| import OAI.Analysis.CircleDomains.Rigidity.MatchedCommonPoints |
| import OAI.Analysis.CircleDomains.Rigidity.RemovableAreaZero |
| import OAI.Analysis.CircleDomains.Sobolev.RealCoordinateLines |
|
|
| namespace OAI |
|
|
| /-! The common point set is contained in the removable boundary and |
| therefore has null intersections with almost every coordinate line. -/ |
| open Set Function Filter MeasureTheory |
| open scoped Topology ENNReal |
| noncomputable section |
| namespace CircleDomainRigidity |
| open FiniteTransfer |
| attribute [local instance] sphereMetric sphereBorel |
| local instance : BorelSpace Sphere := ⟨rfl⟩ |
| namespace MatchedConformalEnds |
| variable {G : Set Sphere} {f : Sphere → Sphere} (D : MatchedConformalEnds G f) |
|
|
| theorem commonPointPlane_subset_frontier |
| (hG : IsCircleDomain G) (hinf : OnePoint.infty ∈ G) |
| (hf : IsConformalEquivalence f G (f '' G)) |
| (hH : IsCircleDomain (f '' G)) (hinf' : OnePoint.infty ∈ f '' G) |
| {z : ℂ} (hz : z ∈ D.commonPointPlane) : (z : Sphere) ∈ frontier G := by |
| have hs := (D.commonPointPlane_singletons hG hinf hf hH hinf' hz).1 |
| have hproper : Gᶜ ≠ univ := by |
| intro he |
| exact (show OnePoint.infty ∈ Gᶜ from he.symm ▸ mem_univ _) hinf |
| obtain ⟨p,hp,hpf⟩ := component_meets_frontier hG.1.isClosed_compl hproper hz.1 |
| have hpz : p = (z : Sphere) := mem_singleton_iff.mp (hs ▸ hp) |
| simpa only [frontier_compl,hpz] using hpf |
|
|
| theorem commonPointPlane_line_null |
| (hprincipal : PrincipalRealBeltrami) |
| (hG : IsCircleDomain G) (hinf : OnePoint.infty ∈ G) |
| (hf : IsConformalEquivalence f G (f '' G)) |
| (hH : IsCircleDomain (f '' G)) (hinf' : OnePoint.infty ∈ f '' G) |
| (hrem : IsConformallyRemovable (frontier G)) (i : Fin 2) : |
| ∀ᵐ y ∂volume, volume {x | complexCoordinateLine i y x ∈ D.commonPointPlane} = 0 := by |
| have hn : OnePoint.infty ∈ (frontier G)ᶜ := by |
| intro hx |
| exact Set.disjoint_left.mp (disjoint_frontier_iff_isOpen.mpr hG.1) hx hinf |
| have hvol := removable_complement_volume_zero hprincipal isClosed_frontier.isOpen_compl hn |
| (by simpa only [compl_compl] using hrem) |
| have havoid : ∀ᵐ z ∂volume, planeSphere z ∉ frontier G := by |
| apply ae_iff.mpr |
| convert hvol using 1 |
| congr 1 |
| have hc : ∀ᵐ z ∂volume, Complex.equivRealProdCLM.symm z ∉ D.commonPointPlane := by |
| have hmp := euclideanPlaneCoordinates_symm_measurePreserving |
| filter_upwards [hmp.quasiMeasurePreserving.ae havoid] with z hz |
| intro hcommon |
| apply hz |
| convert D.commonPointPlane_subset_frontier hG hinf hf hH hinf' hcommon using 1 |
| simp only [planeSphere,complexPlaneCoordinates,ContinuousLinearEquiv.trans_apply, |
| ContinuousLinearEquiv.apply_symm_apply] |
| filter_upwards [ae_ae_realCoordinateLine hc i] with y hy |
| simpa only [not_not,complexCoordinateLine] using (ae_iff.mp hy) |
|
|
| end MatchedConformalEnds |
| end CircleDomainRigidity |
|
|
| end |
|
|
| end OAI |
|
|
