| import Mathlib |
| import OAI.Geometry.Cannon.Conformal.ClosureCoeBall |
| import OAI.Geometry.Cannon.Modulus.TangentDiskMeet |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace CannonMobius |
| open Set Metric Topology CannonTileTopology |
| open scoped OnePoint |
|
|
| theorem finite_packing_at_vertex {ν : Type*} (z : ν → ℂ) (r : ν → ℝ) |
| (hr : ∀ j,0<r j) (hsep : ∀ i j,i≠j → r i+r j≤dist (z i) (z j)) |
| (N : Sphere ≃ₜ Sphere) (hN : N∈biholomorphic) (pole : ν) |
| (hpole : N (z pole:Sphere)=∞) : |
| ∃ c : {j : ν // j≠pole} → ℂ,∃ t : {j : ν // j≠pole} → ℝ, |
| (∀ j,0<t j) ∧ |
| (∀ j,N '' ((↑) '' ball (z j.val) (r j.val) : Set Sphere)=(↑) '' ball (c j) (t j)) ∧ |
| (∀ j,N '' ((↑) '' closedBall (z j.val) (r j.val) : Set Sphere)=(↑) '' closedBall (c j) (t j)) ∧ |
| Pairwise (fun i j=>Disjoint (ball (c i) (t i)) (ball (c j) (t j))) := by |
| have hex (j : {j : ν // j≠pole}) : ∃ c : ℂ,∃ t : ℝ,0<t ∧ |
| N '' ((↑) '' ball (z j.val) (r j.val) : Set Sphere)=(↑) '' ball c t ∧ |
| N '' ((↑) '' closedBall (z j.val) (r j.val) : Set Sphere)=(↑) '' closedBall c t := by |
| apply biholomorphic_finite_disk N hN (z j.val) (hr j.val) |
| rintro ⟨p,⟨q,hq,rfl⟩,hep⟩ |
| have hqpole : q=z pole:=OnePoint.coe_injective (N.injective (hep.trans hpole.symm)) |
| subst q |
| have hs:=hsep pole j.val j.property.symm |
| have hq':=mem_closedBall.mp hq |
| linarith [hr pole] |
| choose c t ht hb hc using hex |
| refine ⟨c,t,ht,hb,hc,?_⟩ |
| intro i j hij |
| have hn : i.val≠j.val:=fun h=>hij (Subtype.ext h) |
| have hd:=packing_balls_disjoint z r hsep hn |
| have hd':=hd.image OnePoint.coe_injective.injOn (subset_univ _) (subset_univ _) |
| have hd'':=hd'.image N.injective.injOn (subset_univ _) (subset_univ _) |
| rw [hb i,hb j] at hd'' |
| exact hd''.of_image |
| end CannonMobius |
|
|
| end |
| end |
|
|
| end OAI |
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|
