Theorems

The theorem cards below are fast hypothesis-to-conclusion maps. They identify the exact object or initial data, the geometric and analytic assumptions, the logical type of the conclusion, and the boundary beyond which the result says nothing. Follow the linked proof page whenever a theorem is carrying weight in an argument; the cards are designed to prevent a wrong theorem from being applied, not to replace its proof.

Choose the theorem by its conclusion

Result	Starting object	Essential control	Conclusion	Not guaranteed
Hawking–Penrose Singularity Theorems	A trapped surface, an expanding Cauchy hypersurface, or one of the combined theorem’s trapped alternatives	The variant-specific convergence, causality, compactness, genericity, regularity, and global hypotheses	Existence of an incomplete causal geodesic, with causal type and time direction fixed only when the selected variant fixes them	Curvature blowup, a singular point, an event horizon, or inextendibility
Black-Hole Area Theorem	Ordered spacelike cross-sections of a classical future event horizon	Null convergence, horizon regularity, and a stated global continuation or predictability package	The later cross-section has no smaller area	A local flux law, an apparent-horizon theorem, or a generalized-entropy law
Black-Hole Rigidity and Uniqueness	A static or stationary black-hole exterior in one precisely stated theorem class	Dimension, asymptotic structure, matter model, global regularity, horizon connectedness and degeneracy, plus analyticity, bifurcation, or near-Kerr control where required	Schwarzschild or Kerr–Newman classification, or a local symmetry, only for the selected row	A dimension-blind no-hair theorem, unrestricted smooth uniqueness, or classification with arbitrary matter or AdS asymptotics
Positive Mass Theorems	A complete asymptotically flat Riemannian manifold or Einstein initial-data set in a theorem-specific class	Dimension, decay and regularity, scalar-curvature or dominant-energy condition, admissible boundary, and spin where the proof requires it	Nonnegative ADM mass or future-causal ADM energy–momentum, with only the row’s stated equality result	Charged, asymptotically AdS, Bondi, quasi-local, or horizon-boundary refinements
Existence and Uniqueness of the Maximal Globally Hyperbolic Development	Constraint-satisfying Einstein initial data in a stated regularity class	Local well-posedness, gauge and constraint propagation, and geometric uniqueness	A universal globally hyperbolic development, unique up to an isometry fixing the data	Geodesic completeness or inextendibility among arbitrary spacetime extensions

How to use a theorem card

Match hypotheses before matching conclusions. In particular, keep a geometric Ricci-convergence condition distinct from a matter energy condition unless the field equation and theory have been specified. Check dimension, differentiability, boundary conditions, asymptotics, time orientation, and the meaning of “maximal” or “horizon” before importing the result into a new problem.

Counterexamples and limiting cases are diagnostic. Compact de Sitter slices show why a Penrose variant needs its noncompact-Cauchy hypothesis; quantum stress energy explains why a classical event-horizon area theorem cannot be relabelled a semiclassical law; black rings and scalar-haired black holes block dimension-blind or matter-blind uniqueness claims; and charged or AdS data need more than an ordinary asymptotically flat positive-mass theorem. Extensions beyond a Cauchy horizon likewise show why maximal globally hyperbolic development is not absolute inextendibility. The linked Library treatments give the complete proof architecture and the exact scope of each check.