Alphabeta MathSupporting prerequisites · 02 — Local compactness of full weighted Hölder inputs
Presents 21 supporting prerequisites, covering dependency levels 2–5.
1 · Prerequisites
- The Axiom of Choice
- Weighted Hölder norms on a three-dimensional exterior
- Weighted exterior elliptic domains and admissible weights
- Arzelà--Ascoli for real C(K) under Countable Choice and Dependent Choice: compact closure iff equicontinuous and pointwise bounded
- The mean value inequality: if f : [a,b] → ℝᵐ is continuous and differentiable on (a,b) with ‖ f'‖₂ ≤ M, then ‖ f(b)-f(a)‖₂ ≤ M(b-a)
- If continuously differentiable functions converge at one point and their derivatives converge uniformly on a closed interval, then the functions converge uniformly to a differentiable function whose derivative is the derivative limit
- 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
- 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
- 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)
- All-jet reflection parity from an exact local boost
- Spacelike sections, maximality and lapse–shift data
- Metric compatibility, torsion freedom and the metric-selected connection class
- Sum, product, quotient, and chain rules from difference quotients
- The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)
- Normal graphs and the nonlinear mean-curvature map
- Ellipticity and zero-slope forcing of the stationary maximal flux
- One extra pointwise derivative supplies scaled Hölder control
- Differentiated asymptotic bounds and parameter uniformity
- Hölder spaces C^k,α, closure and interior scaled norms, and C^k,α domains
- The derivative of the metric density and the chart formula for the divergence of a vector field
- Matrix inversion preserves Cᵏ regularity where the determinant is nonzero
- If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative
- Continuous mixed partials of order k are invariant under permutations
- Cotangent space and cotangent bundle as a disjoint union
- Tangent and cotangent bundles extend over a boundary
- Relative stationary deformation and extension obstruction
- The bracket of two Killing fields is Killing
- Causal exterior, chronological boundaries and Cauchy data
- Openness of the chronological future and past, openness of the chronological relation, and closure of future and past sets
- Stationary Lorentzian data and tensor conventions
- Diffeomorphisms and local diffeomorphisms of manifolds
- Local and global flows generated by a vector field
- Chart independence, concatenation, reparametrization and transitivity for causal curves
- The timelike set has exactly two convex components, described in orthonormal coordinates
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- A subset of ℝ is compact if and only if it is closed and bounded
- Extreme value theorem: a continuous real function on a nonempty compact subset of ℝ attains a greatest and a least value
- Asymptotic mass and axial angular-momentum conventions
- Riemannian volume density
- Integral of a compactly supported smooth density
- Hamed Barzegar, Piotr T. Chruściel and Michael Hörzinger, Energy in higher-dimensional spacetimes · Section II (II.1)–(II.4), printed p.3; Section III A (III.13), p.5, (III.18)–(III.19), p.6; (III.34), p.7; Section IV B (IV.16)–(IV.20), pp.12–13, (IV.29)–(IV.30), p.16; Section V (V.1)–(V.5), pp.19–20; specialize to K=0, n=3, flat background and Ric=0.
