openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /AnalyticFiniteUniformization.lean
| import OAI.Analysis.CircleDomains.Reflection.AnalyticRoundCapImages | |
| import OAI.Analysis.CircleDomains.Reflection.AnalyticStraighteningHolomorphic | |
| import OAI.Analysis.CircleDomains.Topology.ExteriorHomeomorphGerm | |
| import OAI.Analysis.CircleDomains.Uniformization.ConformalInverse | |
| import OAI.Analysis.CircleDomains.Topology.BoundedRoundDisks | |
| namespace OAI | |
| /-! Unconditional uniformization of a finite analytic Jordan configuration. | |
| The conformal map is the actual principal solution composed with the smooth | |
| rounding comparison; the round cap images are proved from reflection. -/ | |
| open Set Function Metric | |
| open scoped Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| lemma exists_analytic_finite_uniformization {I : Type*} [Finite I] | |
| (C : FiniteJordanConfiguration I) (ha : ∀i,IsAnalyticJordanCap (C.hole i)) : | |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), | |
| IsCircleDomain V ∧ IsConformalEquivalence f C.region V := by | |
| obtain ⟨D⟩ := exists_analytic_reflection_data C ha | |
| obtain ⟨s,hs,hcaps⟩ := D.exists_round_cap_images | |
| let H : ℂ ≃ₜ ℂ := D.comparison.trans D.principal | |
| let V : Set Sphere := H.onePointCongr '' C.region | |
| let K : Set ℂ := ⋃i,closure (jordanInterior (C.hole i)) | |
| have hK : IsCompact K := isCompact_iUnion fun i => isCompact_closure_jordanInterior (C.hole i) | |
| have hr : C.region=(((↑) : ℂ → Sphere) '' K)ᶜ := by | |
| change (⋃i,((↑) : ℂ → Sphere) '' closure (jordanInterior (C.hole i)))ᶜ=_ | |
| rw [image_iUnion] | |
| have hconf : IsConformalOn H.onePointCongr C.region := by | |
| rw [hr] | |
| apply exterior_homeomorph_conformalOn H hK | |
| intro z hz | |
| have hp : (z : Sphere)∈C.region := by | |
| rw [hr] | |
| simpa only [mem_compl_iff,OnePoint.coe_injective.mem_set_image] using hz | |
| exact (D.straightening_holomorphic z hp).differentiableAt | |
| ((C.region_open.preimage OnePoint.continuous_coe).mem_nhds hp) | |
| have hcomp : Vᶜ=⋃i,((↑) : ℂ → Sphere) '' closedBall (D.principal (D.center i)) (s i) := by | |
| have hh := rounding_spherical_exterior_image C H (fun i => D.principal (D.center i)) s hcaps | |
| change (H.onePointCongr '' C.region)ᶜ=_ | |
| rw [hh,compl_compl] | |
| have hdisj := rounding_round_disks_disjoint C H (fun i => D.principal (D.center i)) s hcaps | |
| have hV : IsCircleDomain V := by | |
| refine ⟨H.onePointCongr.isOpenMap _ C.region_open, | |
| C.region_connected.image _ H.onePointCongr.continuous.continuousOn,?_⟩ | |
| intro p hp | |
| have hp' := hp | |
| rw [hcomp] at hp' | |
| obtain ⟨i,hi⟩ := mem_iUnion.mp hp' | |
| right | |
| rw [hcomp,connectedComponentIn_finite_disjoint_closed | |
| (fun i => ((↑) : ℂ → Sphere) '' closedBall (D.principal (D.center i)) (s i)) | |
| (fun i => ((isCompact_closedBall _ _).image OnePoint.continuous_coe).isClosed) | |
| (fun i => (convex_closedBall (D.principal (D.center i)) (s i)).isPreconnected.image _ | |
| OnePoint.continuous_coe.continuousOn) hdisj hi] | |
| exact isRoundClosedDisk_coe_closedBall _ (hs i) | |
| exact ⟨V,H.onePointCongr,hV,isConformalEquivalence_of_injOn hconf C.region_open | |
| H.onePointCongr.injective.injOn⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |