Alphabeta MathArticle items · 18 — Asymptotically flat positive-q ends and cylindrical limit
Presents 28 items from the article and its immediate prerequisites, covering dependency levels 10–19.
1 · Prerequisites
- Uniform transformed component estimates
- Uniform denominator and one-form estimates
- Mass-one convention and Harrison denominator
- Single-potential construction of the Harrison one-form family
- Regular non-degenerate stationary vacuum spacetime
- Stationary vacuum model and asymptotic normalization
- Differentiated asymptotic bounds and parameter uniformity
- Round sphere model geometry
- Surface area and flux on a sphere, with scalar integrals on a hemisphere
- Riemannian volume density
- Riemannian volume is the radon measure of the riemannian density
- Riemannian speed and length
- Riemannian distance on a connected manifold
- A specified asymptotically Euclidean ADM end
- The pullback of a covariant tensor field
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Sum, product, quotient, and chain rules from difference quotients
- A closed three-dimensional ball of radius r≥0 has volume 4π r³/3
- The Axiom of Choice
- Stationary Harrison one-form and metric
- Killing two-form and normalized Ernst notation
- The exterior derivative by the invariant vector-field formula
- Closedness, periods and smooth global choice of W0
- Mass-one Ernst and radial primitive asymptotics
- Coordinate components of the transformed metric
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Flattening a smooth fixed boundary preserves the elliptic graph equation
- Tangential differentiated equations admit classical difference quotients
- Boundary Schauder applied to tangential quotients gains one derivative
- Ellipticity recovers every remaining normal derivative
- Interior Schauder estimate for uniformly elliptic equations
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- 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
- AC implies DC implies countable choice
- Normal graphs and the nonlinear mean-curvature map
- Local and global flows generated by a vector field
- 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
- Smooth charts, atlases, and structures with boundary
- Smooth maps between manifolds with boundary
- Immersions and embeddings for manifolds with boundary
- Embedded smooth submanifolds with boundary
- A fixed smooth-domain extension that preserves smooth parameters
- 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)
- Continuous mixed partials of order k are invariant under permutations
- Explicit compactly supported smooth cutoffs
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- The Euclidean implicit function theorem with derivative formula
- 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
- Stationary maximal graphs and normalized graph heights
- Proposition 2.1: Einstein–Maxwell equations and regularity
- Harrison electromagnetic form
- Energy-condition inputs for the transformed geometry
- One extra pointwise derivative supplies scaled Hölder control
- The maximal equation upgrades pointwise order-four graph decay to weighted order five
- Induced metric and graph normal inherit the required weighted regularity
- The graph second form and electrovacuum Jacobi potential have weighted order four
- The transverse lapse is a positive weighted Dirichlet multiplier
- A timelike transverse field gives a fixed-boundary admissible graph neighborhood
- Stationary parameter families have a decaying reference-slice residual
- The global Dirichlet mean-curvature map is continuously differentiable
- Weighted Hölder norms on a three-dimensional exterior
- Weighted exterior elliptic domains and admissible weights
- Mean-curvature variation and the stationary lapse equation
- Spacelike sections, maximality and lapse–shift data
- Horizon expansion variation and Einstein–Maxwell curvature
- Horizon null frame, flux densities and stability operator notation
- Surface measure, flux densities and null normal gauge
- Principal eigenfunctions and MOTS spectral stability
- Admissible domains for outward normal graphs
- Canonical outward normals and null gauge on a graph
- Outgoing expansion depends smoothly on the second graph data
- A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value
- Lemma 3.3: common spacelike reference collar
- Local Hawking and axial fields, boost and pole
- Preservation of bifurcate horizons and surface gravity
- Normal boost action and invariant two-forms at a fixed surface
- Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor
- Through each point there is a unique maximal integral curve
- The fundamental theorem on flows
- Stationary Lorentzian data and tensor conventions
- Exterior Schrödinger maximum principles, Hopf growth and Harnack
- If det(A) is a unit, then A⁻¹=det(A)⁻¹adj(A)
- Invertibility, Lorentz signature and stationary norm
- Normalized outward null normal at large coordinate spheres
- Null expansions, screen quotients and focal data
- The determinant differential is D det(A)[H]=tr(adj(A)H) at every matrix, and Jacobi's formula holds on the invertible locus
- Uniform annular Schauder bootstrap for spacelike graph equations
- Linear fourth derivative bound with controlled realized coefficients
- Semicontinuous extreme value theorem on compact Euclidean sets
- The Hessian is negative semidefinite at an interior local maximum
- Trace of a product of positive semidefinite matrices is nonnegative
- Remark 1.3: lapse bounds and stationary product
