| import OAI.Analysis.CircleDomains.Uniformization.MarkedJointPhaseLimit |
| import OAI.Analysis.CircleDomains.Topology.NondegeneratePhaseCircle |
| import OAI.Analysis.CircleDomains.Topology.SymmetricAnnulusConnected |
| import OAI.Analysis.CircleDomains.Selection.AnnularOuterNonconstant |
| import OAI.Analysis.CircleDomains.Uniformization.HurwitzUnivalentProof |
|
|
| namespace OAI |
|
|
| /-! The actual reflected phase limits are univalent across the original |
| boundary and have nonzero limiting circle radius. -/ |
| open Set Function Filter Metric Complex |
| open scoped Topology OnePoint |
| noncomputable section |
| namespace CircleDomainRigidity |
| open FiniteTransfer |
| namespace FiniteJordanConfiguration |
| variable {m:ℕ} {C:FiniteJordanConfiguration (Fin (m+1))} {i:Fin (m+1)} |
|
|
| lemma PhaseCollar.limit_injective (P:PhaseCollar C i) {s:ℕ→ℕ} {F:ℂ→ℂ} |
| (hl:TendstoLocallyUniformlyOn (fun n => P.reflected (s n)) F atTop |
| (symmetricCircleAnnulus P.radius)) |
| (hi:InjOn F (symmetricCircleAnnulus P.radius∩{z | 1<‖z‖})) : |
| InjOn F (symmetricCircleAnnulus P.radius) := by |
| apply hurwitzUnivalentLimit _ _ _ (isOpen_symmetricCircleAnnulus P.radius) |
| (isConnected_symmetricCircleAnnulus P.radius_gt) |
| (Eventually.of_forall (fun n => ⟨P.reflected_holomorphic (s n),P.reflected_injective (s n)⟩)) hl |
| exact not_constant_of_injective_outer_annulus P.radius_gt hi |
|
|
| structure RegularPhaseLimit (P:PhaseCollar C i) where |
| subsequence : ℕ→ℕ |
| boundaryMap : ℂ→ℂ |
| map : ℂ→ℂ |
| center : ℂ |
| radius : ℝ |
| strictMono : StrictMono subsequence |
| radius_pos : 0<radius |
| centers_tendsto : Tendsto (fun n => selectedCenter C (cyclicSchedule m) |
| (visitTime (subsequence n) i)) atTop (𝓝 center) |
| radii_tendsto : Tendsto (fun n => selectedRadius C (cyclicSchedule m) |
| (visitTime (subsequence n) i)) atTop (𝓝 radius) |
| boundary_tendsto : TendstoLocallyUniformlyOn (fun n => P.reflected (subsequence n)) |
| boundaryMap atTop (symmetricCircleAnnulus P.radius) |
| map_tendsto : TendstoLocallyUniformlyOn (fun n (z:ℂ) => finiteCoord |
| (cumulativeMap C (cyclicSchedule m) (visitTime (subsequence n) i+1) (z:Sphere))) |
| map atTop C.planeRegion |
| holomorphic : DifferentiableOn ℂ boundaryMap (symmetricCircleAnnulus P.radius) |
| injective : InjOn boundaryMap (symmetricCircleAnnulus P.radius) |
| derivative_ne : ∀z∈symmetricCircleAnnulus P.radius,deriv boundaryMap z≠0 |
| normalized : IsPrincipalNormalized (extendAtInfinity map) |
| conformal : IsConformalEquivalence (extendAtInfinity map) C.region (extendAtInfinity map '' C.region) |
| agrees : ∀z∈symmetricCircleAnnulus P.radius,1<‖z‖ → boundaryMap z=map (P.coordinate z) |
| circle : ∀z∈symmetricCircleAnnulus P.radius,‖z‖=1 → ‖boundaryMap z-center‖=radius |
|
|
| lemma PhaseCollar.regularPhaseLimit_nonempty (P:PhaseCollar C i) : |
| Nonempty (RegularPhaseLimit P) := by |
| obtain ⟨s,F,G,a,ρ,hs,hρ,hc,hr,hF,hG,hd,hn,hg,he,hi,hcircle⟩ := P.exists_marked_joint_limit |
| have hpos := circle_radius_pos_of_injective_outer P.radius_gt hρ hd hi hcircle |
| have hfi := P.limit_injective hF hi |
| exact ⟨{ |
| subsequence := s |
| boundaryMap := F |
| map := G |
| center := a |
| radius := ρ |
| strictMono := hs |
| radius_pos := hpos |
| centers_tendsto := hc |
| radii_tendsto := hr |
| boundary_tendsto := hF |
| map_tendsto := hG |
| holomorphic := hd |
| injective := hfi |
| derivative_ne := fun z hz => holomorphicInjOn_deriv_ne_zero |
| (isOpen_symmetricCircleAnnulus P.radius) hd hfi hz |
| normalized := hn |
| conformal := hg |
| agrees := he |
| circle := hcircle }⟩ |
|
|
| end FiniteJordanConfiguration |
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
