Alphabeta MathArticle items · 08 — Periods and primitives on a spherical exterior
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 0–1.
1 · Prerequisites
- The Axiom of Choice
- Closed and exact differential forms
- De rham cohomology
- The pullback of a differential form
- The exterior derivative commutes with pullback
- De rham cohomology of spheres
- De rham cohomology is smooth homotopy invariant
- Finite chart localization of compactly supported top forms, including boundary
- Nonzero total integral obstructs exactness on a closed manifold
- Smooth manifolds and their smooth charts
- Diffeomorphisms and local diffeomorphisms of manifolds
- Products of smooth manifolds have a canonical product smooth structure
- The n-sphere with its standard smooth atlas
- Oriented smooth manifolds and oriented charts
- Embedded submanifolds and slice charts
- A smooth differential k-form
- The wedge product of differential forms
- Riemannian metric and riemannian manifold
- The Riemannian gradient is the metric dual of the differential
- Laplace–Beltrami operator as the trace of the Hessian
- Riemannian volume density
- Gradient hessian and divergence connection formulas
- Riemannian divergence theorem
- Smooth partitions of unity exist on manifolds with boundary
- Orientation-free density integration and its properties
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Cauchy–Schwarz: |⟨ u,v⟩|≤‖ u‖‖ v‖, with equality exactly for linearly dependent vectors
- The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
- The boundary tangent space is the boundary-tangent hyperplane
- Pullback of a riemannian metric is riemannian exactly for immersions
- Positive smooth densities give Radon volume
- Intrinsic density measure and its chart restriction
- Measurable integration extends smooth density integration
- On Borel subsets of ℝᵐ⁺ⁿ, the product λₘ × λₙ agrees with λₘ₊ₙ
- Lebesgue measure is sigma-finite, and every metrically bounded subset of ℝⁿ has finite outer measure
- Positive-codimension immersed submanifolds are null
- Monotonicity and nonnegative homogeneity of the nonnegative integral
- Riemannian divergence
- A box in ℝⁿ with parameters aᵢ≤ bᵢ is Lebesgue measurable of measure ∏_i<n(bᵢ-aᵢ), whichever of its faces are included
- Embedded smooth submanifolds with boundary
- Immersions and embeddings for manifolds with boundary
- Stationary Lorentzian data and tensor conventions
- Null expansions, screen quotients and focal data
- Geodesic of an affine connection
- Causal curves, chronological and causal reachability, and inextendibility
- S. Alexakis, A. D. Ionescu and S. Klainerman, Hawking’s local rigidity theorem without analyticity · §1 setup, pp. 1–2; §2, pp. 4–7, equations (2.1)–(2.4), Definition 2.1 and equation (2.11) with its following sentence.
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Killing fields of a Lorentzian metric
- Local and global flows generated by a vector field
- Asymptotically flat spacelike end and derivative decay
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Complex self-dual forms and metric-dependent contraction
- Interior product of a form by a vector field
- The metric volume form, current flux through 3-faces and finite-chain charges
- Smooth embeddings
- Ionescu–Klainerman, On the uniqueness of smooth, stationary black holes in vacuum · §4.2, (4.11), (4.15), (4.17) and the normalization paragraph after (4.18), printed pp. 22–23; §4.3, (4.19)–(4.21), (4.27), Definition 4.3 (4.29), printed pp. 23–24; §7, (7.3), printed p. 35; Appendix A.3, opening curvature-component formula, printed p. 54 (PDF p. 54).
- Differentiated asymptotic bounds and parameter uniformity
- Spacelike initial data, the constraint equations, developments and maximal developments
- Stationary-axisymmetric geometry and circularity domain
- H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §8.1, equation (8.2), printed p.105, and the orientation example preceding (8.11), printed p.106; §8.2, equation (8.25), printed p.109; §8.4, equations (8.43)–(8.45), printed p.113.
- Bifurcate Killing horizons, rotation and normalization
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Riemannian volume form on an oriented manifold
- Lorentzian exterior calculus and Hodge star
- Integration on an oriented embedded submanifold
- H. S. Reall, Black Holes, Part III lecture notes, 15 March 2024 · §8.1, (8.14) and the following coordinate-sphere definition, printed p.107 (PDF p.108); §8.2, (8.18)–(8.24), printed p.108 (PDF p.109); §9.1, (9.1)–(9.2) and preceding normalization paragraph, printed p.114 (PDF p.115).
- Chronology, causality, strong causality, global hyperbolicity and time functions
- Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Section 2.6, Definition 2.16, printed/PDF p. 33; section 4.4, opening cone-order convention and Definition 4.89, printed/PDF p. 137.
- E. Gourgoulhon, 3+1 Formalism and Bases of Numerical Relativity, arXiv:gr-qc/0703035v1 · Section 3.2.1, printed p. 39, PDF page 39, opening paragraph.