- Constant homothety preserves the original vacuum and maximal section
- Stationary end normalization from quotient compactification
- Extending normalized end coordinates while preserving the section
- Mass-one convention and Harrison denominator
- Properties of normal coordinates at the center
- R. Beig and B. Schmidt, Time-Independent Gravitational Fields · §2.1 (2.33)–(2.34), printed p.7; §2.3 (2.72), p.11; §3.1 (3.5), (3.15)–(3.17), (3.22)–(3.25), pp.17–20
- S. Dain, Initial data for stationary space-times near space-like infinity · §2, (17)–(19), Theorem 2.1 and following normal-coordinate paragraph, printed p.4; Lemma 2.4 and (21)–(22), pp.5–6
- Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form
- The double of a smooth manifold with boundary
- The double has a well-defined smooth structure
- Smooth collars of a manifold boundary
- Riemannian metric and riemannian manifold
- The pullback of a covariant tensor field
- Fundamental theorem of riemannian geometry
- Christoffel formula for the levi civita connection
- Coordinate formula for the curvature tensor
- Scalar curvature
- Cᵏ maps and multi-index derivative notation in Euclidean space
- Smoothness of a tensor field is equivalent to smooth coordinate components
- Taylor's theorem with an integral remainder
- 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 selected exterior is connected and causally convex
- Regular non-degenerate stationary vacuum spacetime
- The push-up rule: p << q and q <= r imply p << r, together with the timelike-piece criterion
- A tensor field is flow-invariant exactly when its Lie derivative vanishes
- Cauchy sets are closed, split the manifold into past, set and future, and are locally Lipschitz graphs
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
- The connected subspaces of ℝ with its usual topology are exactly the order-convex subsets, the published characterisation transported by the identification of the two descriptions of "open in ℝ"
- A continuous image of a connected space is connected, and connectedness is a topological property
- A specified future ingoing Kerr extension and its time orientation
- Null generators and once-only cross-sections
- The Axiom of Countable Choice (AC_ω)
- Saul A. Teukolsky, The Kerr Metric (arXiv:1410.2130v2) · §2, equations (2.1), (2.5)–(2.9), PDF pp. 2–3
- The exterior Cauchy component has a complete asymptotically Euclidean closure
- Null expansions, screen quotients and focal data
- Local null normal pairs and the positive screen quotient
- Stationary maximal hypersurface existence with a horizon boundary
- Integer-order Sobolev spaces and their norms
- Killing fields of a Lorentzian metric
- Robert Beig and Piotr T. Chruściel, Killing Initial Data · Section 2, printed pp. 3–4 (PDF pages 3–4), equations (2.3)–(2.13), especially (2.6), (2.10), (2.13).
- Riemannian volume is the radon measure of the riemannian density
- The nonnegative Lebesgue integral
- The integral of a nonnegative simple function
- A maximal stationary exterior has future access to its far tail
- Causal curves, chronological and causal reachability, and inextendibility
- A decaying stationary time gauge gives bounded section displacement
- Radial primitive and five differentiated decay bounds
- Ernst sublevels and the Mars–Simon tensor interface
- Stationary lifting of a spatial primitive
- Vacuum Killing and Ernst differential identities
- Real and imaginary parts, complex conjugation, and modulus
- Christoffel symbol transformation law
- The pullback of a differential form
- The exterior derivative commutes with pullback
- The exterior derivative squares to zero
- Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model
- Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields
- The inverse metric, musical isomorphisms and index raising
- Oriented smooth manifolds and oriented charts
- Pullback of a riemannian metric as a tensor
- A smooth differential k-form
- Closed and exact differential forms
- Interior product of a form by a vector field
- The exterior derivative by the invariant vector-field formula
- The wedge product of differential forms
- The Lie derivative of a tensor field
- Complex self-dual forms and metric-dependent contraction
- 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
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
Lemma
Local compactness of full weighted Hölder inputs
Lemma
Weighted product and uniform potential tails on the full input space
Lemma
Reflection parities of canonical section data and stationary lapse–shift
Lemma
Stationary maximal graphs have an explicit uniformly spacelike flux
Lemma
Weighted differentiation and multiplication estimates
Lemma
Uniqueness forces an extended axial field to commute
Lemma
Timelike transfer supplies access to every sufficiently distant stationary cylinder
Theorem
External ADM–Komar equality on ordinary stationary vacuum ends
Lemma
Mass-one end coordinates on the unchanged maximal section
Lemma
Smooth vacuum constraint data obtained by reflection of a canonical end section
Lemma
Smooth positive inversion and composition in the weighted multiplier class
Lemma
Stationary end access and future orientation need no maximal slice
Lemma
The outer Kerr null hypersurface and its future generator
Lemma
Zero boundary expansions without maximality
Lemma
Supplied CW logarithmic maximal ends have weak weighted initial data
Lemma
Extrinsic curvature decay on the original stationary end
Lemma
Smooth reflection of the stationary vacuum Killing initial data
Lemma
Vacuum exterior-section constraints imply the dominant energy condition
Lemma
Past access to the far tail by exact time reversal
Lemma
A smooth radial primitive can be returned to a weak end gauge
Lemma