| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.HilbertAtlas |
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|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
| lemma integral_atlas {f : ℕ × Euc n → ℝ} (hf : Integrable f q.atlasMeasure) : |
| ∫ w, f w ∂q.atlasMeasure = ∑' i, ∫ z, q.density i z*f (i,z) ∂volume.restrict (q.chart i).domain := by |
| rw [atlasMeasure,integral_sum_measure hf] |
| apply tsum_congr |
| intro i |
| rw [(measurableEmbedding_prodMk_left i).integral_map] |
| rw [coordinateMeasure,integral_withDensity_eq_integral_toReal_smul₀ |
| (q.density_integrable i).aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| simp only [ENNReal.toReal_ofReal (q.density_nonneg i _),smul_eq_mul] |
|
|
| lemma ae_atlas_of_chart {p : ℕ × Euc n → Prop} |
| (h : ∀ i, ∀ᵐ z ∂volume.restrict (q.chart i).domain, p (i,z)) : ∀ᵐ w ∂q.atlasMeasure, p w := by |
| apply Measure.ae_sum_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr |
| exact (withDensity_absolutelyContinuous _ _).ae_le (h i) |
|
|
| lemma ae_gradient_eq {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| ∀ᵐ w ∂q.atlasMeasure, q.gradient hu w = q.normalizedCovector w.1 u w.2 := |
| (q.memLp_gradientFunction hu).coeFn_toLp |
|
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| lemma ae_gradient_bound {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| ∀ᵐ w ∂q.atlasMeasure, ‖q.gradient hu w‖ ≤ K := by |
| filter_upwards [q.ae_gradient_eq hu,q.ae_gradientFunction_bound hu] with w he hb |
| rw [he] |
| exact hb |
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal MatrixOrder |
|
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| namespace CAT0Fillings |
| lemma dot_mulVec_sqrt_cross {n : ℕ} (P : Matrix (Fin n) (Fin n) ℝ) (hP : P.PosSemidef) |
| (u v : Fin n → ℝ) : |
| u ⬝ᵥ P.mulVec v = (CFC.sqrt P).mulVec u ⬝ᵥ (CFC.sqrt P).mulVec v := by |
| have hsym : (CFC.sqrt P).transpose = CFC.sqrt P := |
| (Matrix.isHermitian_iff_isSymm.mp (CFC.sqrt_nonneg P).posSemidef.isHermitian).eq |
| conv_lhs => rw [← CFC.sq_sqrt P hP.nonneg,pow_two, ← Matrix.mulVec_mulVec] |
| rw [Matrix.dotProduct_mulVec, ← Matrix.mulVec_transpose, hsym] |
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
| lemma ae_normalizedCovector_inner (i : ℕ) (u v : X → ℝ) : |
| ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| inner ℝ (q.normalizedCovector i u z) (q.normalizedCovector i v z) = |
| q.covector i u z ⬝ᵥ (q.gram i z)⁻¹.mulVec (q.covector i v z) := by |
| filter_upwards [q.differential i] with z hz |
| have hG : (q.gram i z).PosDef := polarizationMatrix_posDef _ _ hz.2.2 hz.2.1 |
| simp only [normalizedCovector,EuclideanSpace.inner_eq_star_dotProduct,star_trivial] |
| rw [dotProduct_comm] |
| exact (dot_mulVec_sqrt_cross _ hG.inv.posSemidef _ _).symm |
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| end OAI |
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|
