Alphabeta MathArticle items · 17 — The zero-parameter spatial cylinder and its limiting area
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 8–10.
1 · Prerequisites
- Differentiated asymptotic bounds and parameter uniformity
- Uniform spatial components and the cylinder scale
- Riemannian volume form on an oriented manifold
- A regular level set is an embedded submanifold
- Pullback of a riemannian metric is riemannian exactly for immersions
- Oriented smooth manifolds and oriented charts
- Chart integral with its orientation sign
- Change of variables for an injective C¹ map on a compact Jordan set
- Linearity, monotonicity, the absolute-value estimate and coordinate-slice additivity for the Riemann integral in ℝᵐ
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Sum, product, quotient, and chain rules from difference quotients
- Flattening a smooth fixed boundary preserves the elliptic graph equation
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Cᵏ maps and multi-index derivative notation in Euclidean space
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Clairaut--Schwarz theorem for continuous second partial derivatives
- 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)
- 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
- An asymptotically flat stationary end bounds null speed by Killing energy
- Compact normalized source data give a target-independent energy bound
- The Axiom of Choice
- Stationary Lorentzian data and tensor conventions
- Spacelike sections, maximality and lapse–shift data
- The surviving source-normal ray has the outgoing affine normalization
- Smooth maps are continuous
- 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 closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Geodesic of an affine connection
- Time orientation, future and past timelike/null/causal vectors, and the closed cones
- Time reversal supplies a source-normal generator without interior source focal points
- Null optical comparison with affine and screen conventions
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Null expansions, screen quotients and focal data
- Metric compatibility, torsion freedom and the metric-selected connection class
- First contact lies on the chronological and causal future boundary
- Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points
- The differential of the squared-displacement function and the position vector of the reversed geodesic
- Local causality in a convex normal ball: causal reachability is the exponential cone, the Lorentzian length is bounded by the displacement radius, and null endpoints force the null geodesic
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Compactness gives an earliest translated-cylinder causal contact
- Causal exterior, chronological boundaries and Cauchy data
- Local null normal pairs and the positive screen quotient
- Temporal functions are time functions, and time functions forbid closed causal curves
- Boundary-constant-preserving smooth approximation
- Riemannian Lipschitz coarea on compact interior bands
- Transferring area-controlled band energy through smooth approximation
- Choosing a smooth separating regular level from an average bound
- Lipschitz height functions, cut measure and Jacobians
- Smooth partition of unity on a manifold with boundary
- Riemannian metric and riemannian manifold
- Smooth charts, atlases, and structures with boundary
- Connected rotating degenerate vacuum comparison data
- Ordered horizon rods and twist-jump labels
- Harmonic-map and Weyl reconstruction comparison data
- Exact mixed-rod boundary class for the balance theorem
- A regular weight-one axis has zero logarithmic angle defect
- Adopted Einstein field equation in the direct-equation presentation
- Amsel–Horowitz–Marolf–Roberts, Uniqueness of Extremal Kerr and Kerr-Newman Black Holes · §2, Theorem 1, printed p.2; opening proof and footnote 2, printed p.3
- Han–Khuri–Weinstein–Xiong, Kerr Black Hole Uniqueness · §1, equations (1.1)–(1.11), source-class paragraph and Theorem 1.1; §2.1 variational hypothesis and (A1)–(A3), (2.3)–(2.10); §2.3 Proposition 2.1(i)–(iv), (2.12)–(2.26).
- Chruściel and Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; introduction describing missing global-reduction inputs; §3.3 Proposition 3.1 and §3.4; §3.6, (3.35) and Theorem 3.4, printed pp.20,22
- Chruściel–Costa–Heusler, Stationary Black Holes: Uniqueness and Beyond · §3.2.3, printed p.14; §3.2.7, Theorem 3.2, printed p.15
- Stationary vacuum model and asymptotic normalization
- Geometric data for the analytic multiple-horizon comparison
- Canonical stationary end expansions give the precise source slice decay
- Cofinal stationary end representatives have the same causal exterior
- Positive smooth horizon area and the exact degenerate angular-momentum test
- The exhaustive analytic horizon-component case split
- Stationary vacuum normal form and compatible regularity section
- Imported geometric analytic connectedness and degenerate-component exclusions
- Huang–Li–Zhu, Nonexistence of Multiple Black Holes in Stationary Analytic Vacuum Spacetimes · §1, complete multiple-horizon classification discussion from the paragraph beginning “For the general case” through the Main Theorem and final unnumbered Corollary; (H1)–(H3) specify its geometric class.
