openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /InjectiveModulusContinuation.lean
| import OAI.Analysis.CircleDomains.Topology.ContinuationBoundaryValues | |
| import OAI.Analysis.CircleDomains.Reflection.InjectiveCircleReflectionSides | |
| namespace OAI | |
| /-! The actual boundary continuation is locally injective and has no | |
| critical boundary point when the original upper-side map is injective. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_injective_continuation_of_modulus_trace_sides {R : ℝ} {f : ℂ → ℂ} | |
| (hR : 0<R) (hd : DifferentiableOn ℂ f (upperSemidisk R)) | |
| (hi : InjOn f (upperSemidisk R)) (hu : ∀z∈upperSemidisk R,‖f z‖<1) | |
| (ht : ∀p:ℂ,‖p‖≤R → p.im=0 → | |
| Tendsto (fun z => ‖f z‖) (𝓝[upperSemidisk R] p) (𝓝 1)) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F f (V∩{z:ℂ | 0<z.im}) ∧ (∀z∈V,deriv F z≠0) ∧ | |
| ∀z∈V,‖F z‖<1 ↔ 0<z.im := by | |
| obtain ⟨V₀,F₀,hV₀,h00,hV₀R,hF₀d,hF₀e⟩ := | |
| exists_local_holomorphic_continuation_of_modulus_trace hR hd hu ht | |
| have hF₀r : ∀z∈V₀,z.im=0 → ‖F₀ z‖=1 := | |
| fun z hz hz0 => norm_eq_one_on_real_continuation hV₀ hV₀R hF₀d.continuousOn hF₀e ht hz hz0 | |
| have hn0 : ‖F₀ 0‖=1 := hF₀r 0 h00 (by simp) | |
| let a : ℂ := -conj (F₀ 0) | |
| have ha : ‖a‖=1 := by simpa only [a,norm_neg,norm_conj] using hn0 | |
| have ha0 : a≠0 := norm_ne_zero_iff.mp (by rw [ha];norm_num) | |
| have hval : a*F₀ 0 = -1 := by | |
| dsimp only [a] | |
| rw [neg_mul,←normSq_eq_conj_mul_self,normSq_eq_norm_sq,hn0] | |
| norm_num | |
| let q : ℂ → ℂ := fun z => a*F₀ z | |
| have hqc : ContinuousAt q 0 := ((hF₀d 0 h00).differentiableAt (hV₀.mem_nhds h00)).continuousAt.const_mul a | |
| have hq0 : q 0≠1 := by change a*F₀ 0≠1;rw [hval];norm_num | |
| have hnb : V₀∩{z:ℂ | q z≠1} ∈ 𝓝 (0:ℂ) := | |
| inter_mem (hV₀.mem_nhds h00) (hqc.eventually (isOpen_ne.mem_nhds hq0)) | |
| obtain ⟨r,hr,hrsub⟩ := Metric.mem_nhds_iff.mp hnb | |
| let U := ball (0:ℂ) r | |
| have hU : IsOpen U := isOpen_ball | |
| have hs : ∀z∈U,conj z∈U := by | |
| intro z hz | |
| simpa only [U,mem_ball_zero_iff,norm_conj] using hz | |
| have hqcont : ContinuousOn q (U∩{z:ℂ | 0≤z.im}) := | |
| (hF₀d.continuousOn.mono (fun _ hz => (hrsub hz.1).1)).const_mul a | |
| have hqd : ∀z∈U,0<z.im → DifferentiableAt ℂ q z := by | |
| intro z hz _ | |
| exact ((hF₀d z (hrsub hz).1).differentiableAt (hV₀.mem_nhds (hrsub hz).1)).const_mul a | |
| have hqr : ∀z∈U,z.im=0 → ‖q z‖=1 := by | |
| intro z hz hz0 | |
| change ‖a*F₀ z‖=1 | |
| rw [norm_mul,ha,hF₀r z (hrsub hz).1 hz0,one_mul] | |
| have hqp : ∀z∈U,0≤z.im → q z≠1 := fun z hz _ => (hrsub hz).2 | |
| have hzi : ∀z∈U,0<z.im → z∈upperSemidisk R := | |
| fun z hz hz0 => ⟨hV₀R (hrsub hz).1,hz0⟩ | |
| have hqu : ∀z∈U,0<z.im → ‖q z‖<1 := by | |
| intro z hz hz0 | |
| change ‖a*F₀ z‖<1 | |
| rw [norm_mul,ha,one_mul,hF₀e ⟨(hrsub hz).1,hz0⟩] | |
| exact hu z (hzi z hz hz0) | |
| have hqi : InjOn q (U∩{z:ℂ | 0<z.im}) := by | |
| intro z hz w hw he | |
| have hzw : F₀ z=F₀ w := mul_left_cancel₀ ha0 he | |
| rw [hF₀e ⟨(hrsub hz.1).1,hz.2⟩,hF₀e ⟨(hrsub hw.1).1,hw.2⟩] at hzw | |
| exact hi (hzi z hz.1 hz.2) (hzi w hw.1 hw.2) hzw | |
| obtain ⟨V,H,hV,h0,hVU,hHd,hHi,hHe,hHsides⟩ := exists_local_injective_circle_reflection_sides | |
| hU hs hqcont hqd hqi hqu hqr hqp (mem_ball_self hr) (by simp) | |
| let F : ℂ → ℂ := fun z => a⁻¹*H z | |
| have hFd : DifferentiableOn ℂ F V := hHd.const_mul _ | |
| have hFi : InjOn F V := by | |
| intro z hz w hw he | |
| exact hHi hz hw (mul_left_cancel₀ (inv_ne_zero ha0) he) | |
| refine ⟨V,F,hV,h0,fun z hz => hV₀R (hrsub (hVU hz)).1,hFd,hFi,?_, | |
| fun z hz => holomorphicInjOn_deriv_ne_zero hV hFd hFi hz,?_⟩ | |
| · intro z hz | |
| change a⁻¹*H z=f z | |
| rw [hHe ⟨hz.1,(show 0<z.im from hz.2).le⟩] | |
| change a⁻¹*(a*F₀ z)=f z | |
| rw [←mul_assoc,inv_mul_cancel₀ ha0,one_mul] | |
| exact hF₀e ⟨(hrsub (hVU hz.1)).1,hz.2⟩ | |
| · intro z hz | |
| change ‖a⁻¹*H z‖<1 ↔ 0<z.im | |
| rw [norm_mul,norm_inv,ha,inv_one,one_mul] | |
| exact hHsides z hz | |
| lemma exists_injective_continuation_of_modulus_trace {R : ℝ} {f : ℂ → ℂ} | |
| (hR : 0<R) (hd : DifferentiableOn ℂ f (upperSemidisk R)) | |
| (hi : InjOn f (upperSemidisk R)) (hu : ∀z∈upperSemidisk R,‖f z‖<1) | |
| (ht : ∀p:ℂ,‖p‖≤R → p.im=0 → | |
| Tendsto (fun z => ‖f z‖) (𝓝[upperSemidisk R] p) (𝓝 1)) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ InjOn F V ∧ | |
| EqOn F f (V∩{z:ℂ | 0<z.im}) ∧ ∀z∈V,deriv F z≠0 := by | |
| obtain ⟨V,F,hV,h0,hVR,hFd,hFi,hFe,hFn,_⟩ := | |
| exists_injective_continuation_of_modulus_trace_sides hR hd hi hu ht | |
| exact ⟨V,F,hV,h0,hVR,hFd,hFi,hFe,hFn⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |