openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Reflection /MovingCircleReflectionLimit.lean
| import OAI.Analysis.CircleDomains.Reflection.MovingCircleReflection | |
| import OAI.Analysis.CircleDomains.Uniformization.MontelProof | |
| namespace OAI | |
| /-! Compactness of the explicit reflected maps on a fixed two-sided | |
| collar. The bound is independent of the reciprocals of the target radii. -/ | |
| open Set Function Filter Metric Complex | |
| open scoped Topology ComplexConjugate | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| lemma exists_moving_circle_reflection_limit {U:Set ℂ} {f:ℕ → ℂ → ℂ} | |
| {c:ℕ → ℂ} {r:ℕ → ℝ} {M:ℝ} | |
| (hU:IsOpen U) (hz:∀z∈U,z≠0) (hs:∀z∈U,unitCircleReflection z∈U) | |
| (hr:∀n,0<r n) | |
| (hc:∀n,ContinuousOn (f n) (U∩{z | 1≤‖z‖})) | |
| (hd:∀n z,z∈U → 1<‖z‖ → DifferentiableAt ℂ (f n) z) | |
| (ho:∀n z,z∈U → 1≤‖z‖ → r n≤‖f n z-c n‖) | |
| (hb:∀n z,z∈U → ‖z‖=1 → ‖f n z-c n‖=r n) | |
| (hM:∀n,‖c n‖+r n≤M) | |
| (hf:∀n z,z∈U → 1≤‖z‖ → ‖f n z‖≤M) : | |
| ∃s:ℕ → ℕ, ∃F:ℂ → ℂ,StrictMono s ∧ | |
| TendstoLocallyUniformlyOn (fun n => circleReflectionExtension (c (s n)) (r (s n)) (f (s n))) | |
| F atTop U ∧ DifferentiableOn ℂ F U := by | |
| have hhol (n:ℕ) : DifferentiableOn ℂ (circleReflectionExtension (c n) (r n) (f n)) U := | |
| circleReflectionExtension_differentiableOn (hr n) hU hz hs (hc n) (hd n) (ho n) (hb n) | |
| obtain ⟨s,F,hsm,hlim⟩ := montelExhaustion (fun _ => U) | |
| (fun n => circleReflectionExtension (c n) (r n) (f n)) (fun _ => hU) | |
| (fun _ _ _ => Subset.refl _) hhol (by | |
| intro K hK hKU | |
| refine ⟨M,?_⟩ | |
| intro n z _ hzU | |
| exact circleReflectionExtension_norm_le (hr n) hz hs (hM n) | |
| (fun z hzU hzo => ⟨ho n z hzU hzo,hf n z hzU hzo⟩) hzU) | |
| simp only [iUnion_const] at hlim | |
| exact ⟨s,F,hsm,hlim,hlim.differentiableOn (Eventually.of_forall (fun n => hhol (s n))) hU⟩ | |
| lemma moving_circle_reflection_limit_boundary {U:Set ℂ} {f:ℕ → ℂ → ℂ} | |
| {c:ℕ → ℂ} {r:ℕ → ℝ} {s:ℕ → ℕ} {F:ℂ → ℂ} {a:ℂ} {ρ:ℝ} | |
| (hr:∀n,0<r n) (hs:StrictMono s) | |
| (hlim:TendstoLocallyUniformlyOn | |
| (fun n => circleReflectionExtension (c (s n)) (r (s n)) (f (s n))) F atTop U) | |
| (hc:Tendsto c atTop (𝓝 a)) (hρ:Tendsto r atTop (𝓝 ρ)) | |
| (hb:∀n z,z∈U → ‖z‖=1 → ‖f n z-c n‖=r n) | |
| {z:ℂ} (hz:z∈U) (hz1:‖z‖=1) : ‖F z-a‖=ρ := by | |
| have hp := hlim.tendsto_at hz | |
| have hnorm := (hp.sub (hc.comp hs.tendsto_atTop)).norm | |
| have he (n:ℕ) : | |
| ‖circleReflectionExtension (c (s n)) (r (s n)) (f (s n)) z-c (s n)‖=r (s n) := by | |
| rw [circleReflectionExtension_outer (hr (s n)) hz1.ge] | |
| exact hb (s n) z hz hz1 | |
| apply tendsto_nhds_unique hnorm | |
| exact (hρ.comp hs.tendsto_atTop).congr' (Eventually.of_forall (fun n => (he n).symm)) | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |