File size: 2,489 Bytes
9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | 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
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