File size: 3,439 Bytes
a97875e | 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 74 | 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
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