated expansion
- The asymptotic metric flux of Kerr equals its mass parameter
- Kerr exterior tensor including the extremal parameter range
- The Kerr Metric · §2, equations (2.1), (2.5)–(2.9), PDF pp. 2–3
- Normalized Komar data on a nonrotating horizon section
- Killing-horizon derivative contracted with the binormal
- Riemannian volume form on an oriented manifold
- Linearity and additivity of the form integral
- The tubular neighbourhood theorem in a smooth ambient manifold
- De rham cohomology of spheres
- De rham cohomology is smooth homotopy invariant
- De rham cohomology
- Closed and exact differential forms
- Identifying the spherical Cauchy portion without a stationary lapse
- AC supplies the countable and dependent choices used in Banach integration
- A smooth bump between concentric Euclidean balls
- A smooth function with zero differential is constant on each connected component
- Collar barrier against finite section-time boundary arrival
- Proper end radius and exclusion of finite-time spatial infinity
- A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- The interior is an open smooth n-manifold
- 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 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
- A bifurcate future horizon has an oriented defining pair near its section
- Small outward normal graphs have controlled inward null initial jets
- Weighted exterior elliptic domains and admissible weights
- Fredholm and Banach continuation interfaces
- Differentiated asymptotic bounds and parameter uniformity
- Scalar exterior inverse from exhaustion and uniform annular estimates
- Compact extra components contribute index zero
- Compact multiplication by a faster-decaying potential
- Fredholm index is stable under compact perturbations
- Adopted Einstein field equation in the direct-equation presentation
- Subextremal Kerr exterior in length coordinates
- 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
- Pathwise exhaustion by anchored Ernst sublevels
- Stationary transport of vanishing from a section
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §1.2 AF, SBS, T, equations (1.4)–(1.7), printed pp. 6–7; Main Theorem, printed p. 8
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §8, Lemma 8.1 and (8.2), printed p. 41; sublevel definition and Propositions 8.2–8.3, printed p. 42; §8.2, printed pp. 48–49; completion (8.1)
- Compact sets admit relatively compact open neighborhoods and finite chart pieces
- Compact timelike-length bounds restrict to open subdomains
- A temporal covector has uniform compact negativity and orientation margins
- Inversion is uniformly stable on compact chart families
- A compact temporal covector persists under metric perturbation
- A. F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Appendix B, Lemma B.2, equation (B.2)
- Rescaling a null generator preserves zero expansion
- Null second fundamental form descends to the screen quotient
- Metric transport along a null generator
- A branch-tangent Killing field restricts to the bifurcation sphere
- Periodic isometries and invariant normal averaging
- S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking’s local rigidity theorem without analyticity · §1 setup and Theorems 1.1–1.2 (pp. 1–2); §2 Definition 2.1 (p. 6); §5 Propositions 5.1–5.2 (pp. 16–17)
- The Killing derivative is an exact closed two-form
- Ricci contraction of the Killing derivative
- Vanishing divergence implies closed Hodge dual
- Killing flow preserves its complex two-form
- Closed Ernst contraction and the real potential
- Staticity, axisymmetry and black-hole regularity hypotheses
- Riemannian volume of a compactly supported smooth density
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §2.1 and §§2.3–2.4 (PDF pp. 5–6); §2.5.3, equations (2.20)–(2.21) (PDF p. 10); §2.6, Definition 2.1 (PDF pp. 10–11); §3.2.1 (PDF p. 13); §3.2.7, Theorem 3.2 (PDF p. 15).
- Radu small-ADM global and bifurcation package
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 GR (1.1), definition of J=|S|, mass-one paragraph before (1.2), and Theorem 1.1
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 definitions of h and k, maximality equation and smooth-extension sentence immediately after SBS
- AIK perturbative global regularity and bifurcation package
- Mars–Simon tensor and its smooth weighted extension
- AIK perturbative effective constant and its coordinate data
- Local frame fields, coordinate frames and tetrad fields
- Gram–Schmidt turns every finite independent list into an orthonormal list with the same successive spans
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · §1.1 PK (1.4); §2.1 (2.5); §2.3 (2.18)–(2.19)
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
LemmaDependency level 4
Weighted cutoff maximum gives a uniform normal angle
LemmaDependency level 4
The entire target segment stays on the chronological boundary
LemmaDependency level 4
Positive-lapse stationary graphs give temporal level functions and closed restrictions
LemmaDependency level 4
A complete stationary flow transverse positively to a Cauchy section is a product
LemmaDependency level 4
Induced metric and graph normal inherit the required weighted regularity
LemmaDependency level 4
The stationary product extends the local boost and circle to a complete collar
LemmaDependency level 4
Spacelike normal projection is locally invertible with positive boundary jet
LemmaDependency level 4
A working partial-Cauchy collar satisfies the source GR section equalities
LemmaDependency level 4
The closed selected domain has smooth endpoint extensions
LemmaDependency level 4
Harrison equation residuals are continuous functions of stationary two-jets
LemmaDependency level 4
Regular stationary event branches supply the local AIK horizon pair
LemmaDependency level 4
The zero Harrison parameter follows from continuity on one fixed sphere
LemmaDependency level 4
Positive-parameter Harrison charge limits in the original end chart
LemmaDependency level 4
A causal barrier identifies the future Kerr event horizon
LemmaDependency level 4
The Kerr end is an ADM rest frame
LemmaDependency level 4
Constant surface gravity gives the normalized horizon integral
LemmaDependency level 4
Extending a closed one-form potential across a bifurcation sphere
LemmaDependency level 4
Combining the collar and end barriers into interior compact confinement
LemmaDependency level 4
The inward null family reaches the future local horizon in short affine time
LemmaDependency level 4
Exterior Dirichlet parametrix and compact decaying potentials
Physics theoremDependency level 4
Ionescu–Klainerman local Kerr criterion with full interface
TheoremDependency level 4
Compact timelike-length bounds and local temporal functions
TheoremDependency level 4
Hawking and rotational fields near a bifurcate sphere
TheoremDependency level 4
Vacuum Killing and Ernst differential identities
DefinitionDependency level 5
Smooth global data for the axisymmetric comparison
DefinitionDependency level 5
Connected rotating degenerate vacuum comparison data
DefinitionDependency level 5
The ADM-source mass and angular-momentum parameters
DefinitionDependency level 5
Maximality and smooth boundary in the ADM theorem
DefinitionDependency level 5