- Regular non-degenerate stationary vacuum spacetime
- Single-potential construction of the Harrison one-form family
- Killing two-form and normalized Ernst notation
- Complex self-dual forms and metric-dependent contraction
- Remark 1.3: lapse bounds and stationary product
- Mass-one Ernst and radial primitive asymptotics
- Periods and primitives on a spherical exterior
- Equivariant global Harrison primitive under weak original data
- Local Hawking and axial fields, boost and pole
- Smooth transverse horizon branches are the null axes of the normal tube
- Radial primitive and five differentiated decay bounds
- The general Stokes theorem
- Homology of spheres
- Homotopic maps induce the same map on singular homology
- Closed Ernst form and existence of normalized potential
- Vacuum Killing and Ernst differential identities
- Lorentzian exterior calculus and Hodge star
- Interior product on forms is a graded antiderivation
- Fundamental class of a compact oriented manifold
- Top homology of a connected manifold
- Interior product of a form by a vector field
- Cartan's magic formula
- A smooth function with zero differential is constant on each connected component
- Mass-one convention and Harrison denominator
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- Positive stationary mass from the bifurcate vacuum boundary
- Lie derivative commutes with the exterior derivative
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- The Lie derivative of a vector field equals the Lie bracket
- The exterior derivative commutes with pullback
- The flow of a vector field tangent to a closed embedded submanifold preserves it
- Smooth rigidity from a quantitatively small horizon field
- Subextremal Kerr exterior in length coordinates
- AIK perturbative global regularity and bifurcation package
- Ernst and self-dual curvature data in the AIK convention
- Mars–Simon tensor and its smooth weighted extension
- AIK perturbative effective constant and its coordinate data
- The weighted AIK perturbation hypothesis PK
- Radu nontrapping global regularity package
- Event-horizon bifurcation data for the nontrapping theorem
- Stationary orbit space with its quotient topology
- Zero stationary energy along null geodesics
- The precise maximal zero-energy nontrapping predicate
- Radu small-ADM global and bifurcation package
- The ADM-source mass and angular-momentum parameters
- Maximality and smooth boundary in the ADM theorem
- The actual geometric constant in the small-ADM theorem
- Quantifiers and parameter gap in the ADM smallness hypothesis
- S. Alexakis, A. D. Ionescu and S. Klainerman, Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces · §1.1 setup, GR/SBS/PK (1.1)–(1.4), Main Theorem; §2.1 (2.1)–(2.5); §2.3 (2.13)–(2.20)
- Alexandru F. Radu, On rigidity of stationary black holes under a nontrapping assumption · §1.1 setup, GR (1.1), SBS, quotient-space definition, NT and Theorem 1.1; §1.2 first paragraph
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 setup, GR/SBS, maximality, (1.1)–(1.3) and Theorem 1.1
- The first possible shift and lapse logarithms vanish by indicial projection
- Leading stationary modes and a spatial normalization of the same maximal end
- Supplied CW logarithmic maximal ends have weak weighted initial data
- Stokes theorem for smooth singular chains
- Fundamental theorem of riemannian geometry
- The pullback of a covariant tensor field
- Weighted Hölder spaces and their Dirichlet kernels are Banach
- A timelike transverse field gives a fixed-boundary admissible graph neighborhood
- Compact-core Hölder superposition gives a continuously differentiable graph map
- Stationarity supplies the weighted end differentiability estimates
- Mean-curvature variation and the stationary lapse equation
- Normal graphs and the nonlinear mean-curvature map
- Weighted exterior elliptic domains and admissible weights
- C k map between Banach spaces
- Fréchet derivative between Banach spaces
- The standard product norms on a finite product of normed spaces
- Finite products of Banach spaces are Banach
- Boundary Schauder estimate for the Dirichlet problem
