| import OAI.Analysis.CircleDomains.Topology.FiniteJordanModelInput |
| import OAI.Analysis.CircleDomains.Modulus.NormalizedUniformization |
|
|
| namespace OAI |
|
|
| /-! Circle-domain uniformization from the stated geometric, analytic, |
| and probability hypotheses for local and finite models. -/ |
| open Set Function Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
| open FiniteTransfer |
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| variable [MeasurableSpace C(PathTime, Sphere)] [BorelSpace C(PathTime, Sphere)] |
|
|
| theorem koebeCircleDomainConjecture_of_constructed_quotient |
| (hcross : UnitDiskAlternatingContinua) |
| (hplane : PlanarLipschitzFiberCoarea) |
| (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability) |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) (hsep : UnitDiskCrosscutSeparation) |
| (hcollar : SmoothRegularJordanCollars) (hselect : BorelPathRelationSelection) |
| (hgreen : UnorientedFiniteCircleGreenFormula) (harea : ExteriorCoefficientAreaTheorem) |
| (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization) |
| (hmontel : MontelExhaustion) (hhurwitz : HurwitzUnivalentLimit) (hzero : HurwitzZeroFreeLimit) |
| (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C → |
| HasSimpleArcsInConnectedOpens C) |
| (hstandard : ∀ e : PathCodeData Sphere Sphere, [StandardBorelSpace (BasePathCode e)] → |
| StandardBorelProbabilityLaws (BasePathCode e)) |
| (hpath : PolishProbabilityLaws C(PathTime, Sphere)) |
| (hdual : ∀ (e : PathCodeData Sphere Sphere) |
| (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] |
| [StandardBorelSpace (ConstrainedCostCode e b c ν)], |
| MeasureImageDuality (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I)) |
| (hlift : ∀ (e : PathCodeData Sphere Sphere) |
| (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] |
| [StandardBorelSpace (ConstrainedCostCode e b c ν)], |
| MeasurePlanLifting (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I)) |
| : KoebeCircleDomainConjecture := by |
| intro U hU hconn |
| obtain ⟨x, hx⟩ := hconn.nonempty |
| let N := pointToInfinity x |
| have hG : IsOpen (N '' U) := N.isOpenMap U hU |
| have hGconn : IsConnected (N '' U) := hconn.image _ N.continuous.continuousOn |
| have hinf : OnePoint.infty ∈ N '' U := ⟨x, hx, pointToInfinity_apply_self x⟩ |
| have hN := isConformalEquivalence_pointToInfinity x U |
| by_cases hwhole : N '' U = univ |
| · exact ⟨N '' U, N, hwhole ▸ isCircleDomain_univ, hN⟩ |
| · obtain ⟨V, f, hV, hf⟩ := circle_uniformization_of_infty_mem_of_shrinking hcross hplane |
| hcoarea hLusin hsard hsch hsep hcollar hselect hgreen harea hcompact hfinite |
| hmontel hhurwitz hzero harcs hstandard hpath hdual hlift hG hGconn hinf hwhole |
| exact ⟨V, f ∘ N, hV, hN.trans hf⟩ |
|
|
|
|
| theorem koebeCircleDomainConjecture_of_constructed_contours |
| (_hmoore : MooreSphereDecomposition) |
| (hcross : UnitDiskAlternatingContinua) |
| (hplane : PlanarLipschitzFiberCoarea) |
| (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability) |
| (hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies) (hsep : UnitDiskCrosscutSeparation) |
| (hcollar : SmoothRegularJordanCollars) (hselect : BorelPathRelationSelection) |
| (hgreen : UnorientedFiniteCircleGreenFormula) (harea : ExteriorCoefficientAreaTheorem) |
| (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization) |
| (hmontel : MontelExhaustion) (hhurwitz : HurwitzUnivalentLimit) (hzero : HurwitzZeroFreeLimit) |
| (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C → |
| HasSimpleArcsInConnectedOpens C) |
| (hstandard : ∀ e : PathCodeData Sphere Sphere, [StandardBorelSpace (BasePathCode e)] → |
| StandardBorelProbabilityLaws (BasePathCode e)) |
| (hpath : PolishProbabilityLaws C(PathTime, Sphere)) |
| (hdual : ∀ (e : PathCodeData Sphere Sphere) |
| (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] |
| [StandardBorelSpace (ConstrainedCostCode e b c ν)], |
| MeasureImageDuality (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I)) |
| (hlift : ∀ (e : PathCodeData Sphere Sphere) |
| (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] |
| [StandardBorelSpace (ConstrainedCostCode e b c ν)], |
| MeasurePlanLifting (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I)) |
| : KoebeCircleDomainConjecture := by |
| exact koebeCircleDomainConjecture_of_constructed_quotient hcross hplane hcoarea hLusin hsard hsch hsep hcollar hselect hgreen harea hcompact hfinite hmontel hhurwitz hzero harcs hstandard hpath hdual hlift |
|
|
| theorem koebeCircleDomainConjecture_of_published_inputs |
| (hmoore : MooreSphereDecomposition) |
| (hcross : UnitDiskAlternatingContinua) |
| (hplane : PlanarLipschitzFiberCoarea) |
| (hcoarea : WeightedPlanarScalarCoarea) (hLusin : FiniteTorusAnalyticMeasurability) |
| (hsard : SmoothPlanarSard) (hclass : CompactOneManifoldClassification) |
| (hsch : PlanarSchoenflies) (hsep : UnitDiskCrosscutSeparation) |
| (hcollar : SmoothRegularJordanCollars) (hselect : BorelPathRelationSelection) |
| (hgreen : UnorientedFiniteCircleGreenFormula) (harea : ExteriorCoefficientAreaTheorem) |
| (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteCircleUniformization) |
| (hmontel : MontelExhaustion) (hhurwitz : HurwitzUnivalentLimit) (hzero : HurwitzZeroFreeLimit) |
| (harcs : ∀ C : Set Sphere, IsCompact C → LocallyConnectedSpace C → |
| HasSimpleArcsInConnectedOpens C) |
| (hstandard : ∀ e : PathCodeData Sphere Sphere, [StandardBorelSpace (BasePathCode e)] → |
| StandardBorelProbabilityLaws (BasePathCode e)) |
| (hpath : PolishProbabilityLaws C(PathTime, Sphere)) |
| (hdual : ∀ (e : PathCodeData Sphere Sphere) |
| (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] |
| [StandardBorelSpace (ConstrainedCostCode e b c ν)], |
| MeasureImageDuality (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I)) |
| (hlift : ∀ (e : PathCodeData Sphere Sphere) |
| (I : Set ℕ) [PolishSpace I] (b : I → Sphere) (c : I → ℝ≥0∞) |
| (ν : Measure Sphere) [IsFiniteMeasure ν] |
| [StandardBorelSpace (ConstrainedCostCode e b c ν)], |
| MeasurePlanLifting (ConstrainedCostCode e b c ν) (PathCostSpace Sphere I)) |
| : KoebeCircleDomainConjecture := by |
| have _compatibilityInput : CompactOneManifoldClassification := hclass |
| clear hclass _compatibilityInput |
| exact koebeCircleDomainConjecture_of_constructed_contours hmoore hcross hplane hcoarea hLusin hsard hsch hsep hcollar hselect hgreen harea hcompact (finiteCircleUniformization_jordanModels hfinite) hmontel hhurwitz hzero harcs hstandard hpath hdual hlift |
|
|
| end CircleDomainRigidity |
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| end |
|
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| end OAI |
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