openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /FiniteCompletionUniformization.lean
| import OAI.Analysis.CircleDomains.Uniformization.FiniteUniformization | |
| namespace OAI | |
| /-! | |
| # Finite uniformization of the actual terminal-disk complement | |
| The complementary components are exactly the closed physical terminal | |
| caps. In particular they are finite and nondegenerate. The published | |
| finite circle theorem therefore applies after compact metric | |
| uniformization; the resulting map is conformal on the retained core. | |
| -/ | |
| open Set Function Filter | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity.FiniteTransfer | |
| def closedTerminalCap (H : ℂ ≃ₜ ℂ) : Set Sphere := | |
| ((↑) : ℂ → Sphere) '' closure (jordanInterior H) | |
| def openCompletedRegion {I : Type*} (H : I → ℂ ≃ₜ ℂ) : Set Sphere := | |
| (⋃ i, closedTerminalCap (H i))ᶜ | |
| theorem isCompact_closedTerminalCap (H : ℂ ≃ₜ ℂ) : IsCompact (closedTerminalCap H) := | |
| (isCompact_closure_jordanInterior H).image OnePoint.continuous_coe | |
| theorem isPreconnected_closedTerminalCap (H : ℂ ≃ₜ ℂ) : IsPreconnected (closedTerminalCap H) := | |
| (isConnected_jordanInterior H).isPreconnected.closure.image _ OnePoint.continuous_coe.continuousOn | |
| theorem nontrivial_closedTerminalCap (H : ℂ ≃ₜ ℂ) : (closedTerminalCap H).Nontrivial := by | |
| have hzero : H 0 ∈ closure (jordanInterior H) := by | |
| rw [closure_jordanInterior] | |
| exact ⟨0, by simp, rfl⟩ | |
| have hone : H 1 ∈ closure (jordanInterior H) := by | |
| rw [closure_jordanInterior] | |
| exact ⟨1, by simp, rfl⟩ | |
| refine ⟨(H 0 : ℂ), ⟨H 0, hzero, rfl⟩, (H 1 : ℂ), ⟨H 1, hone, rfl⟩, ?_⟩ | |
| intro heq | |
| exact zero_ne_one (H.injective (OnePoint.coe_injective heq)) | |
| theorem disjoint_closedTerminalCaps {I : Type*} (H : I → ℂ ≃ₜ ℂ) | |
| (hdisj : Pairwise (Disjoint on fun i => closure (jordanInterior (H i)))) : | |
| Pairwise (Disjoint on fun i => closedTerminalCap (H i)) := by | |
| intro i j hij | |
| apply Set.disjoint_left.mpr | |
| rintro p ⟨x, hx, rfl⟩ ⟨y, hy, heq⟩ | |
| have hyx : y = x := OnePoint.coe_injective heq | |
| exact Set.disjoint_left.mp (hdisj hij) hx (hyx ▸ hy) | |
| theorem isOpen_openCompletedRegion {I : Type*} [Finite I] (H : I → ℂ ≃ₜ ℂ) : | |
| IsOpen (openCompletedRegion H) := | |
| (isClosed_iUnion_of_finite fun i => (isCompact_closedTerminalCap (H i)).isClosed).isOpen_compl | |
| theorem isConnected_openCompletedRegion {I : Type*} [Finite I] (H : I → ℂ ≃ₜ ℂ) | |
| (hdisj : Pairwise (Disjoint on fun i => closure (jordanInterior (H i)))) : | |
| IsConnected (openCompletedRegion H) := by | |
| let S : Set ℂ := (⋃ i, closure (jordanInterior (H i)))ᶜ | |
| have hS : IsPreconnected S := isPreconnected_exterior_finite_jordanInteriors H hdisj | |
| have hinf : (∞ : Sphere) ∈ closure (((↑) : ℂ → Sphere) '' S) := by | |
| have hmem : ∀ᶠ z : ℂ in coclosedCompact ℂ, z ∈ S := by | |
| rw [coclosedCompact_eq_cocompact] | |
| exact (isCompact_iUnion fun i => isCompact_closure_jordanInterior (H i)).compl_mem_cocompact | |
| exact mem_closure_of_tendsto OnePoint.tendsto_coe_infty | |
| (hmem.mono fun z hz => ⟨z, hz, rfl⟩) | |
| have hsub : ((↑) : ℂ → Sphere) '' S ⊆ openCompletedRegion H := by | |
| rintro _ ⟨z, hz, rfl⟩ hb | |
| obtain ⟨i, x, hx, heq⟩ := mem_iUnion.mp hb | |
| have hxz : x = z := OnePoint.coe_injective heq | |
| exact hz (mem_iUnion.mpr ⟨i, hxz ▸ hx⟩) | |
| have hupper : openCompletedRegion H ⊆ closure (((↑) : ℂ → Sphere) '' S) := by | |
| intro p hp | |
| cases p using OnePoint.rec with | |
| | infty => exact hinf | |
| | coe z => | |
| apply subset_closure | |
| refine ⟨z, ?_, rfl⟩ | |
| intro hz | |
| obtain ⟨i, hi⟩ := mem_iUnion.mp hz | |
| exact hp (mem_iUnion.mpr ⟨i, z, hi, rfl⟩) | |
| refine ⟨⟨∞, ?_⟩, (hS.image _ OnePoint.continuous_coe.continuousOn).subset_closure hsub hupper⟩ | |
| intro hb | |
| obtain ⟨i, z, _, heq⟩ := mem_iUnion.mp hb | |
| exact OnePoint.coe_ne_infty z heq | |
| theorem uniformize_finite_completion_after_homeomorphism | |
| (hfinite : FiniteCircleUniformization) | |
| {I : Type*} [Finite I] (H : I → ℂ ≃ₜ ℂ) | |
| (hdisj : Pairwise (Disjoint on fun i => closure (jordanInterior (H i)))) | |
| {W : Set Sphere} (e : Sphere ≃ₜ Sphere) | |
| (hW : W ⊆ openCompletedRegion H) (he : IsConformalOn e W) : | |
| ∃ (V : Set Sphere) (f : Sphere → Sphere), | |
| IsCircleDomain V ∧ BijOn f (openCompletedRegion H) V ∧ IsConformalOn f W ∧ | |
| IsConformalEquivalence (f ∘ e.symm) (e '' openCompletedRegion H) V := by | |
| apply circle_model_after_homeomorphism hfinite | |
| (isOpen_openCompletedRegion H) (isConnected_openCompletedRegion H hdisj) | |
| (fun i => closedTerminalCap (H i)) | |
| (fun i => (isCompact_closedTerminalCap (H i)).isClosed) | |
| (fun i => isPreconnected_closedTerminalCap (H i)) (disjoint_closedTerminalCaps H hdisj) | |
| (fun i => nontrivial_closedTerminalCap (H i)) (compl_compl _) e hW he | |
| end CircleDomainRigidity.FiniteTransfer | |
| end | |
| end OAI | |