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- Explicit compactly supported smooth cutoffs
- The fundamental theorem of calculus and integration by parts
- AC implies DC implies countable choice
- For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice
- Continuous mixed partials of order k are invariant under permutations
- Bifurcate Killing horizons, rotation and normalization
- Stationary Harrison one-form and metric
- Positivity of the Harrison denominator and horizon bound
- Invertibility, Lorentz signature and stationary norm
- Normal boost action and invariant two-forms at a fixed surface
- Local and global flows generated by a vector field
- The Lie derivative of a tensor field
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The exterior derivative by the invariant vector-field formula
- Normalized Komar data on a nonrotating horizon section
- The normal binormal is invariant under oriented boosts
- Killing-horizon derivative contracted with the binormal
- Constant surface gravity gives the normalized horizon integral
- Degeneracy forces zero mass when global Komar transport applies
- A nonzero degenerate generator cannot vanish at a regular bifurcation section
- Rotation, surface gravity and horizon degeneracy
- Kerr metric, parameters, exterior and extremality
- Staticity, axisymmetry and black-hole regularity hypotheses
- Geometric electrovacuum equations and charge convention
- Kerr-Newman metric and potential on the exact Boyer-Lindquist domain
- Full-sphere exterior extension of the Kerr-Newman datum
- Regular ingoing KN metric, field and sphere axes
- KN Killing generators, surface gravity, area and potential
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- Embedded submanifolds admit local defining submersions
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- P. T. Chruściel and L. Nguyen, A uniqueness theorem for degenerate Kerr-Newman black holes · Theorem 1.1, printed p.1; §2 opening paragraph, p.2; §3.2 opening paragraph, p.11
- P. T. Chruściel and J. L. Costa, On uniqueness of stationary vacuum black holes · Definition 1.1, printed p.2; Theorem 1.3, pp.3–4; §2.1, (2.5)–(2.8), p.6; Theorem 2.3, p.10; §2.2, (2.9)–(2.10), p.7; Theorem 4.5 and (4.15), p.18, with end-matching conclusion of its proof, p.22; §8 remark 2, p.55
- P. T. Chruściel, J. L. Costa and M. Heusler, Stationary Black Holes: Uniqueness and Beyond · §7.2, printed pp.38–39, especially (7.8) and its horizon-flux explanation; §2.5.3, pp.8–10; §3.3.1–3.3.2, including Theorem 3.3, p.16
- H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §8.2, (8.18)–(8.24), printed p.108 (PDF p.109); §8.4, paragraph after (8.43), printed p.113 (PDF p.114)
- T. Adamo and E. T. Newman, The Kerr-Newman metric: A Review · §3.1, (3.1)–(3.2), printed p.13; §3.3, (3.18)–(3.21), pp.17–18, and extremality paragraph on p.18
- Coordinate components of the transformed metric
- 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
- Harrison electromagnetic form
- A common positive Harrison domain across the compact sphere
- Stationary flow-box data for a nonzero Killing field
- Harrison equation residuals are continuous functions of stationary two-jets
- Passing a local Harrison identity through compatible stationary jets
- Extending smooth equation residuals from a dense nonzero-field locus
- Transverse embedded submanifolds intersect in the expected codimension
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · Appendix A.2, (A.12)–(A.25), and paragraph immediately following (A.25); spacetime formulas (3.11), (3.24)
- Smooth invariant Harrison primitive near the horizon and at infinity
- Regular Harrison Einstein–Maxwell family
- Mass-one Ernst decay and uniform Harrison denominator bounds
- The differentiated spatial decay supplied by the Harrison lift
- Positive-parameter normalized time and spatial ends
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 (1.1); §3.2 (3.19); §3.3 (3.28)–(3.29)
- Surface measure, flux densities and null normal gauge
- Outward unit variation in the normalized null basis
- Agreement of expansion and normal-connection conventions
- General curvature coefficient in the outward hypersurface gauge
- Electrovac contraction in the normalized normal plane
