openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /ModulusBoundaryContinuation.lean
| import OAI.Analysis.CircleDomains.Reflection.SemidiskModulusExtension | |
| import OAI.Analysis.CircleDomains.Reflection.CircleSchwarzReflection | |
| namespace OAI | |
| /-! A bounded holomorphic disk-valued map with modulus tending to one | |
| at a straight boundary has an actual holomorphic continuation there. | |
| Complex boundary continuity is constructed, rather than assumed. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_holomorphic_continuation_of_modulus_trace {R c : ℝ} {f : ℂ → ℂ} | |
| (hR : 0<R) (hc : 0<c) (hc1 : c≤1) | |
| (hd : DifferentiableOn ℂ f (upperSemidisk R)) | |
| (hlower : ∀z∈upperSemidisk R,c≤‖f z‖) | |
| (hupper : ∀z∈upperSemidisk R,‖f z‖<1) | |
| (htrace : ∀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/4) ∧ | |
| DifferentiableOn ℂ F V ∧ EqOn F f (V∩{z : ℂ | 0<z.im}) := by | |
| obtain ⟨F,hFc,hFeq,hFn⟩ := exists_continuous_extension_of_modulus_trace | |
| hR hc hc1 hd hlower hupper htrace | |
| have hFn0 : ‖F 0‖=1 := hFn 0 (by simpa using (show 0<R/4 by positivity)) (by simp) | |
| let a : ℂ := -conj (F 0) | |
| have ha : ‖a‖=1 := by simpa only [a,norm_neg,norm_conj] using hFn0 | |
| have ha0 : a≠0 := norm_ne_zero_iff.mp (by rw [ha];norm_num) | |
| have hval : a*F 0 = -1 := by | |
| dsimp only [a] | |
| rw [neg_mul,←normSq_eq_conj_mul_self,normSq_eq_norm_sq,hFn0] | |
| norm_num | |
| let q : ℂ → ℂ := fun z => a*F z | |
| have hqc : Continuous q := continuous_const.mul hFc | |
| have hq0 : q 0≠1 := by change a*F 0≠1;rw [hval];norm_num | |
| have hnb : ball (0:ℂ) (R/4)∩{z : ℂ | q z≠1} ∈ 𝓝 (0:ℂ) := | |
| inter_mem (ball_mem_nhds _ (by positivity)) | |
| (hqc.continuousAt.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 hqd : ∀z∈U,0<z.im → DifferentiableAt ℂ q z := by | |
| intro z hz hz0 | |
| have hzS : z∈upperSemidisk (R/4) := ⟨(hrsub hz).1,hz0⟩ | |
| have heq : F =ᶠ[𝓝 z] f := by | |
| filter_upwards [(upperSemidisk_open (R/4)).mem_nhds hzS] with w hw | |
| exact hFeq hw | |
| have hzR := upperSemidisk_mono (by linarith : R/4≤R) hzS | |
| exact (heq.differentiableAt_iff.mpr | |
| ((hd z hzR).differentiableAt ((upperSemidisk_open R).mem_nhds hzR))).const_mul a | |
| have hqr : ∀z∈U,z.im=0 → ‖q z‖=1 := by | |
| intro z hz hz0 | |
| change ‖a*F z‖=1 | |
| rw [norm_mul,ha,hFn z (mem_ball_zero_iff.mp (hrsub hz).1) hz0,one_mul] | |
| have hqp : ∀z∈U,0≤z.im → q z≠1 := fun z hz _ => (hrsub hz).2 | |
| obtain ⟨V,H,hV,h0,hVU,hHd,hHeq⟩ := exists_local_circle_reflection hU hs hqc.continuousOn | |
| hqd hqr hqp (mem_ball_self hr) (by simp) | |
| refine ⟨V,fun z => a⁻¹*H z,hV,h0,fun z hz => (hrsub (hVU hz)).1, | |
| hHd.const_mul _,?_⟩ | |
| intro z hz | |
| change a⁻¹*H z=f z | |
| rw [hHeq ⟨hz.1,(show 0<z.im from hz.2).le⟩] | |
| change a⁻¹*(a*F z)=f z | |
| rw [←mul_assoc,inv_mul_cancel₀ ha0,one_mul] | |
| exact hFeq ⟨(hrsub (hVU hz.1)).1,hz.2⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |