| import OAI.Analysis.CircleDomains.Uniformization.MobiusConformal |
|
|
| namespace OAI |
|
|
| /-! |
| # Conformal point-to-infinity normalization |
|
|
| The explicit normalizer and its inverse are conformal in every sphere |
| chart. Postcomposition therefore preserves the conformal equivalence, |
| not just the topology and the round complementary components. |
| -/ |
|
|
| open Set Function |
| open scoped OnePoint |
|
|
| noncomputable section |
|
|
| namespace CircleDomainRigidity |
|
|
| theorem affineSphere_symm_eq (a b : ℂ) (ha : a ≠ 0) : |
| (affineSphere a b ha).symm = affineSphere a⁻¹ (-b / a) (inv_ne_zero ha) := by |
| apply Homeomorph.ext |
| intro p |
| cases p using OnePoint.rec with |
| | infty => rfl |
| | coe z => |
| change (((z - b) / a : ℂ) : Sphere) = ((a⁻¹ * z + -b / a : ℂ) : Sphere) |
| congr 1 |
| ring |
|
|
| theorem isConformalAt_affineSphere_symm (a b : ℂ) (ha : a ≠ 0) (p : Sphere) : |
| IsConformalAt (affineSphere a b ha).symm p := by |
| rw [affineSphere_symm_eq] |
| exact isConformalAt_affineSphere _ _ _ p |
|
|
| theorem isConformalAt_pointToInfinity (p q : Sphere) : |
| IsConformalAt (pointToInfinity p) q := by |
| cases p using OnePoint.rec with |
| | infty => exact isConformalAt_id q |
| | coe z => |
| change IsConformalAt (CannonMobius.inversion ∘ affineSphere 1 (-z) one_ne_zero) q |
| exact (isConformalAt_inversion _).comp (isConformalAt_affineSphere _ _ _ q) |
|
|
| theorem isConformalAt_pointToInfinity_symm (p q : Sphere) : |
| IsConformalAt (pointToInfinity p).symm q := by |
| cases p using OnePoint.rec with |
| | infty => exact isConformalAt_id q |
| | coe z => |
| change IsConformalAt ((affineSphere 1 (-z) one_ne_zero).symm ∘ CannonMobius.inversion) q |
| exact (isConformalAt_affineSphere_symm _ _ _ _).comp (isConformalAt_inversion q) |
|
|
| theorem isConformalEquivalence_of_homeomorph (e : Sphere ≃ₜ Sphere) (U : Set Sphere) |
| (he : IsConformalOn e U) (hi : IsConformalOn e.symm (e '' U)) : |
| IsConformalEquivalence e U (e '' U) := by |
| refine ⟨he, mapsTo_image _ _, e.symm, hi, ?_, ?_, ?_⟩ |
| · rintro _ ⟨p, hp, rfl⟩ |
| simpa only [e.symm_apply_apply] using hp |
| · intro p _ |
| exact e.symm_apply_apply p |
| · intro p _ |
| exact e.apply_symm_apply p |
|
|
| theorem isConformalEquivalence_pointToInfinity (p : Sphere) (U : Set Sphere) : |
| IsConformalEquivalence (pointToInfinity p) U (pointToInfinity p '' U) := |
| isConformalEquivalence_of_homeomorph _ U (fun q _ => isConformalAt_pointToInfinity p q) |
| (fun q _ => isConformalAt_pointToInfinity_symm p q) |
|
|
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
