openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /JordanCircularizationCompactness.lean
| 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 | |