| import OAI.Geometry.EinsteinFour.GeodesicGeometry12 |
| import OAI.Geometry.EinsteinFour.GeodesicGeometry14 |
|
|
| namespace OAI |
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| /-! Actual half-conformally-flat model identification. All analytic, |
| index, injectivity, and metric-model inputs are supplied by proved geometry. -/ |
| namespace PositiveEinsteinFour.Geometry |
| open EinsteinFour |
| open scoped Manifold ContDiff |
| noncomputable section |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] [T3Space M] [CompactSpace M] |
| [MeasurableSpace M] [BorelSpace M] [Nonempty M] |
| [SecondCountableTopology M] [PreconnectedSpace M] |
| {g : SmoothMetric M} {o : FrameOrientation g} |
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| theorem actual_halfFlat_positive_harmonic_dimension_one (hRic : RicciThree g) |
| (hpositive : HasPositiveSectionalCurvature g) |
| (hminus : ∀ x, minusWeyl g x (o.frame x) = 0) |
| (hplus : ∃ x, plusWeyl g x (o.frame x) ≠ 0) : |
| Module.finrank ℝ (smoothHarmonicPositive (o := o) hRic) = 1 := |
| halfFlat_positive_harmonic_dimension_one_of_injective_ball hRic hpositive hminus hplus |
| (actual_frameExponential_injOn_sqrt_three g o hRic hpositive) |
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| theorem actual_halfFlat_models (hRic : RicciThree g) |
| (hpositive : HasPositiveSectionalCurvature g) |
| (hminus : ∀ x, minusWeyl g x (o.frame x) = 0) : |
| ∃ a : ℝ, 0 < a ∧ (ScaledIsometricTo g a roundDistance ∨ |
| ScaledIsometricTo g a fubiniStudyDistance) := by |
| by_cases hp : ∀ x, plusWeyl g x (o.frame x) = 0 |
| · exact ⟨1,by norm_num,Or.inl (actual_round_model hRic hpositive hp hminus)⟩ |
| · have hn : ∃ x, plusWeyl g x (o.frame x) ≠ 0 := not_forall.mp hp |
| exact ⟨1/2,by norm_num,Or.inr (actual_complex_projective_model hRic hminus |
| (actual_halfFlat_positive_harmonic_dimension_one hRic hpositive hminus hn))⟩ |
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| variable [ConnectedSpace M] |
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| omit [CompactSpace M] [SecondCountableTopology M] [ConnectedSpace M] [PreconnectedSpace M] in |
| theorem actual_derdzinski_hitchin (g : SmoothMetric M) (o : FrameOrientation g) |
| (hRic : RicciThree g) : DerdzinskiHitchinInput g o := by |
| intro _ _ _ _ _ hpositive hblock |
| rcases hblock with hplus | hminus |
| · have hminus : ∀ x, minusWeyl g x (o.reversed.frame x) = 0 := by |
| intro x |
| apply (minusNorm_eq_zero_iff o.reversed x).mp |
| rw [minusNorm_reversed] |
| exact (plusNorm_eq_zero_iff o x).mpr (hplus x) |
| exact actual_halfFlat_models hRic hpositive hminus |
| · exact actual_halfFlat_models hRic hpositive hminus |
|
|
| end |
| end PositiveEinsteinFour.Geometry |
|
|
| namespace PositiveEinsteinFour |
| open EinsteinFour _root_.OAI.PositiveEinsteinFour.Geometry |
| open scoped Manifold ContDiff |
| noncomputable section |
| universe u |
|
|
| /-- Every connected closed oriented positively curved Einstein four-manifold is, |
| up to positive scaling and isometry, the round S4 or Fubini–Study CP2. -/ |
| theorem orientedClassification (X : ClosedOrientedMetricFour.{u}) |
| (hEinstein : IsEinstein X.metric) (hpositive : HasPositiveSectionalCurvature X.metric) : |
| ∃ a : ℝ, 0 < a ∧ (ScaledIsometricTo X.metric a roundDistance ∨ |
| ScaledIsometricTo X.metric a fubiniStudyDistance) := by |
| let : MeasurableSpace X.manifold := borel X.manifold |
| let : BorelSpace X.manifold := ⟨rfl⟩ |
| exact models_of_actual_model_inputs X.metric X.orientation |
| actual_derdzinski_hitchin hEinstein hpositive |
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|
| variable (M : Type u) [TopologicalSpace M] [ChartedSpace FourSpace M] |
| [IsManifold I4 ∞ M] [T2Space M] [SecondCountableTopology M] |
| [CompactSpace M] [ConnectedSpace M] |
|
|
| /-- Every closed connected positively curved Einstein four-manifold is, |
| up to positive scaling and isometry, standard S4, CP2, or RP4. -/ |
| theorem classificationOn : ClassificationOn M := by |
| apply classificationOn_of_model_inputs |
| refine ⟨?_⟩ |
| intro X |
| let : MeasurableSpace X.manifold := borel X.manifold |
| let : BorelSpace X.manifold := ⟨rfl⟩ |
| intro g o hRic |
| exact actual_derdzinski_hitchin g o hRic |
|
|
| end |
| end PositiveEinsteinFour |
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| end OAI |
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|
