openAI-math-dataset / lean /OAI /Analysis /CircleDomains /Uniformization /ConformalJordanCapImage.lean
| import OAI.Analysis.CircleDomains.Uniformization.ConformalFiniteEmbedding | |
| import OAI.Analysis.CircleDomains.Topology.JordanDiskImages | |
| import OAI.Analysis.CircleDomains.Uniformization.FiniteCompletionUniformization | |
| import OAI.Analysis.CircleDomains.Topology.PlanarSchoenfliesProof | |
| namespace OAI | |
| /-! A compact Jordan hole lying in the domain of a conformal open embedding | |
| has another actual Jordan hole as its image. -/ | |
| open Set Function | |
| open scoped Topology OnePoint | |
| noncomputable section | |
| namespace CircleDomainRigidity | |
| open FiniteTransfer | |
| lemma exists_conformal_jordan_cap_image | |
| {f : Sphere → Sphere} {U : Set Sphere} (hf : IsConformalOn f U) | |
| (hU : IsOpen U) (hi : InjOn f U) (hinf : ∞ ∈ U) (hfix : f ∞=∞) | |
| (H : ℂ ≃ₜ ℂ) (hHU : closedTerminalCap H ⊆ U) : | |
| ∃ K : ℂ ≃ₜ ℂ, f '' closedTerminalCap H = closedTerminalCap K := by | |
| obtain ⟨hd,ho,hij⟩ := hf.finite_open_embedding hU hi hinf hfix | |
| have hsub : closure (jordanInterior H) ⊆ ((↑) : ℂ → Sphere) ⁻¹' U := | |
| fun z hz => hHU ⟨z,hz,rfl⟩ | |
| obtain ⟨K,_hKi,hKc,_hKf⟩ := exists_jordan_image_disk planarSchoenflies | |
| (hU.preimage OnePoint.continuous_coe) hd.continuousOn ho hij H hsub | |
| refine ⟨K,?_⟩ | |
| rw [closedTerminalCap, ← finite_part_sphere_image | |
| (fun z hz => finite_image_of_fixes_infty hi hinf hfix (hsub hz)), hKc] | |
| rfl | |
| end CircleDomainRigidity | |
| end | |
| end OAI | |