File size: 2,822 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 66 67 | import OAI.Analysis.CircleDomains.Reflection.UnitAnalyticJordanParametrization
import OAI.Analysis.CircleDomains.Transfer.GlobalCoreExtension
namespace OAI
/-! A fixed analytic source collar for a chosen actual hole. Its outer
side lies in the multiply connected domain and avoids every other cap. -/
open Set Function Filter Metric Complex
open scoped Topology OnePoint
noncomputable section
namespace CircleDomainRigidity
open FiniteTransfer
namespace FiniteJordanConfiguration
variable {I:Type*} [Finite I]
lemma exists_analytic_boundary_collar (C:FiniteJordanConfiguration I) (i:I)
(ha:IsAnalyticJordanCap (C.hole i)) :
∃R:ℝ, ∃β:ℂ→ℂ, 1<R ∧
DifferentiableOn ℂ β (symmetricCircleAnnulus R) ∧
InjOn β (symmetricCircleAnnulus R) ∧
β '' sphere 0 1=frontier (jordanInterior (C.hole i)) ∧
(∀z∈symmetricCircleAnnulus R,1<‖z‖ → (β z:Sphere)∈C.region) ∧
(∀z∈symmetricCircleAnnulus R,1≤‖z‖ → (β z:Sphere)∈closure C.region) := by
obtain ⟨S,β,hS,hDS,hβ,hi,hclosed,hboundary⟩ :=
exists_unit_analytic_jordan_parametrization ha
let B : Set ℂ := ⋃j:{j:I // j≠i},closure (jordanInterior (C.hole j))
have hB : IsClosed B := isClosed_iUnion_of_finite fun _ => isClosed_closure
let T := S∩β ⁻¹' Bᶜ
have hT : IsOpen T := hβ.continuousOn.isOpen_inter_preimage hS hB.isOpen_compl
have hcircle : sphere (0:ℂ) 1⊆T := by
intro z hz
refine ⟨hDS (sphere_subset_closedBall hz),?_⟩
intro hzB
obtain ⟨j,hj⟩ := mem_iUnion.mp hzB
have hzi : β z∈closure (jordanInterior (C.hole i)) :=
frontier_subset_closure (hboundary ▸ mem_image_of_mem β hz)
exact Set.disjoint_left.mp (C.disjoint j.property.symm) hzi hj
obtain ⟨R,hR,hRT⟩ := exists_symmetricCircleAnnulus_subset hT hcircle
have hout : ∀z∈symmetricCircleAnnulus R,1<‖z‖ → (β z:Sphere)∈C.region := by
intro z hz hzn hp
obtain ⟨j,w,hw,he⟩ := mem_iUnion.mp hp
have hwb : w=β z := OnePoint.coe_injective he
subst w
by_cases hji : j=i
· subst j
rw [←hclosed] at hw
obtain ⟨w,hw,he⟩ := hw
have hwz : w=z := hi (hDS hw) (hRT hz).1 he
rw [hwz] at hw
exact (not_le_of_gt hzn) (mem_closedBall_zero_iff.mp hw)
· exact (hRT hz).2 (mem_iUnion.mpr ⟨⟨j,hji⟩,hw⟩)
refine ⟨R,β,hR,hβ.mono (hRT.trans inter_subset_left),
hi.mono (hRT.trans inter_subset_left),hboundary,hout,?_⟩
intro z hz hzn
rcases eq_or_lt_of_le hzn with he|he
· apply C.boundary_subset_closure_region i
rw [frontier_closedTerminalCap]
exact ⟨β z,hboundary ▸ mem_image_of_mem β (mem_sphere_zero_iff_norm.mpr he.symm),rfl⟩
· exact subset_closure (hout z hz he)
end FiniteJordanConfiguration
end CircleDomainRigidity
end
end OAI
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