- Collar neighborhood theorem
- 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 product of finitely many compact spaces is compact in the product topology
- 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 closed subspace of a compact space is compact, and a finite union of compact subspaces is compact
- 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
- 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
- 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
- Rigidity of stationary black holes with small ADM angular momentum · Section 5.1, equations (5.4)–(5.5); Section 5.2, equations (5.15), (5.22); Section 5.3, Lemma 5.1 and equation (5.24)
- Comparison spheres and their stationary cylinders
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- The inverse metric, musical isomorphisms and index raising
- The future unit normal, induced metric and second fundamental form of a spacelike hypersurface
- A compact spacelike surface has a normal exponential tube
- Circle-invariant normal coordinates linearize a bifurcate Killing boost
- The parametrized implicit function theorem with Cᵏ regularity
- Newton–Leibniz needs only continuity on [a,b], differentiability on (a,b), and a Riemann-integrable extension of the interior derivative
- A continuous function on [a,b] is Riemann integrable, by Heine-Cantor and Riemann's criterion
- 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
- Tube lemma: if K is compact and an open N ⊆ X × Z contains K × {z₀}, then N contains K × W for some open W ∋ z₀
- Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on [a,b] takes every value between f(a) and f(b)
- Smooth transverse horizon branches are the null axes of the normal tube
- A smooth spacelike reference section is a bounded rapidity graph
- The stationary product extends the local boost and circle to a complete collar
- Weighted Hölder spaces and their Dirichlet kernels are Banach
- A proper Euclidean local diffeomorphism has finite diffeomorphic sheets near every target point
- Bifurcate Killing horizons, rotation and normalization
- An existing stationary graph supplies global projection before any estimate
- Stationary coefficients in a transverse collar retain the tangential motion
- Uniform stationary return persists under arbitrary tangential drift
- Collar comparison and an exact end projection imply properness
- A proper stationary projection with the correct end has one sheet
- Compact metric families have a common spacelike collar with fixed boost rate
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- Diffeomorphisms and local diffeomorphisms of manifolds
- The smooth inverse function theorem on manifolds
- Causal exterior, chronological boundaries and Cauchy data
- Stationary transfer of timelike curves
- A maximal stationary exterior has future access to its far tail
- Sufficiently far stationary end points can be routed to a chosen cylinder
- An open chronological exterior is causally convex without a timelike-end assumption
- Piecewise timelike transfer yields arrival on the cylinder under far-tail access
- Future timelike cylinder translations make their chronological pasts increase
- Pointwise eventual chronological entry is uniform on compact subsets
- Closed Ernst form and existence of normalized potential
- Transverse embedded submanifolds intersect in the expected codimension
- Lemma 3.1: weighted Dirichlet Laplacian isomorphism
- Implicit function theorem for Banach spaces
- Fredholm and Banach continuation interfaces
- The derivative of the metric density and the chart formula for the divergence of a vector field
- The fundamental theorem of calculus and integration by parts
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Existence of strictly convex normal neighborhoods: small coordinate balls are normal neighborhoods of each of their points
- 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
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- The joint exponential map: smoothness, invertible differential at the diagonal, and the small-velocity two-point inverse
- The geodesic initial-value problem and the exponential map of the metric-selected connection
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- 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
- Every open cover of a compact metric space has a Lebesgue number: a δ > 0 such that every nonempty subset of diameter less than δ lies inside a single member of the cover
- Strongly causal complete stationary ambient gluing around the exterior
- Canonical bifurcation collar joined to the original exterior Cauchy section
- The exterior is the interior domain of dependence in the glued ambient
- Differentiated stationary end bounds imply the weighted Hölder interface
- Stationary maximal hypersurface existence with a horizon boundary
- Closure of the exterior Cauchy section meets only the double horizon
- Convergent causal ends are extendible, maximal geodesics have no limit, and Cauchy sets are acausal
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- Temporal functions are time functions, and time functions forbid closed causal curves
- Equivalence of the absolutely-continuous and continuous causal conventions, and one-sided inextendible extensions
- Embedded submanifolds and slice charts
- E. Minguzzi, Lorentzian causality theory, Living Reviews in Relativity 22 (2019), article 3 · Section 3.2, Definition 3.29, Remark 3.31, Proposition 3.33 (3.8), and Corollary 3.34, printed/PDF pp. 93–95.
