openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /ExteriorDiskNormalization.lean
| import OAI.Analysis.CircleDomains.Reflection.ConformalBoundaryChart | |
| namespace OAI | |
| /-! Explicit Möbius normalization of a finite circular exterior to the | |
| unit disk. This converts the actual circularization maps to the target | |
| used by the boundary-continuation argument. -/ | |
| open Set Function Filter Metric | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| def exteriorDiskNormalization (c : ℂ) (r : ℝ) (hr : 0<r) : Sphere ≃ₜ Sphere := | |
| (affineSphere (r:ℂ) c (by exact_mod_cast hr.ne')).symm.trans CannonMobius.inversion | |
| lemma affineSphere_image_unit_closedBall (c : ℂ) {r : ℝ} (hr : 0<r) : | |
| affineSphere (r:ℂ) c (by exact_mod_cast hr.ne') '' | |
| (((↑) : ℂ → Sphere) '' closedBall 0 1) = | |
| ((↑) : ℂ → Sphere) '' closedBall c r := by | |
| rw [closedBall_eq_affine_image hr.le c,image_image,image_image] | |
| congr 1 | |
| funext z | |
| rw [affineSphere_coe] | |
| congr 1 | |
| simp only [Complex.real_smul,mul_comm,add_comm] | |
| lemma exteriorDiskNormalization_image (c : ℂ) {r : ℝ} (hr : 0<r) : | |
| exteriorDiskNormalization c r hr '' | |
| ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ) = | |
| ((↑) : ℂ → Sphere) '' ball 0 1 := by | |
| let A := affineSphere (r:ℂ) c (by exact_mod_cast hr.ne') | |
| have hA : A '' (((↑) : ℂ → Sphere) '' closedBall 0 1)= | |
| ((↑) : ℂ → Sphere) '' closedBall c r := affineSphere_image_unit_closedBall c hr | |
| have hAi : A.symm '' ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ)= | |
| ((((↑) : ℂ → Sphere) '' closedBall 0 1)ᶜ) := by | |
| rw [←hA,←A.image_compl,image_image] | |
| simp only [A.symm_apply_apply, image_id'] | |
| change (CannonMobius.inversion ∘ A.symm) '' _ = _ | |
| rw [image_comp,hAi,←inversion_preimage_unitDisk, | |
| image_preimage_eq _ CannonMobius.inversion.surjective] | |
| lemma isConformalEquivalence_exteriorDiskNormalization (c : ℂ) {r : ℝ} (hr : 0<r) : | |
| IsConformalEquivalence (exteriorDiskNormalization c r hr) | |
| ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ) | |
| (((↑) : ℂ → Sphere) '' ball 0 1) := by | |
| have he := isConformalEquivalence_of_homeomorph (exteriorDiskNormalization c r hr) | |
| ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ) | |
| (fun p _ => (isConformalAt_inversion _).comp | |
| (isConformalAt_affineSphere_symm (r:ℂ) c (by exact_mod_cast hr.ne') p)) | |
| (fun p _ => (isConformalAt_affineSphere (r:ℂ) c (by exact_mod_cast hr.ne') _).comp | |
| (isConformalAt_inversion p)) | |
| rwa [exteriorDiskNormalization_image c hr] at he | |
| lemma circular_exterior_boundary_chart_continuation {U : Set Sphere} | |
| {f : Sphere → Sphere} {κ : ℂ → ℂ} {R r : ℝ} {c : ℂ} | |
| (hU : IsOpen U) | |
| (hf : IsConformalEquivalence f U ((((↑) : ℂ → Sphere) '' closedBall c r)ᶜ)) | |
| (hr : 0<r) (hR : 0<R) (hκ : DifferentiableOn ℂ κ (upperSemidisk R)) | |
| (hm : ∀z∈upperSemidisk R,(κ z : Sphere)∈U) | |
| (hc : ∀p:ℂ,‖p‖≤R → p.im=0 → ContinuousAt κ p) | |
| (hb : ∀p:ℂ,‖p‖≤R → p.im=0 → (κ p : Sphere)∉U) : | |
| ∃ (V : Set ℂ) (F : ℂ → ℂ), IsOpen V ∧ (0:ℂ)∈V ∧ V⊆ball 0 R ∧ | |
| DifferentiableOn ℂ F V ∧ | |
| EqOn F (fun z => finiteCoord (exteriorDiskNormalization c r hr (f (κ z : Sphere)))) | |
| (V∩{z : ℂ | 0<z.im}) := | |
| conformal_disk_boundary_chart_continuation hU | |
| (hf.trans (isConformalEquivalence_exteriorDiskNormalization c hr)) hR hκ hm hc hb | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |