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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 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 | import OAI.Analysis.CircleDomains.Uniformization.CyclicCircularization
import OAI.Analysis.CircleDomains.Uniformization.OmittedLimits
import OAI.Analysis.CircleDomains.Uniformization.HurwitzZeroFreeProof
import OAI.Analysis.CircleDomains.Modulus.NormalizedLimitEquivalence
import OAI.Analysis.CircleDomains.Sobolev.HolomorphicInjOnDerivative
namespace OAI
/-! A phase of the actual cyclic construction has a conformal normal
limit which omits the entire limiting selected disk, including a
possible radius-zero disk. This does not yet identify the other holes. -/
open Set Function Metric Filter Asymptotics Bornology
open scoped Topology OnePoint
noncomputable section
namespace CircleDomainRigidity.FiniteJordanConfiguration
variable {I : Type*} [Finite I]
lemma planeRegion_nontrivial (C : FiniteJordanConfiguration I) : C.planeRegion.Nontrivial := by
obtain ⟨R,hR,hsource⟩ := C.exists_exterior_radius
have hmem (t : ℝ) (ht : R≤t) : (t : ℂ) ∈ C.planeRegion := by
apply hsource
simpa only [OnePoint.coe_injective.mem_set_image,mem_ball,dist_zero_right,
Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hR.le.trans ht),
mem_compl_iff,not_lt] using ht
refine ⟨(R : ℂ),hmem R le_rfl,((R+1 : ℝ) : ℂ),hmem (R+1) (by linarith),?_⟩
intro he
have := Complex.ofReal_injective he
linarith
lemma not_constant_of_injOn (C : FiniteJordanConfiguration I) {F : ℂ → ℂ}
(hi : InjOn F C.planeRegion) : ¬ ∃a, ∀z∈C.planeRegion,F z=a := by
rintro ⟨a,ha⟩
obtain ⟨x,hx,y,hy,hne⟩ := C.planeRegion_nontrivial
exact hne (hi hx hy ((ha x hx).trans (ha y hy).symm))
omit [Finite I] in
lemma region_eq_plane_with_infty (C : FiniteJordanConfiguration I) :
C.region=((↑) : ℂ → Sphere) '' C.planeRegion ∪ {∞} := by
ext p
cases p using OnePoint.rec <;> simp [planeRegion,C.infty_mem_region]
lemma conformal_limit (C : FiniteJordanConfiguration I) {F : ℂ → ℂ}
(hd : DifferentiableOn ℂ F C.planeRegion) (hi : InjOn F C.planeRegion)
(hn : (fun z => F z-z) =O[cobounded ℂ] (fun z => z⁻¹)) :
IsConformalEquivalence (extendAtInfinity F) C.region (extendAtInfinity F '' C.region) := by
obtain ⟨R,hR,hsource⟩ := C.exists_exterior_radius
rw [C.region_eq_plane_with_infty]
apply normalized_limit_conformal_equivalence C.planeRegion_open hd hi
(fun z hz => holomorphicInjOn_deriv_ne_zero C.planeRegion_open hd hi hz) hn
intro z hz
change R ≤ ‖z‖ at hz
apply hsource
simpa only [OnePoint.coe_injective.mem_set_image,mem_ball,dist_zero_right,
mem_compl_iff,not_lt] using hz
lemma selected_disk_omitted (C : FiniteJordanConfiguration I) (schedule : ℕ → I)
(n : ℕ) (z : ℂ) (hz : z ∈ C.planeRegion) :
finiteCoord (cumulativeMap C schedule (n+1) (z : Sphere)) ∉
closedBall (selectedCenter C schedule n) (selectedRadius C schedule n) := by
have hf := cumulativeMap_conformal C schedule (n+1)
have hfin := finite_image_of_fixes_infty hf.bijOn.injOn C.infty_mem_region
(cumulativeMap_principal C schedule (n+1)).1 hz
intro hdisk
have hcap : cumulativeMap C schedule (n+1) (z : Sphere) ∈
FiniteTransfer.closedTerminalCap ((iterate C schedule (n+1)).hole (schedule n)) := by
rw [selected_cap_eq]
exact ⟨_,hdisk,coe_finiteCoord hfin⟩
exact hf.bijOn.mapsTo hz (mem_iUnion.mpr ⟨schedule n,hcap⟩)
lemma exists_cyclic_phase_limit {m : ℕ}
(C : FiniteJordanConfiguration (Fin (m+1))) (i : Fin (m+1)) :
∃ (s : ℕ → ℕ) (F : ℂ → ℂ) (c : Fin (m+1) → ℂ) (r : Fin (m+1) → ℝ),
StrictMono s ∧ (∀j,0≤r j) ∧
(∀j,Tendsto (fun n => selectedCenter C (cyclicSchedule m) (visitTime (s n) j))
atTop (𝓝 (c j))) ∧
(∀j,Tendsto (fun n => selectedRadius C (cyclicSchedule m) (visitTime (s n) j))
atTop (𝓝 (r j))) ∧
TendstoLocallyUniformlyOn (fun n (z : ℂ) => finiteCoord
(cumulativeMap C (cyclicSchedule m) (visitTime (s n) i+1) (z : Sphere)))
F atTop C.planeRegion ∧
IsPrincipalNormalized (extendAtInfinity F) ∧
IsConformalEquivalence (extendAtInfinity F) C.region (extendAtInfinity F '' C.region) ∧
((↑) : ℂ → Sphere) '' closedBall (c i) (r i) ⊆
(extendAtInfinity F '' C.region)ᶜ := by
obtain ⟨t,c,r,ht,hr,hc,hrlim,_⟩ := exists_cyclic_selected_disk_limits C
obtain ⟨s,F,hs,hlim,hd,hi,hn⟩ := exists_circularization_normal_limit_along C
(cyclicSchedule m) (fun n => visitTime (t n) i+1)
refine ⟨t ∘ s,F,c,r,ht.comp hs,hr,fun j => (hc j).comp hs.tendsto_atTop,
fun j => (hrlim j).comp hs.tendsto_atTop,hlim,
isPrincipalNormalized_extendAtInfinity hn,C.conformal_limit hd hi hn,?_⟩
rw [C.region_eq_plane_with_infty]
apply sphere_disk_omitted_of_plane_disk
apply omitted_closedBall_limit hurwitzZeroFreeLimit C.planeRegion_open C.planeRegion_connected
(Filter.Eventually.of_forall (fun n => ?_))
(fun n => (selectedRadius_pos C (cyclicSchedule m) (visitTime (t (s n)) i)).le)
(hr i) hlim ((hc i).comp hs.tendsto_atTop) ((hrlim i).comp hs.tendsto_atTop)
(C.not_constant_of_injOn hi)
exact ⟨((cumulativeMap_conformal C (cyclicSchedule m) (visitTime (t (s n)) i+1)).1.finite_open_embedding
C.region_open (cumulativeMap_conformal C (cyclicSchedule m)
(visitTime (t (s n)) i+1)).bijOn.injOn C.infty_mem_region
(cumulativeMap_principal C (cyclicSchedule m) (visitTime (t (s n)) i+1)).1).1,
fun z hz => selected_disk_omitted C (cyclicSchedule m) (visitTime (t (s n)) i) z hz⟩
end CircleDomainRigidity.FiniteJordanConfiguration
end
end OAI
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