File size: 2,708 Bytes
8e4d533 | 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 | import Mathlib
import OAI.Geometry.Cannon.Boundary.BoundaryModulusBounded
import OAI.Geometry.Cannon.Energy.ApproxSelfSimilar
namespace OAI
section
section
section
noncomputable section
namespace CannonRelease
open CannonGraph CannonVisual CannonGeometry CannonNet CannonCells CannonCharts Set Metric
universe v
variable {G : Type v} [Group G] [TopologicalSpace G] [DiscreteTopology G]
variable (D : CayleyData G) {δ : ℕ} (hδ : Thin D.graph δ)
theorem boundary_uniformization_inputs (φ : Boundary D ≃ₜ SphereTwo) :
let _ := visualMetricSpace D hδ
let a := Real.exp (2*visualExponent δ)
let LE := 8*Real.exp (visualExponent δ*(4*(8*δ+6):ℕ))
∀ d κ : ℝ, 0 < d → d < diam (univ : Set (Boundary D))/10 →
d ≤ 1/(LE*3^2) → 2*a < κ →
PathConnectedSpace (Boundary D) ∧ ApproxSelfSimilar (Boundary D) ∧
∃ T : ℕ → Finset (Boundary D), ∃ N : ℕ → ℕ, ∃ B : ℝ, ∀ m,
IsApproximation (fun i : T (N m) => i.val)
(fun i : T (N m) => ball i.val (2*(a^(N m))⁻¹)) ((2:ℝ)^m)⁻¹ κ ∧
modulus (fun i : T (N m) => ball i.val (2*(a^(N m))⁻¹)) (macroFamily d) ≤ B := by
classical
let _ := visualMetricSpace D hδ
dsimp only
intro d κ hd hdia hdc hκ
let a := Real.exp (2*visualExponent δ)
have ha : 1 < a := Real.one_lt_exp_iff.mpr (mul_pos (by norm_num) (visualExponent_pos δ))
let _ : PathConnectedSpace SphereTwo := CannonCharts.sphere_pathConnected
refine ⟨φ.symm.surjective.pathConnectedSpace φ.symm.continuous,
boundary_approx_self_similar D hδ,?_⟩
obtain ⟨T,hT⟩ := exists_net_sequence (X:=Boundary D) ha
have hscale (m : ℕ) : ∃ n : ℕ,
((2:ℝ)^m)⁻¹/a < (a^n)⁻¹ ∧ (a^n)⁻¹ ≤ ((2:ℝ)^m)⁻¹ := by
apply exists_scale_below ha (by positivity)
exact inv_le_one_of_one_le₀ (one_le_pow₀ (by norm_num))
choose N hNlow hNupp using hscale
have hsep (n : ℕ) (i j : T n) (hij : i ≠ j) : (a^n)⁻¹ ≤ dist i.val j.val :=
(hT n).1 i.val i.property j.val j.property (fun hh => hij (Subtype.ext hh))
have hnet (n : ℕ) (x : Boundary D) : ∃ i : T n, dist i.val x ≤ (a^n)⁻¹ := by
obtain ⟨y,hy,hh⟩ := (hT n).2 x
exact ⟨⟨y,hy⟩,hh⟩
have hdia' : d < diam (univ : Set (Boundary D)) := by
have := diam_nonneg (s:= (univ : Set (Boundary D)))
linarith
obtain ⟨B,hB⟩ := boundary_modulus_bounded D hδ φ d hd hdia' hdc
(fun n => ↥(T n)) (fun n i => i.val) hsep hnet
refine ⟨T,N,B,?_⟩
intro m
exact ⟨balls_isApproximation (fun i : T (N m) => i.val) ha (by positivity) hκ
(hNlow m) (hNupp m) (hsep (N m)) (hnet (N m)),hB (N m)⟩
end CannonRelease
end
end
end
end
end OAI
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