openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /InjectiveConformalBoundaryChart.lean
| import OAI.Analysis.CircleDomains.Reflection.ConformalBoundaryChart | |
| import OAI.Analysis.CircleDomains.Reflection.InjectiveModulusContinuation | |
| import OAI.Analysis.CircleDomains.Uniformization.ExteriorDiskNormalization | |
| namespace OAI | |
| /-! The actual conformal equivalence, in an injective source chart, has | |
| an injective holomorphic continuation with nonzero boundary derivative. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma injective_conformal_disk_boundary_chart_sides {U : Set Sphere} | |
| {f : Sphere → Sphere} {κ : ℂ → ℂ} {R : ℝ} | |
| (hU : IsOpen U) | |
| (hf : IsConformalEquivalence f U (((↑) : ℂ → Sphere) '' ball (0:ℂ) 1)) | |
| (hR : 0<R) (hκ : DifferentiableOn ℂ κ (upperSemidisk R)) | |
| (hi : InjOn κ (upperSemidisk R)) | |
| (hm : ∀z∈upperSemidisk R,(κ z : Sphere)∈U) | |
| (hc : ∀p:ℂ,‖p‖≤R → p.im=0 → ContinuousAt κ p) | |
| (hb : ∀p:ℂ,‖p‖≤R → p.im=0 → (κ p : Sphere)∉U) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F (fun z => finiteCoord (f (κ z : Sphere))) (V∩{z : ℂ | 0<z.im}) ∧ | |
| (∀z∈V,deriv F z≠0) ∧ ∀z∈V,‖F z‖<1 ↔ 0<z.im := by | |
| obtain ⟨hd,hu,ht⟩ := conformal_disk_boundary_chart_data hU hf hκ hm hc hb | |
| have hgi : InjOn (fun z => finiteCoord (f (κ z : Sphere))) (upperSemidisk R) := by | |
| intro z hz w hw he | |
| apply hi hz hw | |
| apply OnePoint.coe_injective | |
| apply hf.bijOn.injOn (hm z hz) (hm w hw) | |
| obtain ⟨a,_,ha⟩ := hf.2.1 (hm z hz) | |
| obtain ⟨b,_,hb⟩ := hf.2.1 (hm w hw) | |
| change finiteCoord (f (κ z : Sphere))=finiteCoord (f (κ w : Sphere)) at he | |
| rw [←ha,←hb,finiteCoord_coe,finiteCoord_coe] at he | |
| rw [←ha,←hb,he] | |
| exact exists_injective_continuation_of_modulus_trace_sides hR hd hgi hu ht | |
| lemma injective_conformal_disk_boundary_chart {U : Set Sphere} | |
| {f : Sphere → Sphere} {κ : ℂ → ℂ} {R : ℝ} | |
| (hU : IsOpen U) | |
| (hf : IsConformalEquivalence f U (((↑) : ℂ → Sphere) '' ball (0:ℂ) 1)) | |
| (hR : 0<R) (hκ : DifferentiableOn ℂ κ (upperSemidisk R)) | |
| (hi : InjOn κ (upperSemidisk R)) | |
| (hm : ∀z∈upperSemidisk R,(κ z : Sphere)∈U) | |
| (hc : ∀p:ℂ,‖p‖≤R → p.im=0 → ContinuousAt κ p) | |
| (hb : ∀p:ℂ,‖p‖≤R → p.im=0 → (κ p : Sphere)∉U) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F (fun z => finiteCoord (f (κ z : Sphere))) (V∩{z : ℂ | 0<z.im}) ∧ | |
| ∀z∈V,deriv F z≠0 := by | |
| obtain ⟨V,F,hV,h0,hVR,hFd,hFi,hFe,hFn,_⟩ := | |
| injective_conformal_disk_boundary_chart_sides hU hf hR hκ hi hm hc hb | |
| exact ⟨V,F,hV,h0,hVR,hFd,hFi,hFe,hFn⟩ | |
| lemma injective_circular_exterior_boundary_chart_sides {U : Set Sphere} | |
| {f : Sphere → Sphere} {κ : ℂ → ℂ} {R r : ℝ} {c : ℂ} | |
| (hU : IsOpen U) | |
| (hf : IsConformalEquivalence f U ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ)) | |
| (hr : 0<r) (hR : 0<R) (hκ : DifferentiableOn ℂ κ (upperSemidisk R)) | |
| (hi : InjOn κ (upperSemidisk R)) | |
| (hm : ∀z∈upperSemidisk R,(κ z : Sphere)∈U) | |
| (hc : ∀p:ℂ,‖p‖≤R → p.im=0 → ContinuousAt κ p) | |
| (hb : ∀p:ℂ,‖p‖≤R → p.im=0 → (κ p : Sphere)∉U) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F (fun z => finiteCoord (exteriorDiskNormalization c r hr (f (κ z : Sphere)))) | |
| (V∩{z : ℂ | 0<z.im}) ∧ (∀z∈V,deriv F z≠0) ∧ ∀z∈V,‖F z‖<1 ↔ 0<z.im := | |
| injective_conformal_disk_boundary_chart_sides hU | |
| (hf.trans (isConformalEquivalence_exteriorDiskNormalization c hr)) hR hκ hi hm hc hb | |
| lemma injective_circular_exterior_boundary_chart {U : Set Sphere} | |
| {f : Sphere → Sphere} {κ : ℂ → ℂ} {R r : ℝ} {c : ℂ} | |
| (hU : IsOpen U) | |
| (hf : IsConformalEquivalence f U ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ)) | |
| (hr : 0<r) (hR : 0<R) (hκ : DifferentiableOn ℂ κ (upperSemidisk R)) | |
| (hi : InjOn κ (upperSemidisk R)) | |
| (hm : ∀z∈upperSemidisk R,(κ z : Sphere)∈U) | |
| (hc : ∀p:ℂ,‖p‖≤R → p.im=0 → ContinuousAt κ p) | |
| (hb : ∀p:ℂ,‖p‖≤R → p.im=0 → (κ p : Sphere)∉U) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F (fun z => finiteCoord (exteriorDiskNormalization c r hr (f (κ z : Sphere)))) | |
| (V∩{z : ℂ | 0<z.im}) ∧ ∀z∈V,deriv F z≠0 := by | |
| obtain ⟨V,F,hV,h0,hVR,hFd,hFi,hFe,hFn,_⟩ := | |
| injective_circular_exterior_boundary_chart_sides hU hf hr hR hκ hi hm hc hb | |
| exact ⟨V,F,hV,h0,hVR,hFd,hFi,hFe,hFn⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |