| 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 | |