| import OAI.Analysis.CircleDomains.Reflection.ModulusBoundaryContinuation | |
| namespace OAI | |
| /-! The lower modulus bound in the logarithm argument is automatic | |
| locally from the actual boundary trace. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_local_holomorphic_continuation_of_modulus_trace {R : ℝ} {f : ℂ → ℂ} | |
| (hR : 0<R) (hd : DifferentiableOn ℂ f (upperSemidisk R)) | |
| (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 ∧ | |
| DifferentiableOn ℂ F V ∧ EqOn F f (V∩{z : ℂ | 0<z.im}) := by | |
| have ht := htrace 0 (by simpa using hR.le) (by simp) | |
| have he : ∀ᶠ z in 𝓝[upperSemidisk R] (0:ℂ),(1/2:ℝ)<‖f z‖ := | |
| ht.eventually (eventually_gt_nhds (by norm_num : (1/2:ℝ)<1)) | |
| obtain ⟨δ,hδ,hδbound⟩ := Metric.eventually_nhds_iff.mp (eventually_nhdsWithin_iff.mp he) | |
| let r := min R δ/2 | |
| have hr : 0<r := by dsimp [r];positivity | |
| have hrR : r≤R := by dsimp [r];nlinarith [min_le_left R δ] | |
| have hrδ : r<δ := by dsimp [r];nlinarith [min_le_right R δ] | |
| have hlower : ∀z∈upperSemidisk r,(1/2:ℝ)≤‖f z‖ := by | |
| intro z hz | |
| apply le_of_lt | |
| apply hδbound | |
| · simpa only [dist_zero_right] using (mem_ball_zero_iff.mp hz.1).trans hrδ | |
| · exact upperSemidisk_mono hrR hz | |
| have htr : ∀p:ℂ,‖p‖≤r → p.im=0 → | |
| Tendsto (fun z => ‖f z‖) (𝓝[upperSemidisk r] p) (𝓝 1) := by | |
| intro p hp hp0 | |
| exact (htrace p (hp.trans hrR) hp0).mono_left | |
| (nhdsWithin_mono p (upperSemidisk_mono hrR)) | |
| obtain ⟨V,F,hV,h0,hVr,hFd,hFe⟩ := exists_holomorphic_continuation_of_modulus_trace | |
| hr (by norm_num : (0:ℝ)<1/2) (by norm_num : (1/2:ℝ)≤1) | |
| (hd.mono (upperSemidisk_mono hrR)) hlower | |
| (fun z hz => hupper z (upperSemidisk_mono hrR hz)) htr | |
| exact ⟨V,F,hV,h0,hVr.trans (ball_subset_ball (by linarith)),hFd,hFe⟩ | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |