File size: 2,873 Bytes
9655094 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | 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
|