File size: 2,666 Bytes
8fadf68 | 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 | import Mathlib
namespace OAI
namespace Yau
namespace Target
open Bundle Manifold Matrix Filter
open scoped ContDiff Topology ENNReal
noncomputable section
abbrev Base := Metric.sphere (0 : EuclideanSpace ℝ (Fin 5)) 1
abbrev Manifold5 := Base × Circle
abbrev Model := EuclideanSpace ℝ (Fin 4) × EuclideanSpace ℝ (Fin 1)
abbrev modelWithCorners := (𝓡 4).prod (𝓡 1)
abbrev SmoothMetric := ContMDiffRiemannianMetric modelWithCorners ∞ Model
(fun x : Manifold5 ↦ TangentSpace modelWithCorners x)
abbrev Index := Fin (Module.finrank ℝ Model)
def frame : Module.Basis Index ℝ Model := Module.finBasis ℝ Model
def metricMatrix (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)
(y : Model) : Matrix Index Index ℝ := fun i j ↦
g.inner (c.symm y)
(mfderiv 𝓘(ℝ, Model) modelWithCorners c.symm y (frame i))
(mfderiv 𝓘(ℝ, Model) modelWithCorners c.symm y (frame j))
def volumeFactor (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)
(y : Model) : ℝ := Real.sqrt (metricMatrix g c y).det
def chartLaplacian (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)
(u : Manifold5 → ℝ) (y : Model) : ℝ :=
(volumeFactor g c y)⁻¹ * ∑ i, fderiv ℝ
(fun z ↦ volumeFactor g c z * ∑ j,
(metricMatrix g c z)⁻¹ i j * fderiv ℝ (u ∘ c.symm) z (frame j)) y (frame i)
def nodalMeasure (g : SmoothMetric) (u : Manifold5 → ℝ) : ℝ≥0∞ :=
letI : RiemannianBundle (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) :=
⟨g.toRiemannianMetric⟩
letI : IsContMDiffRiemannianBundle modelWithCorners ∞ Model
(fun x : Manifold5 ↦ TangentSpace modelWithCorners x) := inferInstance
letI : MeasurableSpace Manifold5 := borel Manifold5
letI : BorelSpace Manifold5 := ⟨rfl⟩
letI : IsContinuousRiemannianBundle Model
(fun x : Manifold5 ↦ TangentSpace modelWithCorners x) :=
⟨g.inner, g.toContinuousRiemannianMetric.continuous, fun _ _ _ ↦ rfl⟩
letI : EMetricSpace Manifold5 := EMetricSpace.ofRiemannianMetric modelWithCorners Manifold5
MeasureTheory.Measure.hausdorffMeasure (4 : ℝ) {x | u x = 0}
def MainTarget : Prop :=
∃ (g : SmoothMetric) (lam : ℕ → ℝ) (u : ℕ → Manifold5 → ℝ),
(∀ k, 0 < lam k) ∧
(∀ k, u k ≠ 0) ∧
(∀ k, ContMDiff modelWithCorners 𝓘(ℝ, ℝ) ∞ (u k)) ∧
(∀ k x, -chartLaplacian g (extChartAt modelWithCorners x) (u k) ((extChartAt modelWithCorners x) x) =
lam k * u k x) ∧
Tendsto lam atTop atTop ∧
Tendsto (fun k ↦ nodalMeasure g (u k) / ENNReal.ofReal (Real.sqrt (lam k)))
atTop (𝓝 ⊤)
end
end Target
end Yau
end OAI
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