| import Mathlib |
| import OAI.Geometry.Cannon.Conformal.SphereTile |
|
|
| namespace OAI |
|
|
| section |
| section |
| section |
| noncomputable section |
| namespace CannonMobius |
| open Set Metric Topology |
| open scoped OnePoint |
|
|
| theorem normalize_packing_centers {ν : Type*} |
| (z : ν → ℂ) (r : ν → ℝ) (hr : ∀ v,0<r v) |
| (hsep : ∀ u v,u≠v → r u+r v≤dist (z u) (z v)) |
| (b : Fin 3 → ν) (hb : Function.Injective b) : |
| ∃ N : Sphere ≃ₜ Sphere,N∈biholomorphic ∧ |
| (∀ i,N ((z (b i):ℂ):Sphere)=std i) ∧ |
| (∀ v,IsOpen (N '' ((↑) '' ball (z v) (r v) : Set Sphere))) ∧ |
| Pairwise (fun u v=>Disjoint (N '' ((↑) '' ball (z u) (r u) : Set Sphere)) |
| (N '' ((↑) '' ball (z v) (r v) : Set Sphere))) ∧ |
| (∀ v,∃ i j : Fin 3,i≠j ∧ |
| std i∉N '' ((↑) '' ball (z v) (r v) : Set Sphere) ∧ |
| std j∉N '' ((↑) '' ball (z v) (r v) : Set Sphere)) := by |
| have hz : Function.Injective z := by |
| intro u v huv |
| by_contra hne |
| have hh:=hsep u v hne |
| rw [huv,dist_self] at hh |
| linarith [hr u,hr v] |
| let t : Triple:=⟨fun i=>(z (b i):Sphere),OnePoint.coe_injective.comp (hz.comp hb)⟩ |
| obtain ⟨M,hM,hMt⟩:=exists_triple_normalizer t |
| let N : Sphere ≃ₜ Sphere:=M.symm |
| have hN (i : Fin 3) : N (z (b i):Sphere)=std i := by |
| have H : M (std i)=(z (b i):Sphere):=congrArg (fun t : Triple=>t.val i) hMt |
| dsimp only [N] |
| rw [←H,M.symm_apply_apply] |
| have hout {v : ν} {i : Fin 3} (hi : b i≠v) : |
| std i∉N '' ((↑) '' ball (z v) (r v) : Set Sphere) := by |
| rintro ⟨_,⟨w,hw,rfl⟩,hew⟩ |
| have hcoe : (w:Sphere)=(z (b i):Sphere):=N.injective (hew.trans (hN i).symm) |
| have hew : w=z (b i):=OnePoint.coe_injective hcoe |
| rw [hew] at hw |
| have hh:=hsep (b i) v hi |
| have hh':=mem_ball.mp hw |
| linarith [hr (b i)] |
| refine ⟨N,biholomorphic.inv_mem hM,hN,?_,?_,?_⟩ |
| · intro v |
| exact N.isOpenMap _ (OnePoint.isOpenMap_coe _ isOpen_ball) |
| · intro u v huv |
| apply disjoint_left.mpr |
| rintro q ⟨_,⟨w,hwu,rfl⟩,hwq⟩ ⟨_,⟨w',hwv,rfl⟩,hwq'⟩ |
| have heq : w=w' := OnePoint.coe_injective (N.injective (hwq.trans hwq'.symm)) |
| subst w' |
| have h1:=mem_ball.mp hwu |
| have h2:=mem_ball.mp hwv |
| have ht:=dist_triangle (z u) w (z v) |
| rw [dist_comm (z u) w] at ht |
| have hh:=hsep u v huv |
| linarith |
| · intro v |
| have hx : ∃ i j : Fin 3,i≠j ∧ b i≠v ∧ b j≠v := by |
| by_cases h0 : b 0=v |
| · refine ⟨1,2,by decide,?_,?_⟩ <;> intro h |
| · have hh:=hb (h.trans h0.symm) |
| exact (by decide : (1:Fin 3)≠0) hh |
| · have hh:=hb (h.trans h0.symm) |
| exact (by decide : (2:Fin 3)≠0) hh |
| · by_cases h1 : b 1=v |
| · refine ⟨0,2,by decide,h0,?_⟩ |
| intro h |
| have hh:=hb (h.trans h1.symm) |
| exact (by decide : (2:Fin 3)≠1) hh |
| · exact ⟨0,1,by decide,h0,h1⟩ |
| obtain ⟨i,j,hij,hi,hj⟩:=hx |
| exact ⟨i,j,hij,hout hi,hout hj⟩ |
| end CannonMobius |
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| end |
| end |
| end |
| end |
|
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| end OAI |
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