openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /InjectiveAnalyticCircularization.lean
| import OAI.Analysis.CircleDomains.Reflection.AnalyticCircularization | |
| namespace OAI | |
| /-! The actual cyclic boundary continuations are injective, with | |
| nonzero derivative at every point of their open domains. -/ | |
| open Set Function Metric | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| namespace FiniteJordanConfiguration | |
| variable {I : Type*} [Finite I] {C : FiniteJordanConfiguration I} {j : I} | |
| omit [Finite I] in | |
| lemma Step.injective_analytic_boundary_continuation (q : Step C j) | |
| (ha : IsAnalyticJordanCap (C.hole j)) : | |
| ∃ (α : ℂ → ℂ) (c : ℂ) (r : ℝ), 0<r ∧ | |
| α '' closedBall c r=closure (jordanInterior (C.hole j)) ∧ | |
| ∀p:ℂ,‖p‖=1 → ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F (fun z => finiteCoord (exteriorDiskNormalization q.center q.radius q.radius_pos | |
| (q.map (α (roundExteriorBoundaryChart c r p z) : Sphere)))) | |
| (V∩{z : ℂ | 0<z.im}) ∧ ∀z∈V,deriv F z≠0 := by | |
| obtain ⟨S,α,c,r,hS,hr,hDS,hα,hi,he⟩ := ha | |
| refine ⟨α,c,r,hr,he,?_⟩ | |
| intro p hp | |
| have hf : IsConformalEquivalence q.map | |
| ((((↑) : ℂ → Sphere) '' (α '' closedBall c r))ᶜ) | |
| ((((↑) : ℂ → Sphere) '' closedBall q.center q.radius)ᶜ) := by | |
| simpa only [he,closedTerminalCap] using q.exterior_conformal | |
| exact analytic_jordan_boundary_injective_continuation hS hα hi hr q.radius_pos hp hDS hf | |
| lemma cyclic_stage_injective_boundary_continuation {m : ℕ} | |
| (C : FiniteJordanConfiguration (Fin (m+1))) {n : ℕ} (hn : m+1≤n) : | |
| ∃ (α : ℂ → ℂ) (c : ℂ) (r : ℝ), 0<r ∧ | |
| α '' closedBall c r=closure (jordanInterior ((iterate C (cyclicSchedule m) n).hole | |
| (cyclicSchedule m n))) ∧ | |
| ∀p:ℂ,‖p‖=1 → ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F (fun z => finiteCoord (exteriorDiskNormalization | |
| (selectedCenter C (cyclicSchedule m) n) (selectedRadius C (cyclicSchedule m) n) | |
| (selectedRadius_pos C (cyclicSchedule m) n) | |
| (stageMap C (cyclicSchedule m) n (α (roundExteriorBoundaryChart c r p z) : Sphere)))) | |
| (V∩{z : ℂ | 0<z.im}) ∧ ∀z∈V,deriv F z≠0 := | |
| (chosenStep (iterate C (cyclicSchedule m) n) (cyclicSchedule m n )).injective_analytic_boundary_continuation | |
| (cyclic_iterate_analytic C hn _) | |
| end FiniteJordanConfiguration | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |