| import OAI.Analysis.CircleDomains.Reflection.AnalyticJordanBoundaryContinuation |
| import OAI.Analysis.CircleDomains.Uniformization.JordanCircularizationSequence |
|
|
| namespace OAI |
|
|
| /-! Analytic circle parametrizations of the actual compact Jordan holes. |
| The neighborhood and injectivity are retained so that a later |
| circularization has the proved local continuation. -/ |
| open Set Function Metric |
| open scoped Topology OnePoint |
| noncomputable section |
| namespace CircleDomainRigidity |
| open FiniteTransfer |
|
|
| def IsAnalyticJordanCap (H : ℂ ≃ₜ ℂ) : Prop := |
| ∃ (S : Set ℂ) (α : ℂ → ℂ) (c : ℂ) (r : ℝ), IsOpen S ∧ 0<r ∧ |
| closedBall c r⊆S ∧ DifferentiableOn ℂ α S ∧ InjOn α S ∧ |
| α '' closedBall c r=closure (jordanInterior H) |
|
|
| lemma isAnalyticJordanCap_of_round {H : ℂ ≃ₜ ℂ} {c : ℂ} {r : ℝ} |
| (hr : 0<r) (he : jordanInterior H=ball c r) : IsAnalyticJordanCap H := by |
| refine ⟨univ,id,c,r,isOpen_univ,hr,subset_univ _,differentiableOn_id,injective_id.injOn,?_⟩ |
| rw [image_id,he,closure_ball _ hr.ne'] |
|
|
| lemma IsAnalyticJordanCap.conformal_image {H K : ℂ ≃ₜ ℂ} (hH : IsAnalyticJordanCap H) |
| {f : Sphere → Sphere} {U V : Set Sphere} |
| (hf : IsConformalEquivalence f U V) (hU : IsOpen U) |
| (hinf : ∞∈U) (hfix : f ∞=∞) (hHU : closedTerminalCap H⊆U) |
| (himage : f '' closedTerminalCap H=closedTerminalCap K) : IsAnalyticJordanCap K := by |
| obtain ⟨S,α,c,r,hS,hr,hDS,hα,hi,hαeq⟩ := hH |
| let g : ℂ → ℂ := fun z => finiteCoord (f (z : Sphere)) |
| let P := ((↑) : ℂ → Sphere) ⁻¹' U |
| have hP : IsOpen P := hU.preimage OnePoint.continuous_coe |
| obtain ⟨hgd,_,hgi⟩ := hf.1.finite_open_embedding hU hf.bijOn.injOn hinf hfix |
| let W := S ∩ α ⁻¹' P |
| have hW : IsOpen W := hα.continuousOn.isOpen_inter_preimage hS hP |
| have hDW : closedBall c r⊆W := by |
| intro z hz |
| refine ⟨hDS hz,?_⟩ |
| apply hHU |
| refine ⟨α z,?_,rfl⟩ |
| rw [←hαeq] |
| exact mem_image_of_mem α hz |
| have hfin : ∀z∈closure (jordanInterior H),f (z : Sphere)≠∞ := |
| fun z hz => finite_image_of_fixes_infty hf.bijOn.injOn hinf hfix (hHU ⟨z,hz,rfl⟩) |
| have hgimage : g '' closure (jordanInterior H)=closure (jordanInterior K) := by |
| apply OnePoint.coe_injective.image_injective |
| rw [finite_part_sphere_image hfin] |
| exact himage |
| refine ⟨W,g ∘ α,c,r,hW,hr,hDW,?_,?_,?_⟩ |
| · exact hgd.comp (hα.mono inter_subset_left) (fun _ hz => hz.2) |
| · exact hgi.comp (hi.mono inter_subset_left) (fun _ hz => hz.2) |
| · rw [image_comp,hαeq,hgimage] |
|
|
| namespace FiniteJordanConfiguration |
| variable {I : Type*} [Finite I] {C : FiniteJordanConfiguration I} {j : I} |
|
|
| omit [Finite I] in |
| lemma Step.analytic_selected (q : Step C j) : IsAnalyticJordanCap (q.target.hole j) := |
| isAnalyticJordanCap_of_round q.radius_pos q.selected |
|
|
| omit [Finite I] in |
| lemma Step.analytic_other (q : Step C j) {i : I} (hij : i≠j) |
| (ha : IsAnalyticJordanCap (C.hole i)) : IsAnalyticJordanCap (q.target.hole i) := by |
| let U := (closedTerminalCap (C.hole j))ᶜ |
| have hU : IsOpen U := (isCompact_closedTerminalCap (C.hole j)).isClosed.isOpen_compl |
| have hinf : ∞∈U := by |
| rintro ⟨z,_,he⟩ |
| exact OnePoint.coe_ne_infty z he |
| have hHU : closedTerminalCap (C.hole i)⊆U := by |
| intro x hx hxj |
| exact Set.disjoint_left.mp (disjoint_closedTerminalCaps C.hole C.disjoint hij) hx hxj |
| exact ha.conformal_image q.exterior_conformal hU hinf q.principal.1 hHU (q.others i hij) |
|
|
| lemma iterate_analytic_of_visited (C : FiniteJordanConfiguration I) (schedule : ℕ → I) |
| (n : ℕ) (i : I) (hv : ∃k<n,schedule k=i) : |
| IsAnalyticJordanCap ((iterate C schedule n).hole i) := by |
| induction n with |
| | zero => obtain ⟨k,hk,_⟩ := hv;omega |
| | succ n ih => |
| by_cases hi : i=schedule n |
| · subst i |
| exact (chosenStep (iterate C schedule n) (schedule n)).analytic_selected |
| · apply (chosenStep (iterate C schedule n) (schedule n)).analytic_other hi |
| apply ih |
| obtain ⟨k,hk,hki⟩ := hv |
| have hkn : k≠n := by |
| intro he |
| subst k |
| exact hi hki.symm |
| exact ⟨k,by omega,hki⟩ |
|
|
| end FiniteJordanConfiguration |
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
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|