- Spacelike sections, maximality and lapse–shift data
- Weighted exterior elliptic domains and admissible weights
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- Two time orientations agree or are opposite on each connected set
- 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
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value
- Metric compatibility, torsion freedom and the metric-selected connection class
- The graph of a smooth map is an embedded submanifold
- The derivative of the metric density and the chart formula for the divergence of a vector field
- Affine connection on a smooth manifold
- Christoffel symbols of an affine connection
- The derivative of the determinant along an invertible C¹ matrix curve: (det A)'=det A tr(A⁻¹A')
- If det(A) is a unit, then A⁻¹=det(A)⁻¹adj(A)
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Uniform quadratic-form bounds for positive definite symmetric matrices and smoothness of the inverse
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- 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
- 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)
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, proof of Lemma B.1, printed/PDF pp.38–39; frame bound immediately after (B.8) and tangent-component identity immediately before (B.10)
- The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors
- The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors
- Properties of normal coordinates at the center
- Existence of normal neighborhoods
- Normal neighborhood and normal coordinate chart
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1 statement, printed/PDF pp.35–36; proof, equation (B.8) and following (B.9)–(B.11), pp.38–39
- A compact smooth boundary has a uniform inward normal exponential collar
- Levi civita connection
- Smooth variation and variation field of a curve
- Geodesics have constant speed for a metric-compatible connection
- Riemannian distance on a connected manifold
- Extended riemannian distance on a disconnected manifold
- Riemannian speed and length
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- The smooth inverse function theorem on manifolds
- The Euclidean inverse function theorem
- A local inverse of a Cᵏ regular map is Cᵏ
- Cᵏ Euclidean maps and diffeomorphisms
- Cᵏ Euclidean maps are closed under componentwise algebra and composition
- Coordinate geodesic equation
- Smooth functions on relatively open half-space sets
- Choice-free smooth inverse function theorem in Euclidean space
- Chain rule for smooth half-space maps
- The fundamental theorem on flows
- Equal fixed-surface flow heights imply boundary tangency
- Geometric electrovacuum equations and charge convention
- The exterior derivative by the invariant vector-field formula
- Smooth dependence of ODE solutions on parameters
- The parametrized implicit function theorem with Cᵏ regularity
- Domain and exponential map of a connection
- Riemannian gradient
- Smooth collars of a manifold boundary
- The action of a vector field on smooth functions
- Torsion tensor of an affine connection
- The Lie derivative is a derivation of the tensor algebra
- Tensor Lie derivative agrees with X on functions and bracket on vector fields
- The exterior derivative of a function is its differential
- The Lie bracket of smooth vector fields
- Periodic isometries and invariant normal averaging
- Integral curves of a vector field
- The chain rule for differentials of smooth maps
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The Axiom of Countable Choice (AC_ω)
- A smooth vector field is a smooth section of the tangent bundle
- Smoothness of a vector field is equivalent to smooth coordinate components
- Finite-regularity differential forms, Hamiltonian fields and canonical maps in phase charts
- Improper integrals over unbounded intervals
- Differentiation under an improper multiple integral under an integrable derivative bound
- Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- Comparison tests for improper integrals
- Sum, product, quotient, and chain rules from difference quotients
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Classification of future and past causal vectors by a timelike field
- Fundamental theorem of riemannian geometry
- Christoffel formula for the levi civita connection
- Dual connection
- Yishen Huang, Junbin Li, and Xi-Ping Zhu, Nonexistence of Multiple Black Holes in Stationary Analytic Vacuum Spacetimes · arXiv:2611.03766v1, Section 1, hypothesis (H1), printed p. 2 (PDF page 2), including the little-o convention immediately following its displayed formulas.
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 0
Periods and primitives on a spherical exterior
LemmaDependency level 0
Divergence comparison in a smooth compact boundary collar
DefinitionDependency level 1
Regular bifurcate null horizon data before rigidity
DefinitionDependency level 1
Complete stationary flow and its exterior region
DefinitionDependency level 1
Ernst sublevels and the Mars–Simon tensor interface
DefinitionDependency level 1
Asymptotic mass and axial angular-momentum conventions
DefinitionDependency level 1
Normalized Komar data on a nonrotating horizon section
DefinitionDependency level 1
Causal exterior, chronological boundaries and Cauchy data
DefinitionDependency level 1
Rotation, surface gravity and horizon degeneracy
DefinitionDependency level 1
Killing two-form and normalized Ernst notation
DefinitionDependency level 1
Lapse, shift, induced metric and second fundamental form
DefinitionDependency level 1
Weighted Hölder norms on a three-dimensional exterior
LemmaDependency level 1
Sufficiently far stationary end points can be routed to a chosen cylinder
LemmaDependency level 1
Gaussian maximal graph equation on a compact spacelike slope cone
LemmaDependency level 1
One extra pointwise derivative supplies scaled Hölder control
LemmaDependency level 1
Boost-frame bounds for a smooth observer and arbitrary spacelike normals
LemmaDependency level 1
Pointwise Gauss–Codazzi identities for the normal angle
LemmaDependency level 1
The inward normal coordinate is intrinsic distance from the boundary
LemmaDependency level 1
Mean curvature at identical spacelike graph jets has positive Hessian symbol
LemmaDependency level 1
Ordered timelike flow graphs lie on the corresponding geometric side
LemmaDependency level 1
Vacuum metrics embed as stationary electrovacuum data with zero field
LemmaDependency level 1
Compact metric families have a common spacelike collar with fixed boost rate
LemmaDependency level 1
Affine null-generator profiles vanish exactly on the bifurcation section
LemmaDependency level 1
For a Killing field the acceleration and surface-gravity equations agree
LemmaDependency level 1
Constant combinations preserve horizon symmetries and normalize periods
LemmaDependency level 1
Radial primitive and five differentiated decay bounds
LemmaDependency level 1
Zero stationary energy along null geodesics
LemmaDependency level 1