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