File size: 3,413 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 | import OAI.Analysis.CircleDomains.Transfer.TransportedTerminalBranches
namespace OAI
/-!
# Terminal estimates on an actual finite circle model
The terminal assignment is constructed from the smooth model. The same
branches produced on physical terminal collars have the required summed
cluster-oscillation bound at their assigned target components.
-/
open Set Filter Function
open scoped Topology ContDiff
noncomputable section
namespace CircleDomainRigidity
open FiniteTransfer
theorem exists_finite_model_terminal_estimates
(hsard : SmoothPlanarSard) (hsch : PlanarSchoenflies)
{I J : Type*} [Fintype I] [Fintype J] {n : ℕ}
(T : I → ℂ ≃ₜ ℂ) (H : J → ℂ ≃ₜ ℂ)
(hTd : Pairwise (Disjoint on fun i => closure (jordanInterior (T i))))
(hTH : ∀ i j, Disjoint (closure (jordanInterior (T i))) (closure (jordanInterior (H j))))
{φ φi : Sphere → Sphere} {V : Set Sphere}
(heq : IsSmoothSphereEquivalence φ φi (openCompletedRegion T) V)
(hV : IsOpen V) (hfix : φ OnePoint.infty = OnePoint.infty)
{Ω : Set (ℝ × ℝ)} (O : I → Set (ℝ × ℝ)) (hO : ∀ i, IsOpen (O i))
(hTO : ∀ i, frontier (realJordanInterior (T i)) ⊆ O i)
(M : Fin n → Set (ℝ × ℝ)) (f F : Fin n → (ℝ × ℝ) → UnitAddCircle)
(χ : (ℝ × ℝ) → ℝ) (u : I → Fin n → (ℝ × ℝ) → ℝ)
(S : Fin n → Finset I) (c : I → Fin n → ℝ)
(hu : ∀ l i, i ∈ S l → ContDiffOn ℝ ∞ (u i l) (O i))
(hproj : ∀ l i, i ∈ S l → ∀ x ∈ O i, (u i l x : UnitAddCircle) = f l x)
(hχone : ∀ i x, x ∈ O i → χ x = 1)
(N : ℕ) {δ η : ℝ} (hδ : 0 < δ) (hδh : δ < reciprocalGridStep N / 4)
(θ : Fin n → ℝ)
(hF : IsMaskedFilledCircleFamily (fun j => realJordanInterior (H j)) Ω M
(circleCutoffFamily f χ N δ θ) F)
(hc : ∀ l i, i ∈ S l → 0 ≤ c i l)
(hbound : ∀ l i, i ∈ S l → ∀ x ∈ frontier (jordanInterior (T i)),
∀ y ∈ frontier (jordanInterior (T i)),
|shiftedStaircase (reciprocalGridStep N) δ (θ l) (u i l (Complex.equivRealProdCLM x)) -
shiftedStaircase (reciprocalGridStep N) δ (θ l) (u i l (Complex.equivRealProdCLM y))| ≤
c i l + 4 * reciprocalGridStep N)
(hsmall : reciprocalGridStep N * Real.sqrt (Fintype.card I : ℝ) < η) :
∃ (a : TerminalComponentAssignment T φ V) (w : I → Fin n → (ℝ × ℝ) → ℝ),
(∀ l i, i ∈ S l → ContDiffOn ℝ ∞ (w i l) (O i) ∧
∀ᶠ x in 𝓝ˢ (frontier (realJordanInterior (T i))), (w i l x : UnitAddCircle) = F l x) ∧
(∀ l, Real.sqrt (∑ i ∈ S l,
max (Metric.diam (w i l '' frontier (realJordanInterior (T i))) - c i l) 0 ^ 2) < 4 * η) ∧
(∀ l, Real.sqrt (∑ i ∈ S l, max (Metric.diam
(componentClusterValues V (a.point i) (sphereRealFunction (w i l) ∘ φi)) - c i l) 0 ^ 2)
< 4 * η) := by
obtain ⟨a⟩ := exists_terminal_component_assignment hsard hsch T hTd heq hV hfix
obtain ⟨w, hw, hexcess⟩ := physical_masked_terminal_branches T H hTH O hO hTO
M f F χ u S c hu hproj hχone N hδ hδh θ hF hc hbound hsmall
refine ⟨a, w, hw, hexcess, ?_⟩
intro l
exact (a.real_cluster_excess_le heq (S l) (fun i => w i l) (fun i => c i l)
O hO hTO (fun i hi => (hw l i hi).1.continuousOn)).trans_lt (hexcess l)
end CircleDomainRigidity
end
end OAI
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