| import OAI.Analysis.CircleDomains.Uniformization.LabelledCoreIteration |
|
|
| namespace OAI |
|
|
| /-! Global representatives of the compact core correspondences. They |
| agree with the actual conformal maps throughout the open source. -/ |
| open Set Function Filter Metric |
| open scoped Topology OnePoint |
| noncomputable section |
| namespace CircleDomainRigidity |
| open FiniteTransfer |
| namespace FiniteJordanConfiguration |
| variable {I:Type*} [Finite I] |
|
|
| def coreExtension {C D:FiniteJordanConfiguration I} |
| (e:closure C.region ≃ₜ closure D.region) (p:Sphere) : Sphere := by |
| classical |
| exact if hp:p∈closure C.region then (e ⟨p,hp⟩:Sphere) else p |
|
|
| omit [Finite I] in |
| lemma coreExtension_eq {C D:FiniteJordanConfiguration I} |
| (e:closure C.region ≃ₜ closure D.region) {p:Sphere} (hp:p∈closure C.region) : |
| coreExtension e p=(e ⟨p,hp⟩:Sphere) := by simp only [coreExtension,dite_eq_left hp] |
|
|
| omit [Finite I] in |
| lemma coreExtension_continuousOn {C D:FiniteJordanConfiguration I} |
| (e:closure C.region ≃ₜ closure D.region) : |
| ContinuousOn (coreExtension e) (closure C.region) := by |
| rw [continuousOn_iff_continuous_domRestrict] |
| have he : (closure C.region).domRestrict (coreExtension e)=fun p => (e p:Sphere) := by |
| funext p |
| exact coreExtension_eq e p.property |
| rw [he] |
| exact continuous_subtype_val.comp e.continuous |
|
|
| omit [Finite I] in |
| lemma coreExtension_injOn {C D:FiniteJordanConfiguration I} |
| (e:closure C.region ≃ₜ closure D.region) : InjOn (coreExtension e) (closure C.region) := by |
| intro p hp q hq he |
| rw [coreExtension_eq e hp,coreExtension_eq e hq] at he |
| exact congrArg Subtype.val (e.injective (Subtype.val_injective he)) |
|
|
| lemma exists_global_cumulative_core_extension (C:FiniteJordanConfiguration I) |
| (ha:∀i,IsAnalyticJordanCap (C.hole i)) (schedule:ℕ→I) (n:ℕ) : |
| ∃E:Sphere→Sphere, ContinuousOn E (closure C.region) ∧ InjOn E (closure C.region) ∧ |
| EqOn E (cumulativeMap C schedule n) C.region ∧ |
| E '' closure C.region=closure (iterate C schedule n).region ∧ |
| ∀i,E '' frontier (closedTerminalCap (C.hole i))= |
| frontier (closedTerminalCap ((iterate C schedule n).hole i)) := by |
| obtain ⟨e,he,hl⟩ := exists_cumulative_labelled_core_equiv C ha schedule n |
| refine ⟨coreExtension e,coreExtension_continuousOn e,coreExtension_injOn e,?_,?_,?_⟩ |
| · intro p hp |
| rw [coreExtension_eq e (subset_closure hp)] |
| exact he _ hp |
| · apply Subset.antisymm |
| · rintro _ ⟨p,hp,rfl⟩ |
| rw [coreExtension_eq e hp] |
| exact (e ⟨p,hp⟩).property |
| · intro p hp |
| obtain ⟨q,hq⟩ := e.surjective ⟨p,hp⟩ |
| refine ⟨q,q.property,?_⟩ |
| rw [coreExtension_eq e q.property] |
| exact congrArg Subtype.val hq |
| · intro i |
| have hm : coreExtension e '' frontier (closedTerminalCap (C.hole i))= |
| Subtype.val '' (e '' C.boundaryInCore i) := by |
| apply Subset.antisymm |
| · rintro _ ⟨p,hp,rfl⟩ |
| refine ⟨e ⟨p,C.boundary_subset_closure_region i hp⟩, |
| ⟨⟨p,C.boundary_subset_closure_region i hp⟩,hp,rfl⟩,?_⟩ |
| exact (coreExtension_eq e (C.boundary_subset_closure_region i hp)).symm |
| · rintro _ ⟨_,⟨q,hq,rfl⟩,rfl⟩ |
| exact ⟨q,hq,coreExtension_eq e q.property⟩ |
| rw [hm,hl i,(iterate C schedule n).boundaryInCore_image] |
|
|
| end FiniteJordanConfiguration |
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
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|
