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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 | import OAI.Analysis.CircleDomains.Uniformization.JordanConfigurationExterior
import OAI.Analysis.CircleDomains.Sobolev.ExteriorCoefficientAreaProof
import OAI.Analysis.CircleDomains.Topology.FiniteCircleGreenProof
import OAI.Analysis.CircleDomains.Uniformization.MontelProof
import OAI.Analysis.CircleDomains.Uniformization.HurwitzUnivalentProof
namespace OAI
/-! Normal-family compactness for the actual successive circularizations.
The source exterior radius supplies uniform bounds for every cumulative
map and every omitted component, without assuming a circular limit. -/
open Set Function Metric Filter Asymptotics Bornology
open scoped Topology OnePoint
noncomputable section
namespace CircleDomainRigidity.FiniteJordanConfiguration
variable {I : Type*} [Finite I]
lemma circularization_uniform_bounds (C : FiniteJordanConfiguration I) (schedule : ℕ → I)
{R : ℝ} (hR : 0<R)
(hsource : ((((↑) : ℂ → Sphere) '' ball 0 R)ᶜ) ⊆ C.region) (n : ℕ) :
(∀ p ∈ (iterate C schedule n).regionᶜ,
p ∈ ((↑) : ℂ → Sphere) '' closedBall 0 (3*R)) ∧
(∀ z : ℂ, z ∈ C.planeRegion → ‖z‖≤4*R →
‖finiteCoord (cumulativeMap C schedule n (z : Sphere))‖≤5*R) ∧
(∀ z : ℂ, 2*R≤‖z‖ →
‖finiteCoord (cumulativeMap C schedule n (z : Sphere))-z‖≤2*R^2/‖z‖) := by
let harea := exteriorCoefficientAreaTheorem planarSchoenflies unorientedFiniteCircleGreenFormula
have hconf := cumulativeMap_conformal C schedule n
obtain ⟨fi,heq⟩ := hconf.exists_smooth_inverse C.region_open (iterate C schedule n).region_open
have hext : ((((↑) : ℂ → Sphere) '' closedBall 0 R)ᶜ) ⊆ C.region :=
(compl_subset_compl.mpr (image_mono ball_subset_closedBall)).trans hsource
have hc : IsConformalOn (cumulativeMap C schedule n)
((((↑) : ℂ → Sphere) '' closedBall 0 R)ᶜ) := fun p hp => hconf.1 p (hext hp)
obtain ⟨homit,hsmall⟩ := normalized_model_uniform_bound harea heq
(iterate C schedule n).region_open hR hsource hc (cumulativeMap_principal C schedule n)
refine ⟨homit,?_,normalized_sphere_model_exterior_decay harea heq hR hsource hc
(cumulativeMap_principal C schedule n)⟩
intro z hz hzn
obtain ⟨w,hw,he⟩ := hsmall z hz hzn
have hew : w=finiteCoord (cumulativeMap C schedule n (z : Sphere)) := by
simpa only [finiteCoord_coe] using congrArg finiteCoord he
simpa only [hew,mem_closedBall,dist_zero_right] using hw
lemma exists_circularization_normal_limit_along (C : FiniteJordanConfiguration I)
(schedule : ℕ → I) (t : ℕ → ℕ) :
∃ (s : ℕ → ℕ) (F : ℂ → ℂ), StrictMono s ∧
TendstoLocallyUniformlyOn (fun n (z : ℂ) =>
finiteCoord (cumulativeMap C schedule (t (s n)) (z : Sphere))) F atTop C.planeRegion ∧
DifferentiableOn ℂ F C.planeRegion ∧ InjOn F C.planeRegion ∧
(fun z => F z-z) =O[cobounded ℂ] (fun z => z⁻¹) := by
obtain ⟨R,hR,hsource⟩ := C.exists_exterior_radius
have hfin (n : ℕ) := (cumulativeMap_conformal C schedule n).1.finite_open_embedding
C.region_open (cumulativeMap_conformal C schedule n).bijOn.injOn C.infty_mem_region
(cumulativeMap_principal C schedule n).1
have hext : {z : ℂ | 2*R≤‖z‖} ⊆ C.planeRegion := by
intro z hz
apply hsource
simpa only [OnePoint.coe_injective.mem_set_image,mem_ball,dist_zero_right,
mem_compl_iff,not_lt] using (show R≤‖z‖ by change 2*R≤‖z‖ at hz; linarith)
obtain ⟨s,F,hs,hlim,hd,hi,hn,_⟩ := exists_normalized_univalent_limit montelExhaustion
hurwitzUnivalentLimit hR (fun _ => C.planeRegion_open)
(fun _ => C.planeRegion_connected) (fun _ _ _ => Subset.rfl)
(fun n => (hfin (t n)).1) (fun n => (hfin (t n)).2.2) hext
(fun n z hz hzn => (circularization_uniform_bounds C schedule hR hsource (t n)).2.1 z hz hzn)
(fun n => (circularization_uniform_bounds C schedule hR hsource (t n)).2.2)
exact ⟨s,F,hs,by simpa only [iUnion_const] using hlim,
by simpa only [iUnion_const] using hd,by simpa only [iUnion_const] using hi,hn⟩
lemma exists_circularization_normal_limit (C : FiniteJordanConfiguration I) (schedule : ℕ → I) :
∃ (s : ℕ → ℕ) (F : ℂ → ℂ), StrictMono s ∧
TendstoLocallyUniformlyOn (fun n (z : ℂ) =>
finiteCoord (cumulativeMap C schedule (s n) (z : Sphere))) F atTop C.planeRegion ∧
DifferentiableOn ℂ F C.planeRegion ∧ InjOn F C.planeRegion ∧
(fun z => F z-z) =O[cobounded ℂ] (fun z => z⁻¹) :=
exists_circularization_normal_limit_along C schedule id
end CircleDomainRigidity.FiniteJordanConfiguration
end
end OAI
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