openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /FiniteJordanImageUniformization.lean
| import OAI.Analysis.CircleDomains.Topology.FiniteJordanUniformization | |
| import OAI.Analysis.CircleDomains.Topology.PlaneHomeomorphFromSphere | |
| namespace OAI | |
| /-! Finite Jordan uniformization also applies after an arbitrary sphere | |
| homeomorphism. A conformal normalization makes its image of infinity the | |
| point at infinity, leaving an actual plane homeomorphism of all caps. -/ | |
| open Set Function | |
| open scoped Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| lemma spherePlaneHomeomorph_onePointCongr (e : Sphere ≃ₜ Sphere) | |
| (he : e OnePoint.infty=OnePoint.infty) : | |
| (spherePlaneHomeomorph e he).onePointCongr=e := by | |
| ext p | |
| cases p using OnePoint.rec with | |
| | infty => exact he.symm | |
| | coe z => exact coe_spherePlaneMap e he z | |
| lemma trans_closed_jordan_image (H P : ℂ ≃ₜ ℂ) : | |
| closure (jordanInterior (H.trans P))=P '' closure (jordanInterior H) := by | |
| change closure ((P ∘ H) '' _) = P '' closure (H '' _) | |
| rw [image_comp,P.image_closure] | |
| lemma exists_finite_jordan_image_uniformization {I : Type*} [Finite I] | |
| (C : FiniteJordanConfiguration I) (e : Sphere ≃ₜ Sphere) : | |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), | |
| IsCircleDomain V ∧ IsConformalEquivalence f (e '' C.region) V := by | |
| let N : Sphere ≃ₜ Sphere := pointToInfinity (e OnePoint.infty) | |
| let E : Sphere ≃ₜ Sphere := e.trans N | |
| have hE : E OnePoint.infty=OnePoint.infty := by | |
| exact pointToInfinity_apply_self (e OnePoint.infty) | |
| let P : ℂ ≃ₜ ℂ := spherePlaneHomeomorph E hE | |
| have hP : P.onePointCongr=E := spherePlaneHomeomorph_onePointCongr E hE | |
| let C' : FiniteJordanConfiguration I := { | |
| hole := fun i => (C.hole i).trans P | |
| disjoint := by | |
| intro i j hij | |
| change Disjoint (closure (jordanInterior ((C.hole i).trans P))) | |
| (closure (jordanInterior ((C.hole j).trans P))) | |
| rw [trans_closed_jordan_image,trans_closed_jordan_image] | |
| exact (C.disjoint hij).image P.injective.injOn (subset_univ _) (subset_univ _) } | |
| have hcap (i : I) : E '' closedTerminalCap (C.hole i)=closedTerminalCap (C'.hole i) := by | |
| rw [←hP,closedTerminalCap,CannonSphereReflection.onePoint_image] | |
| change ((↑) : ℂ → Sphere) '' (P '' closure (jordanInterior (C.hole i)))=_ | |
| rw [←trans_closed_jordan_image] | |
| rfl | |
| have hreg : C'.region=E '' C.region := by | |
| change (⋃i,closedTerminalCap (C'.hole i))ᶜ=E '' (⋃i,closedTerminalCap (C.hole i))ᶜ | |
| rw [E.image_compl,image_iUnion] | |
| simp_rw [hcap] | |
| have hreg' : C'.region=N '' (e '' C.region) := by | |
| rw [hreg,image_image] | |
| rfl | |
| obtain ⟨V,f,hV,hf⟩ := exists_finite_jordan_uniformization C' | |
| rw [hreg'] at hf | |
| exact ⟨V,f ∘ N,hV,(isConformalEquivalence_pointToInfinity (e OnePoint.infty) | |
| (e '' C.region)).trans hf⟩ | |
| lemma uniformize_constructed_finite_completion_after_homeomorphism | |
| {I : Type*} [Finite I] (H : I → ℂ ≃ₜ ℂ) | |
| (hd : Pairwise (Disjoint on fun i => closure (jordanInterior (H i)))) | |
| {W : Set Sphere} (e : Sphere ≃ₜ Sphere) | |
| (hW : W ⊆ openCompletedRegion H) (he : IsConformalOn e W) : | |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), | |
| IsCircleDomain V ∧ BijOn f (openCompletedRegion H) V ∧ IsConformalOn f W ∧ | |
| IsConformalEquivalence (f ∘ e.symm) (e '' openCompletedRegion H) V := by | |
| obtain ⟨V,g,hV,hg⟩ := exists_finite_jordan_image_uniformization ⟨H,hd⟩ e | |
| change IsConformalEquivalence g (e '' openCompletedRegion H) V at hg | |
| refine ⟨V,g ∘ e,hV,hg.bijOn.comp e.injective.bijOn_image, | |
| hg.1.comp he (fun p hp => ⟨p,hW hp,rfl⟩),?_⟩ | |
| have heq : (g ∘ e) ∘ e.symm=g := by funext p; simp only [comp_apply,e.apply_symm_apply] | |
| simpa only [heq] using hg | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |