| import Mathlib |
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| namespace OAI |
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| namespace Yau |
| namespace Target |
| open Bundle Manifold Matrix Filter |
| open scoped ContDiff Topology ENNReal |
| noncomputable section |
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| abbrev Base := Metric.sphere (0 : EuclideanSpace ℝ (Fin 5)) 1 |
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| abbrev Manifold5 := Base × Circle |
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| abbrev Model := EuclideanSpace ℝ (Fin 4) × EuclideanSpace ℝ (Fin 1) |
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| abbrev modelWithCorners := (𝓡 4).prod (𝓡 1) |
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| abbrev SmoothMetric := ContMDiffRiemannianMetric modelWithCorners ∞ Model |
| (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) |
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| abbrev Index := Fin (Module.finrank ℝ Model) |
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| def frame : Module.Basis Index ℝ Model := Module.finBasis ℝ Model |
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| 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)) |
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| def volumeFactor (g : SmoothMetric) (c : PartialEquiv Manifold5 Model) |
| (y : Model) : ℝ := Real.sqrt (metricMatrix g c y).det |
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| 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) |
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| 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} |
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| 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 ⊤) |
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|
| end |
| end Target |
| end Yau |
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| end OAI |
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