| import OAI.Analysis.CircleDomains.Topology.FiniteJordanModelInput |
| import OAI.Analysis.CircleDomains.Uniformization.RealFiniteModel |
|
|
| namespace OAI |
|
|
| /-! The same principal model transports both the real barriers and the |
| masked circle coordinates, with exact energy identities for each. -/ |
|
|
| open Set Filter Function MeasureTheory |
| open scoped Topology ContDiff Matrix.Norms.Elementwise |
| noncomputable section |
| namespace CircleDomainRigidity.FiniteTransfer |
|
|
| theorem finite_metric_model_with_mixed_fillings |
| (hcompact : SmoothCompactMetricSphereUniformization) (hfinite : FiniteJordanModelUniformization) |
| {I : Type*} [Finite I] (T : I → ℂ ≃ₜ ℂ) |
| (hdisj : Pairwise (Disjoint on fun i => closure (jordanInterior (T i)))) |
| {K Ω : Set (ℝ × ℝ)} (hKQ : sphericalCore K ⊆ openCompletedRegion T) |
| (hQΩ : realChartDomain (completedRegion T) ⊆ Ω) |
| (g : (ℝ × ℝ) → PlanarMetric) (hg : ContDiff ℝ ∞ g) (hPD : ∀ z, (g z).PosDef) |
| (hfar : ∃ R : ℝ, 0 < R ∧ ∀ z ∉ Metric.closedBall 0 R, g z = sphericalMetricFactor z • 1) |
| (hcore : ∀ᶠ z in 𝓝ˢ K, g z = sphericalMetricFactor z • 1) |
| {J : Type*} {n : ℕ} {D : J → Set (ℝ × ℝ)} {M : Fin n → Set (ℝ × ℝ)} |
| {v F : Fin n → (ℝ × ℝ) → UnitAddCircle} |
| (hF : IsMaskedFilledCircleFamily D Ω M v F) |
| {R : Type*} (W : R → (ℝ × ℝ) → ℝ) |
| (hW : ∀ j, ContDiffOn ℝ ∞ (W j) Ω) : |
| ∃ (V : Set Sphere) (f fi : Sphere → Sphere), IsCircleDomain V ∧ |
| IsSmoothSphereEquivalence f fi (openCompletedRegion T) V ∧ |
| IsConformalOn f (sphericalCore K) ∧ IsPrincipalNormalized f ∧ |
| (∀ l, IsOpen (sphereRealMap f '' (realChartDomain (openCompletedRegion T) ∩ interior (M l))) ∧ |
| SmoothCircleOn (F l ∘ sphereRealMap fi) |
| (sphereRealMap f '' (realChartDomain (openCompletedRegion T) ∩ interior (M l))) ∧ |
| (∀ x ∈ realChartDomain (openCompletedRegion T) ∩ interior (M l), |
| ((F l ∘ sphereRealMap fi) ∘ sphereRealMap f) =ᶠ[𝓝 x] F l)) ∧ |
| (∀ l (s : Set (ℝ × ℝ)), MeasurableSet s → |
| s ⊆ realChartDomain (openCompletedRegion T) ∩ interior (M l) → |
| (∫ y in sphereRealMap f '' s, |
| fieldEnergy (circleGradient (F l ∘ sphereRealMap fi)) (fun _ => 1) y) = |
| ∫ x in s, fieldEnergy (circleGradient (F l)) g x) ∧ |
| (∀ j, ContDiffOn ℝ ∞ (W j ∘ sphereRealMap fi) (realChartDomain V) ∧ |
| ∀ x ∈ realChartDomain (openCompletedRegion T), |
| ((W j ∘ sphereRealMap fi) ∘ sphereRealMap f) =ᶠ[𝓝 x] W j) ∧ |
| (∀ j (s : Set (ℝ × ℝ)), MeasurableSet s → s ⊆ realChartDomain (openCompletedRegion T) → |
| (∫ y in sphereRealMap f '' s, gradientEnergy (W j ∘ sphereRealMap fi) (fun _ => 1) y) = |
| ∫ x in s, gradientEnergy (W j) g x) ∧ |
| (∀ p ∈ openCompletedRegion T, IsMetricConformalAt g f p) := by |
| obtain ⟨V, f, fi, hV, heq, hconf, hmetric, hnorm⟩ := |
| principal_finite_metric_circle_model hcompact hfinite T hdisj hKQ g hg hPD hfar hcore |
| have hinf := hKQ (infty_mem_sphericalCore K) |
| have hO : IsOpen (realChartDomain (openCompletedRegion T)) := |
| isOpen_realChartDomain (isOpen_openCompletedRegion T) |
| have hFO (l : Fin n) : SmoothCircleOn (F l) |
| (realChartDomain (openCompletedRegion T) ∩ interior (M l)) := by |
| intro x hx |
| exact hF.1 l x ⟨hQΩ (openCompletedRegion_subset_completedRegion T hx.1), hx.2⟩ |
| have hWO (j : R) : ContDiffOn ℝ ∞ (W j) (realChartDomain (openCompletedRegion T)) := |
| (hW j).mono (fun _ hx => hQΩ (openCompletedRegion_subset_completedRegion T hx)) |
| have himage : sphereRealMap f '' realChartDomain (openCompletedRegion T) = |
| realChartDomain V := (heq.real_bijOn hinf hnorm.1).image_eq |
| refine ⟨V, f, fi, hV, heq, hconf, hnorm, ?_, ?_, ?_, ?_, hmetric⟩ |
| · intro l |
| exact transported_circle_coordinate heq hV.1 hinf hnorm.1 |
| (hO.inter isOpen_interior) inter_subset_left (F l) (hFO l) |
| · intro l s hs hsO |
| exact integral_transported_circle_coordinate heq hV.1 hinf hnorm.1 |
| (hO.inter isOpen_interior) inter_subset_left (F l) (hFO l) hs hsO |
| hmetric (fun x _ => hPD x) |
| · intro j |
| obtain ⟨_, hsmooth, hpull⟩ := transported_real_coordinate heq hV.1 hinf hnorm.1 |
| hO (Subset.refl _) (W j) (hWO j) |
| exact ⟨himage ▸ hsmooth, hpull⟩ |
| · intro j s hs hsO |
| exact integral_transported_real_coordinate heq hV.1 hinf hnorm.1 hO (Subset.refl _) |
| (W j) (hWO j) hs hsO hmetric (fun x _ => hPD x) |
|
|
| end CircleDomainRigidity.FiniteTransfer |
|
|
| end |
|
|
| end OAI |
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|
