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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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.Analysis.CircleDomains.Reflection.ReflectionPrincipalSolution
import OAI.Analysis.CircleDomains.Selection.FiniteRoundComparison
namespace OAI
/-! Actual data for finite analytic uniformization by invariant Beltrami
coefficients. Conformality of the straightened comparison and roundness of
its boundary images are not fields and remain to be proved. -/
open Set Function Filter MeasureTheory Metric
open scoped Topology ContDiff
noncomputable section
namespace CircleDomainRigidity
open FiniteTransfer CannonBeltrami
attribute [local instance] sphereMetric sphereBorel
local instance : BorelSpace Sphere := ⟨rfl⟩
structure AnalyticReflectionData {I : Type*} [Finite I] (C : FiniteJordanConfiguration I) where
comparison : ℂ ≃ₜ ℂ
center : I → ℂ
radius : I → ℝ
radius_pos : ∀i,0<radius i
comparison_smooth : ContDiff ℝ ∞ (comparison : ℂ → ℂ)
comparison_inverse_smooth : ContDiff ℝ ∞ (comparison.symm : ℂ → ℂ)
cap_image : ∀i,comparison '' closure (jordanInterior (C.hole i))=closedBall (center i) (radius i)
baseCoefficient : ℂ → ℂ
bound : ℝ
bound_nonneg : 0≤bound
bound_lt_one : bound<1
base_bound : ∀z,‖baseCoefficient z‖≤bound
base_equation : ∀z,partialBar comparison.symm z=baseCoefficient z*partialZ comparison.symm z
extension : Sphere → ℂ
extension_measurable : Measurable extension
extension_bound : ∀p,‖extension p‖≤bound
extension_base : EqOn extension (sphereCoefficient 0 baseCoefficient) (comparison.onePointCongr '' C.region)
extension_invariant : ∀i p,extension p=roundReflectionCoefficient (center i) p
(extension (roundDiskReflection (center i) (radius i) (radius_pos i) p))
density : PlaneL2
supportRadius : ℝ
extension_zero : ∀z : ℂ,supportRadius≤‖z‖ → extension (z : Sphere)=0
approximation : PrincipalSmoothApproximation density bound supportRadius
limitData : PrincipalMapLimit approximation
density_equation : ∀ᵐz ∂volume,density z=extension (z : Sphere)*(1+beurling density z)
coefficient_tendsto : ∀ᵐz ∂volume,Tendsto (fun n => approximation.coefficient n z)
atTop (𝓝 (extension (z : Sphere)))
namespace AnalyticReflectionData
variable {I : Type*} [Finite I] {C : FiniteJordanConfiguration I}
def principal (D : AnalyticReflectionData C) : ℂ ≃ₜ ℂ := D.limitData.map
lemma principal_decay (D : AnalyticReflectionData C) :
Tendsto (fun z => D.principal z-z) (Bornology.cobounded ℂ) (𝓝 0) := D.limitData.principal
lemma principal_weak (D : AnalyticReflectionData C) (v : ℂ) (φ : SchwartzMap ℂ ℂ)
(hs : HasCompactSupport (φ : ℂ → ℂ)) :
(∫z,D.principal z*fderiv ℝ (φ : ℂ → ℂ) z v)=
-(∫z,(v+principalDirectionalCorrection D.density v z)*φ z) := D.limitData.weak v φ hs
end AnalyticReflectionData
lemma exists_analytic_reflection_data {I : Type*} [Finite I]
(C : FiniteJordanConfiguration I) (ha : ∀i,IsAnalyticJordanCap (C.hole i)) :
Nonempty (AnalyticReflectionData C) := by
obtain ⟨P,c,r,μ,k,S,hPs,hPis,hr,hcap,hμs,hk,hk1,hS,hb,hz,he⟩ :=
exists_finite_analytic_rounding_coefficient C ha
have hd := rounding_round_disks_disjoint C P c r hcap
obtain ⟨E,T,g,S,L,hEm,hEz,hEb,hEf,hEinv,hg,hc⟩ := exists_round_invariant_principal_approximation_with_coefficient
c r hr hd μ hμs.continuous.measurable hk hk1 hb hz
refine ⟨{
comparison := P
center := c
radius := r
radius_pos := hr
comparison_smooth := hPs
comparison_inverse_smooth := hPis
cap_image := hcap
baseCoefficient := μ
bound := k
bound_nonneg := hk
bound_lt_one := hk1
base_bound := hb
base_equation := he
extension := E
extension_measurable := hEm
extension_bound := hEb
extension_base := ?_
extension_invariant := hEinv
density := g
supportRadius := T
extension_zero := hEz
approximation := S
limitData := L
density_equation := hg
coefficient_tendsto := hc }⟩
rwa [rounding_spherical_exterior_image C P c r hcap]
end CircleDomainRigidity
end
end OAI
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