File size: 3,638 Bytes
a97875e | 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 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | 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
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