openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /AnalyticJordanBoundaryInjection.lean
| import OAI.Analysis.CircleDomains.Topology.AnalyticBoundaryGerms | |
| import OAI.Analysis.CircleDomains.Topology.BoundaryGlobalInjectivity | |
| import OAI.Analysis.CircleDomains.Reflection.ClosedExteriorBoundaryValues | |
| namespace OAI | |
| /-! The actual closed-exterior extension is globally injective, including | |
| its entire analytic Jordan boundary. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| lemma analytic_jordan_closed_exterior_injective {H : ℂ ≃ₜ ℂ} | |
| (hH : IsAnalyticJordanCap H) {f : Sphere → Sphere} {d : ℂ} {s : ℝ} | |
| (hs : 0<s) (hf : IsConformalEquivalence f (closedTerminalCap H)ᶜ | |
| ((((↑) : ℂ → Sphere) '' closedBall d s)ᶜ)) : | |
| let A := (closure (jordanInterior H))ᶜ | |
| let g := fun z : ℂ => finiteCoord (exteriorDiskNormalization d s hs (f (z : Sphere))) | |
| InjOn (extendFrom A g) (closure A) := by | |
| dsimp only | |
| let A := (closure (jordanInterior H))ᶜ | |
| let g := fun z : ℂ => finiteCoord (exteriorDiskNormalization d s hs (f (z : Sphere))) | |
| let E := extendFrom A g | |
| let K := frontier (jordanInterior H) | |
| change InjOn E (closure A) | |
| have hnorm := hf.trans (isConformalEquivalence_exteriorDiskNormalization d hs) | |
| have hzU {z:ℂ} (hz:z∈A) : (z:Sphere)∈(closedTerminalCap H)ᶜ := by | |
| rintro ⟨w,hw,he⟩ | |
| exact hz (OnePoint.coe_injective he ▸ hw) | |
| have hu : ∀z∈A,‖g z‖<1 := by | |
| intro z hz | |
| obtain ⟨w,hw,he⟩ := hnorm.2.1 (hzU hz) | |
| change (w : Sphere)=exteriorDiskNormalization d s hs (f (z : Sphere)) at he | |
| change ‖finiteCoord (exteriorDiskNormalization d s hs (f (z : Sphere)))‖<1 | |
| rw [←he,finiteCoord_coe] | |
| exact mem_ball_zero_iff.mp hw | |
| have hgi : InjOn g A := by | |
| intro z hz w hw he | |
| apply OnePoint.coe_injective | |
| apply hnorm.bijOn.injOn (hzU hz) (hzU hw) | |
| obtain ⟨a,_,ha⟩ := hnorm.2.1 (hzU hz) | |
| obtain ⟨b,_,hb⟩ := hnorm.2.1 (hzU hw) | |
| change finiteCoord ((exteriorDiskNormalization d s hs ∘ f) (z : Sphere))= | |
| finiteCoord ((exteriorDiskNormalization d s hs ∘ f) (w : Sphere)) at he | |
| rw [←ha,←hb,finiteCoord_coe,finiteCoord_coe] at he | |
| rw [←ha,←hb,he] | |
| have hbdy {x:ℂ} (hx:x∈K) : x∈closure A := by | |
| have hxn : x∉jordanInterior H := by | |
| change x∈frontier (jordanInterior H) at hx | |
| rw [(jordanInterior_isOpen H).frontier_eq] at hx | |
| exact hx.2 | |
| simpa only [A,closure_compl,interior_closure_jordanInterior,mem_compl_iff] using hxn | |
| obtain ⟨_,he⟩ := analytic_jordan_closed_exterior_extension hH hs hf | |
| change EqOn E g A at he | |
| have hKb : MapsTo E K (closure (ball (0:ℂ) 1)) := by | |
| intro x hx | |
| rw [closure_ball (0:ℂ) one_ne_zero] | |
| apply mem_closedBall_zero_iff.mpr | |
| exact (analytic_jordan_closed_exterior_boundary_norm hH hs hf hx).le | |
| have hEi : InjOn E K := by | |
| apply injOn_boundary_of_holomorphic_extensions hgi hKb | |
| intro x hx | |
| obtain ⟨O,G,hO,hxO,hGd,hGi,hGe,hside⟩ := analytic_jordan_boundary_germ hH hs hf hx | |
| have hEG : E x=G x := extendFrom_eq_local_extension hO hGd.continuousOn hGe ⟨hxO,hbdy hx⟩ | |
| exact ⟨O,G,hO,hxO,hGd,hGi,hEG.symm,hGe,fun z hz hzB => | |
| (hside z hz).mp (mem_ball_zero_iff.mp hzB)⟩ | |
| have hbd {x:ℂ} (hx:x∈closure A) (hxn:x∉A) : x∈K := by | |
| have hxcl : x∈closure (jordanInterior H) := not_not.mp hxn | |
| have hxout : x∉jordanInterior H := by | |
| simpa only [A,closure_compl,interior_closure_jordanInterior,mem_compl_iff] using hx | |
| change x∈frontier (jordanInterior H) | |
| rw [(jordanInterior_isOpen H).frontier_eq] | |
| exact ⟨hxcl,hxout⟩ | |
| intro x hx y hy hexy | |
| by_cases hxA : x∈A | |
| · by_cases hyA : y∈A | |
| · apply hgi hxA hyA | |
| rwa [he hxA,he hyA] at hexy | |
| · have hxn : ‖E x‖<1 := by rw [he hxA];exact hu x hxA | |
| have hyn : ‖E y‖=1 := analytic_jordan_closed_exterior_boundary_norm hH hs hf (hbd hy hyA) | |
| rw [hexy,hyn] at hxn | |
| exact (lt_irrefl _ hxn).elim | |
| · by_cases hyA : y∈A | |
| · have hyn : ‖E y‖<1 := by rw [he hyA];exact hu y hyA | |
| have hxn : ‖E x‖=1 := analytic_jordan_closed_exterior_boundary_norm hH hs hf (hbd hx hxA) | |
| rw [←hexy,hxn] at hyn | |
| exact (lt_irrefl _ hyn).elim | |
| · exact hEi (hbd hx hxA) (hbd hy hyA) hexy | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |