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9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | 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
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