- C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation
- A C¹ map is locally Lipschitz on compact coordinate subsets
- Fundamental theorem of calculus for absolutely continuous functions
- Dominated convergence
- Causal curves, chronological and causal reachability, and inextendibility
- Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold
- Second-order contact on a closed half-domain has nonnegative Hessian
- Mean curvature at identical spacelike graph jets has positive Hessian symbol
- Equal fixed-surface flow heights imply boundary tangency
- Ordered timelike flow graphs lie on the corresponding geometric side
- Strict mean-curvature barriers cannot have ordered boundary tangency
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- A variable-time flow through a fixed boundary has the boosted unit normal
- Bounded boost times give a uniform Gaussian collar cone
- Spacelike normal projection is locally invertible with positive boundary jet
- Radial comparison gives proper normal projection onto a fixed target collar
- A one-sheet boundary germ propagates across the common normal collar
- 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
- Logarithmic angle inequality through the full range v at least one
- Invariant Ernst data and smooth K-invariant gauge
- 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
- Local existence, uniqueness, and smooth dependence for manifold integral curves
- Compact timelike-length bounds and local temporal functions
- Riemannian metric and riemannian manifold
- Assuming countable choice, every smooth manifold admits a Riemannian metric
- Local comparison of a riemannian metric with the euclidean metric
- A Lipschitz map ℝᵐ→ℝᵐ sends null sets to null sets
- The same maximal slice has the differentiated article end gauge and a limiting old time
- 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
- A complete timelike collar flow is a reference product
- Positivity of the Harrison denominator and horizon bound
- Rigidity of stationary black holes with small ADM angular momentum · Section 5.1, equations (5.4)–(5.5); Section 5.2, equations (5.15), (5.22); Section 5.3, Lemma 5.1 and equation (5.24)
- Boost-frame bounds for a smooth observer and arbitrary spacelike normals
- Pointwise Gauss–Codazzi identities for the normal angle
- Trace-free three-dimensional coercivity for the logarithmic gradient
- Uniform scalar angle inequalities from local background tensors
- The Riemannian gradient is the metric dual of the differential
- Laplace–Beltrami operator as the trace of the Hessian
- Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum, arXiv:2610.09729v1 · Appendix B, Lemma B.1, printed/PDF pp.35–36 (hypotheses); proof pp.38–39, (B.8)–(B.11)
- 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
- Bolzano-Weierstrass: every bounded real sequence has a convergent subsequence
- Rigidity of stationary black holes with small ADM angular momentum · Section 5.1, equations (5.4)–(5.5); Section 5.2, equations (5.15), (5.22); Section 5.3, Lemma 5.1 and equation (5.24)
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 10
Asymptotically flat positive-q ends and cylindrical limit
LemmaDependency level 11
Classical C4-alpha maximal graphs with fixed smooth boundary are smooth
Physics theoremDependency level 11
Mean-curvature linearization, weighted regularity and fixed-boundary graph map
Physics theoremDependency level 11
Negative stability eigenvalue produces outer trapped graphs
Physics theoremDependency level 11
Normal boost action of the Hawking flow
Physics theoremDependency level 11
Transformed lapse equation and bounds
Physics theoremDependency level 11
Uniform coordinate-sphere area and null expansion
Physics theoremDependency level 12
Uniform end estimates indexed only by existing admissible slices
Physics theoremDependency level 12
Proposition 4.1: positive-expansion spheres of limiting area 4π
Physics theoremDependency level 12
Uniform asymptotic slope bound
Physics theoremDependency level 12
Existence and boost-coordinate description of Hawking-flow heights
Physics theoremDependency level 12
Proper stationary projection for continued slices
Physics theoremDependency level 13
Eventual chronological access to translated comparison spheres
Physics theoremDependency level 13
Openness via the weighted implicit-function branch and smooth boundary bootstrap
Physics theoremDependency level 13
Asymptotic maximal graph equation and coefficients
Physics theoremDependency level 14
Radial barrier, absorption and height decay
Physics theoremDependency level 15
Rescaled annular estimates through four derivatives
Physics theoremDependency level 15
Lemma 3.2: compact stationary-height bounds
LemmaDependency level 16
Maximal bifurcation-boundary section under the original regular hypotheses
Physics theoremDependency level 16
Compact-tube timelike length bound
Physics theoremDependency level 16
K-flow barriers with boundary tangency comparison and uniform height bound
Physics theoremDependency level 17
Indexed compact heights and uniform smooth horizon collars
Physics theoremDependency level 17
Boundary angle bound and normal-graph cone
Physics theoremDependency level 17
Local causality and temporal functions on compact tubes
Physics theoremDependency level 17
Remark 1.1: maximal slice and decay
Physics theoremDependency level 18
Compact estimates for arbitrary sequences of individually existing slices
Physics theoremDependency level 18
Angle differential inequality near S
Physics theoremDependency level 19