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7d9acbd | 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 | import Mathlib
namespace OAI
noncomputable section
open Filter MeasureTheory
open scoped BigOperators Topology ENNReal ContDiff
namespace HarmonicCounterexample
abbrev Space (n : ℕ) := Fin n → ℝ
abbrev Mat (n : ℕ) := Matrix (Fin n) (Fin n) ℝ
/-- A genuine smooth positive definite covariant metric tensor on R^n. -/
structure SmoothMetric (n : ℕ) where
coeff : Space n → Mat n
smooth : ∀ i j, ContDiff ℝ ∞ (fun x => coeff x i j)
positive : ∀ x, (coeff x).PosDef
/-- The coordinate directional derivative. -/
def coordDeriv {n : ℕ} (i : Fin n) (f : Space n → ℝ) (x : Space n) : ℝ :=
fderiv ℝ f x (Pi.single i 1)
/-- Levi-Civita connection in the single global Cartesian chart. -/
def christoffel {n : ℕ} (g : SmoothMetric n) (x : Space n)
(k i j : Fin n) : ℝ :=
(1 / 2 : ℝ) * ∑ l, (g.coeff x)⁻¹ k l *
(coordDeriv i (fun y => g.coeff y j l) x +
coordDeriv j (fun y => g.coeff y i l) x -
coordDeriv l (fun y => g.coeff y i j) x)
/-- Ricci tensor, with the convention in which the round sphere has positive Ricci. -/
def ricci {n : ℕ} (g : SmoothMetric n) (x : Space n) (i j : Fin n) : ℝ :=
(∑ k, (coordDeriv k (fun y => christoffel g y k i j) x -
coordDeriv j (fun y => christoffel g y k i k) x)) +
∑ k, ∑ l, (christoffel g x k i j * christoffel g x l k l -
christoffel g x l i k * christoffel g x k j l)
def RicciNonnegative {n : ℕ} (g : SmoothMetric n) : Prop :=
∀ (x v : Space n), 0 ≤ ∑ i, ∑ j, ricci g x i j * v i * v j
/-- Trace of the covariant Hessian, i.e. the Laplace-Beltrami operator. -/
def laplaceBeltrami {n : ℕ} (g : SmoothMetric n) (u : Space n → ℝ)
(x : Space n) : ℝ :=
∑ i, ∑ j, (g.coeff x)⁻¹ i j *
(coordDeriv i (coordDeriv j u) x - ∑ k, christoffel g x k i j * coordDeriv k u x)
/-- Riemannian length of a path on the unit parameter interval. -/
def pathLength {n : ℕ} (g : SmoothMetric n) (γ : ℝ → Space n) : ℝ :=
∫ t in (0 : ℝ)..1, Real.sqrt
(∑ i, ∑ j, g.coeff (γ t) i j * deriv γ t i * deriv γ t j)
/-- Infimum of smooth path lengths, the intrinsic Riemannian distance. -/
def distance {n : ℕ} (g : SmoothMetric n) (x y : Space n) : ℝ :=
sInf {L : ℝ | ∃ γ : ℝ → Space n,
ContDiff ℝ 1 γ ∧ γ 0 = x ∧ γ 1 = y ∧ L = pathLength g γ}
/-- Sequential metric completeness, for the intrinsic distance, not the background one. -/
def Complete {n : ℕ} (g : SmoothMetric n) : Prop :=
∀ z : ℕ → Space n,
(∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ p ≥ N, ∀ q ≥ N, distance g (z p) (z q) < ε) →
∃ x : Space n, ∀ ε : ℝ, 0 < ε → ∃ N : ℕ, ∀ p ≥ N, distance g (z p) x < ε
def EuclideanNearOrigin {n : ℕ} (g : SmoothMetric n) : Prop :=
∃ ε : ℝ, 0 < ε ∧ ∀ x : Space n, (∑ i, x i ^ 2) < ε ^ 2 → g.coeff x = 1
/-- Riemannian measure of the intrinsic open ball, retaining infinite values. -/
def ballVolume {n : ℕ} (g : SmoothMetric n) (R : ℝ) : ℝ≥0∞ :=
∫⁻ x in {x : Space n | distance g 0 x < R},
ENNReal.ofReal (Real.sqrt (g.coeff x).det) ∂volume
/-- The ordinary Euclidean unit ball volume, in Cartesian Lebesgue measure. -/
def omega (n : ℕ) : ℝ≥0∞ :=
volume {x : Space n | (∑ i, x i ^ 2) < 1}
/-- The volume-ratio limit is finite, positive, and strictly below one. -/
def PositiveSubunitAVR {n : ℕ} (g : SmoothMetric n) : Prop :=
∃ a : ℝ, 0 < a ∧ a < 1 ∧
Tendsto (fun R : ℝ => ballVolume g R / (omega n * ENNReal.ofReal (R ^ n)))
atTop (𝓝 (ENNReal.ofReal a))
/-- Membership in the actual space H_k, with the bound required at every point. -/
def HarmonicGrowth {n : ℕ} (g : SmoothMetric n) (k : ℕ) (u : Space n → ℝ) : Prop :=
ContDiff ℝ ∞ u ∧ (∀ x, laplaceBeltrami g u x = 0) ∧
∃ C : ℝ, 0 ≤ C ∧ ∀ x, |u x| ≤ C * (1 + distance g 0 x) ^ k
def euclideanDimension (n k : ℕ) : ℕ :=
(n + k - 1).choose k + (n + k - 2).choose (k - 1)
/-- The manuscript's unconditional main existence claim. -/
def MainClaim : Prop :=
∃ (n k : ℕ) (g : SmoothMetric n), Even n ∧ 8 ≤ n ∧ 2 ≤ k ∧
Complete g ∧ RicciNonnegative g ∧ EuclideanNearOrigin g ∧ PositiveSubunitAVR g ∧
∃ u : Fin (euclideanDimension n k + 1) → (Space n → ℝ),
LinearIndependent ℝ u ∧ ∀ i, HarmonicGrowth g k (u i)
end HarmonicCounterexample
end
end OAI
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