| import OAI.Analysis.CircleDomains.Rigidity.MaskedModelTerminalBranches |
| import OAI.Analysis.CircleDomains.Rigidity.AssembledMaskedModel |
| import OAI.Analysis.CircleDomains.Transfer.TerminalTransfer |
|
|
| namespace OAI |
|
|
| /-! Actual quotient branch neighborhoods induce the physical terminal |
| collars needed by the common masked model. -/ |
|
|
| open Set Function Filter |
| open scoped Topology ContDiff |
| noncomputable section |
| namespace CircleDomainRigidity.FiniteTransfer |
|
|
| theorem marked_model_terminal_control |
| {I J : Type*} [Fintype I] [Fintype J] {n N : ℕ} |
| {G B : Set Sphere} (hG : IsOpen G) {π : Sphere → Sphere} (hπ : Continuous π) |
| {p : I → Sphere} {O : I → Set Sphere} |
| (D : PhysicalTerminalDisks G π B p O) (H : J → ℂ ≃ₜ ℂ) |
| (hTH : ∀ i j, Disjoint (closure (jordanInterior (D.H i))) (closure (jordanInterior (H j)))) |
| {K : Set (ℝ × ℝ)} (Mask : Fin n → Set Sphere) |
| {f : Fin n → (ℝ × ℝ) → UnitAddCircle} {χ : (ℝ × ℝ) → ℝ} |
| {θ : Fin n → ℝ} {ε : ℝ} |
| (Q : MaskedModelData D.H H (realChartDomain G) K (fun l => physicalMask π (Mask l)) f χ N θ ε) |
| (hTC : ∀ i, Disjoint (closedTerminalCap (D.H i)) (cutoffCore χ)) |
| (V : I → Set Sphere) (hV : ∀ i, IsOpen (V i)) (hOV : ∀ i, O i ⊆ V i) |
| (u : I → Fin n → Sphere → ℝ) (S : Fin n → Finset I) (a : I → Fin n → ℝ) |
| (hVM : ∀ l i, i ∈ S l → V i ⊆ Mask l) |
| (hu : ∀ l i, i ∈ S l → ContDiffOn ℝ ∞ |
| (fun x : ℝ × ℝ => u i l (π (Complex.equivRealProdCLM.symm x : Sphere))) |
| (realChartDomain G ∩ realChartDomain (π ⁻¹' V i))) |
| (hproj : ∀ l i, i ∈ S l → ∀ x, |
| x ∈ realChartDomain G ∩ realChartDomain (π ⁻¹' V i) → |
| (u i l (π (Complex.equivRealProdCLM.symm x : Sphere)) : UnitAddCircle) = f l x) |
| (ha : ∀ l i, i ∈ S l → 0 ≤ a i l) |
| (hbound : ∀ l i, i ∈ S l → ∀ x ∈ frontier (jordanInterior (D.H i)), |
| ∀ y ∈ frontier (jordanInterior (D.H i)), |
| |shiftedStaircase (reciprocalGridStep N) |
| (staircaseWidth (reciprocalGridStep N) Q.widthIndex) (θ l) (u i l (π (x : Sphere))) - |
| shiftedStaircase (reciprocalGridStep N) |
| (staircaseWidth (reciprocalGridStep N) Q.widthIndex) (θ l) (u i l (π (y : Sphere)))| ≤ |
| a i l + 4 * reciprocalGridStep N) |
| {η : ℝ} (hsmall : reciprocalGridStep N * Real.sqrt (Fintype.card I : ℝ) < η) : |
| ∃ (E : I → Set (ℝ × ℝ)) (w : I → Fin n → (ℝ × ℝ) → ℝ), |
| (∀ i, IsOpen (E i) ∧ frontier (realJordanInterior (D.H i)) ⊆ E i) ∧ |
| (∀ l i, i ∈ S l → ContDiffOn ℝ ∞ (w i l) (E i) ∧ |
| ∀ᶠ x in 𝓝ˢ (frontier (realJordanInterior (D.H i))), (w i l x : UnitAddCircle) = Q.functions l x) ∧ |
| ∀ l, Real.sqrt (∑ i ∈ S l, |
| max (Metric.diam (w i l '' frontier (realJordanInterior (D.H i))) - a i l) 0 ^ 2) < 4 * η := by |
| let E (i : I) := (realChartDomain G ∩ realChartDomain (π ⁻¹' V i)) ∩ interior {x | χ x = 1} |
| have hE (i : I) : IsOpen (E i) := |
| ((isOpen_realChartDomain hG).inter (isOpen_realChartDomain ((hV i).preimage hπ))).inter isOpen_interior |
| have hDE (i : I) : frontier (realJordanInterior (D.H i)) ⊆ E i := by |
| rw [frontier_realJordanInterior] |
| rintro x ⟨z, hz, rfl⟩ |
| refine ⟨⟨?_, ?_⟩, ?_⟩ |
| · simpa only [realChartDomain, mem_preimage, ContinuousLinearEquiv.symm_apply_apply] using |
| (D.frontier_good i z hz).1 |
| · simpa only [realChartDomain, mem_preimage, ContinuousLinearEquiv.symm_apply_apply] using |
| hOV i (D.closure_subset i (frontier_subset_closure hz)) |
| · exact boundary_in_one_plateau_of_disjoint_cutoffCore (D.H i) χ (hTC i) z hz |
| obtain ⟨w, hw, hexcess⟩ := Q.terminal_branches hTH E hE hDE |
| (fun i l x => u i l (π (Complex.equivRealProdCLM.symm x : Sphere))) S a |
| (fun l i hi => (hu l i hi).mono inter_subset_left) |
| (fun l i hi x hx => hVM l i hi hx.1.2) |
| (fun l i hi x hx => hproj l i hi x hx.1) |
| (fun i x hx => (show x ∈ {y | χ y = 1} from interior_subset hx.2)) ha |
| (fun l i hi x hx y hy => by |
| simpa only [ContinuousLinearEquiv.symm_apply_apply] using hbound l i hi x hx y hy) hsmall |
| exact ⟨E, w, fun i => ⟨hE i, hDE i⟩, hw, hexcess⟩ |
|
|
| end CircleDomainRigidity.FiniteTransfer |
|
|
| end |
|
|
| end OAI |
|
|