- Zero null optics supplies the missing null Ricci condition
- Scalar curvature is twice the sum of sectional curvatures of orthonormal coordinate planes
- Andersson–Mars–Simon, Stability of marginally outer trapped surfaces and existence of marginally outer trapped tubes · §2, printed pp. 5–6 (null normals, second form and expansion); §3, printed p. 7 (immersed-surface variation and varying null normal, before Lemma 3.1); Lemma 3.1, printed p. 7; equations (2)–(7), Definition 3.1 and Lemma 3.2, printed p. 8
- The general Leibniz rule for the n-th derivative of a product
- Periodic isometries and invariant normal averaging
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- A smooth spacelike reference section is a bounded rapidity graph
- The stationary product extends the local boost and circle to a complete collar
- Compact metric families have a common spacelike collar with fixed boost rate
- A complete timelike collar flow is a reference product
- Riemannian gradient
- Domain and exponential map of a connection
- For a nonempty subset of ℝⁿ with n≥1, compactness, closedness and boundedness, pseudocompactness, and attainment of extrema by every continuous real-valued function are equivalent
- Maxwell algebra supplies timelike and null convergence
- The mass-one Killing form has electric seed flux minus four pi
- Harrison spatial two-form differs from the seed by an integrable tail
- Positive-parameter Harrison charge limits in the original end chart
- Closed electric and magnetic two-forms transfer their periods to the horizon
- The zero Harrison parameter follows from continuity on one fixed sphere
- Horizon null frame, flux densities and stability operator notation
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 8
The zero-parameter spatial cylinder and its limiting area
LemmaDependency level 8
Tangential differentiated equations admit classical difference quotients
LemmaDependency level 8
Distant targets force arbitrarily long source-normalized affine segments
LemmaDependency level 8
A sufficiently long focal-free boundary segment contradicts negative outgoing expansion
LemmaDependency level 8
The earliest contact generator is orthogonal to the distant target cut
LemmaDependency level 8
Coarea smoothing with fixed boundary values and area control
Physics theoremDependency level 8
Analytic extremal and multiple-horizon comparison contracts
Physics theoremDependency level 8
Exclusion of regular analytic multiple horizons
Physics theoremDependency level 8
Closedness, periods and smooth global choice of W0
Physics theoremDependency level 8
Uniform denominator and one-form estimates
Physics theoremDependency level 8
Cartan normalization at the bifurcation pole
Physics theoremDependency level 8
Invariant Ernst data and smooth K-invariant gauge
Physics theoremDependency level 8
Reported smooth uniqueness under additional hypotheses
LemmaDependency level 9
The same maximal slice has the differentiated article end gauge and a limiting old time
LemmaDependency level 9
The global Dirichlet mean-curvature map is continuously differentiable
LemmaDependency level 9
Boundary Schauder applied to tangential quotients gains one derivative
Physics theoremDependency level 9
Preservation of bifurcate horizons and surface gravity
Physics theoremDependency level 9
Analytic connected-horizon uniqueness including the degenerate case
Physics theoremDependency level 9
Uniform transformed component estimates
Physics theoremDependency level 9
Proposition 2.1: Einstein–Maxwell equations and regularity
RemarkDependency level 9
Introductory Harrison asymptotic claims
TheoremDependency level 9
General MOTS first variation before electrovac specialization
LemmaDependency level 10
Ellipticity recovers every remaining normal derivative
Physics theoremDependency level 10
Lemma 3.3: common spacelike reference collar
Physics theoremDependency level 10
Normalized outward null normal at large coordinate spheres
Physics theoremDependency level 10
Energy-condition inputs for the transformed geometry
Physics theoremDependency level 10
Proposition 2.1: electric and magnetic fluxes
Physics theoremDependency level 10