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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 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 | import Mathlib
namespace OAI
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace CAT0Fillings
attribute [local instance] Classical.propDecidable
universe u
abbrev Euc (k : ℕ) := EuclideanSpace ℝ (Fin k)
abbrev Functional (X : Type u) (k : ℕ) := (X → ℝ) → (Fin k → X → ℝ) → ℝ
section MetricCurrents
variable {X : Type u} [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
def BoundedLip (b : X → ℝ) : Prop :=
(∃ K : ℝ≥0, LipschitzWith K b) ∧ ∃ M : ℝ, ∀ x, |b x| ≤ M
def Admissible {k : ℕ} (b : X → ℝ) (π : Fin k → X → ℝ) : Prop :=
BoundedLip b ∧ ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)
def Controls {k : ℕ} (T : Functional X k) (μ : Measure X) : Prop :=
∀ (b : X → ℝ) (π : Fin k → X → ℝ), BoundedLip b →
(∀ i, LipschitzWith 1 (π i)) →
|T b π| ≤ ∫ x, |b x| ∂μ
noncomputable def mass {k : ℕ} (T : Functional X k) : ℝ :=
sInf {r : ℝ | ∃ μ : Measure X,
IsFiniteMeasure μ ∧ Controls T μ ∧ r = μ.real univ}
structure IsMetricCurrent {k : ℕ} (T : Functional X k) : Prop where
offDomain : ∀ b π, ¬Admissible b π → T b π = 0
linearFirst : ∀ b₁ b₂ π (a c : ℝ), BoundedLip b₁ → BoundedLip b₂ →
(∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) →
T (fun x => a * b₁ x + c * b₂ x) π = a * T b₁ π + c * T b₂ π
linearCoord : ∀ b π i f (a c : ℝ), Admissible b π →
(∃ K : ℝ≥0, LipschitzWith K f) →
T b (Function.update π i (fun x => a * π i x + c * f x)) =
a * T b π + c * T b (Function.update π i f)
sequentialContinuity : ∀ b π (πs : ℕ → Fin k → X → ℝ), BoundedLip b →
(∀ i, ∃ K : ℝ≥0, ∀ j, LipschitzWith K (πs j i)) →
(∀ i x, Tendsto (fun j => πs j i x) atTop (𝓝 (π i x))) →
Tendsto (fun j => T b (πs j)) atTop (𝓝 (T b π))
locality : ∀ b π, Admissible b π →
(∃ (i : Fin k) (U : Set X) (c : ℝ), IsOpen U ∧ Function.support b ⊆ U ∧
∀ x ∈ U, π i x = c) → T b π = 0
finiteMass : ∃ μ : Measure X, IsFiniteMeasure μ ∧ Controls T μ
noncomputable def boundarySucc {k : ℕ} (T : Functional X (k + 1)) : Functional X k :=
fun b π => if Admissible b π then T (fun _ => 1) (Matrix.vecCons b π) else 0
noncomputable def boundary : {k : ℕ} → Functional X k → Functional X (k - 1)
| 0, _ => fun _ _ => 0
| _ + 1, T => boundarySucc T
structure IntegerChart (X : Type u) [MetricSpace X] (k : ℕ) where
domain : Set (Euc k)
borel : MeasurableSet domain
bounded : Bornology.IsBounded domain
param : domain → X
bilipschitz : ∃ L U : ℝ≥0, LipschitzWith L param ∧ AntilipschitzWith U param
multiplicity : Euc k → ℤ
integrable : Integrable (fun z => (multiplicity z : ℝ)) (volume.restrict domain)
namespace IntegerChart
variable {k : ℕ} (C : IntegerChart X k)
noncomputable def scalar (f : X → ℝ) (z : Euc k) : ℝ :=
if hz : z ∈ C.domain then f (C.param ⟨z, hz⟩) else 0
noncomputable def jacobian (π : Fin k → X → ℝ) (z : Euc k) : ℝ :=
Matrix.det (fun i j =>
(fderivWithin ℝ (C.scalar (π i)) C.domain z) (EuclideanSpace.single j 1))
noncomputable def action : Functional X k :=
fun b π => if Admissible b π then
∫ z in C.domain, (C.multiplicity z : ℝ) * C.scalar b z * C.jacobian π z
else 0
def image : Set X := Set.range C.param
end IntegerChart
def IntegerRectifiable {k : ℕ} (T : Functional X k) : Prop :=
∃ C : ℕ → IntegerChart X k,
Pairwise (fun i j => Disjoint (C i).image (C j).image) ∧
(∀ i, IsMetricCurrent (C i).action) ∧
Summable (fun i => mass (C i).action) ∧
∀ b π, T b π = ∑' i, (C i).action b π
def IsIntegral : (k : ℕ) → Functional X k → Prop
| 0, T => IsMetricCurrent T ∧ IntegerRectifiable T
| _ + 1, T => IsMetricCurrent T ∧ IntegerRectifiable T ∧
IsMetricCurrent (boundarySucc T) ∧ IntegerRectifiable (boundarySucc T)
abbrev IntegralCurrent (X : Type u) [MetricSpace X] [MeasurableSpace X] [BorelSpace X]
(k : ℕ) := { T : Functional X k // IsIntegral k T }
def CompactlySupported {k : ℕ} (T : Functional X k) : Prop :=
∃ K : Set X, IsCompact K ∧
∀ b π, Admissible b π → (∀ x ∈ K, b x = 0) → T b π = 0
def IsCycle {k : ℕ} (T : Functional X k) : Prop := boundary T = fun _ _ => 0
end MetricCurrents
section Geometry
variable (X : Type u) [MetricSpace X]
def IsCAT0 : Prop :=
∃ segment : X → X → ℝ → X,
(∀ x y, segment x y 0 = x ∧ segment x y 1 = y) ∧
(∀ x y s t, s ∈ Icc (0 : ℝ) 1 → t ∈ Icc (0 : ℝ) 1 →
dist (segment x y s) (segment x y t) = |s - t| * dist x y) ∧
(∀ o x y s t, s ∈ Icc (0 : ℝ) 1 → t ∈ Icc (0 : ℝ) 1 →
dist (segment o x s) (segment o y t) ^ 2 ≤
(s * dist o x - t * dist o y) ^ 2 +
s * t * ((dist x y) ^ 2 - (dist o x - dist o y) ^ 2))
end Geometry
noncomputable def omega (k : ℕ) : ℝ :=
(volume (Metric.ball (0 : Euc k) 1)).toReal
noncomputable def sphereArea (n : ℕ) : ℝ := (n + 1 : ℝ) * omega (n + 1)
noncomputable def fillingCoefficient (n : ℕ) : ℝ :=
1 / ((n + 1 : ℝ) * (sphereArea n) ^ (1 / (n : ℝ)))
noncomputable def fillingPower (n : ℕ) : ℝ := (n + 1 : ℝ) / (n : ℝ)
end CAT0Fillings
end OAI
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