File size: 4,313 Bytes
a97875e | 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 | 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
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