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59d1c80 | 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 | import Mathlib
namespace OAI
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
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
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 SharpIntegralFillings
end OAI
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