| import Mathlib | |
| import OAI.Geometry.CAT0Fillings.Tangent.Quadratic | |
| import OAI.Geometry.CAT0Fillings.Density.Chart | |
| import OAI.Geometry.CAT0Fillings.Density.NearEuclidean | |
| namespace OAI | |
| section | |
| open Set Filter MeasureTheory Matrix | |
| open scoped Topology NNReal ENNReal MatrixOrder | |
| namespace CAT0Fillings | |
| namespace IntegerChart | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [CompactSpace X] [Nonempty X] {k : ℕ} | |
| lemma normalized_piece (C : IntegerChart X k) (hC : IsMetricCurrent C.action) | |
| {μ : Measure X} [IsFiniteMeasure μ] (hμ : Controls C.action μ) | |
| {s : Set (Euc k)} (hs : MeasurableSet s) (hss : s ⊆ C.domain) (hs0 : volume s ≠ 0) | |
| (hθ : ∀ᵐ z ∂volume.restrict s, 1 ≤ |(C.multiplicity z : ℝ)|) | |
| (p : Seminorm ℝ (Euc k)) (hp : ∀ v, p v = 0 ↔ v = 0) | |
| (hpara : ∀ u v, p (u+v)^2+p (u-v)^2 = 2*p u^2+2*p v^2) | |
| {ε : ℝ} (hε : 0 < ε) (hε1 : ε < 1) | |
| (hnear : ∀ y ∈ s, ∀ z ∈ s, | |
| (1-ε)*p (y-z) ≤ dist (C.paramExtended y) (C.paramExtended z) ∧ | |
| dist (C.paramExtended y) (C.paramExtended z) ≤ (1+ε)*p (y-z)) : | |
| ∃ (t : Set (Euc k)) (f : Euc k → X), MeasurableSet t ∧ volume t ≠ 0 ∧ Measurable f ∧ | |
| (∀ y ∈ t, ∀ z ∈ t, dist (f y) (f z) ≤ (1+ε)*dist y z) ∧ | |
| Measure.map f (volume.restrict t) ≤ ENNReal.ofReal ((1-ε)⁻¹^k) • μ := by | |
| let b := (EuclideanSpace.basisFun (Fin k) ℝ).toBasis | |
| let Q := polarizationMatrix p b | |
| let J := Real.sqrt Q.det | |
| have hQ : Q.PosDef := polarizationMatrix_posDef b p hp hpara | |
| have hJ : 0 < J := Real.sqrt_pos.mpr hQ.det_pos | |
| obtain ⟨e,he,hdet⟩ := exists_hilbert_norm_equiv p hp hpara | |
| let K : ℝ≥0 := ⟨(1-ε)⁻¹, inv_nonneg.mpr (sub_pos.mpr hε1).le⟩ | |
| let ℓ : Fin k → Euc k →L[ℝ] ℝ := fun i => matrixRowFunctional (CFC.sqrt Q) i | |
| have hrow : ∀ i y (hy : y ∈ s) z (hz : z ∈ s), | |
| |ℓ i y-ℓ i z| ≤ (K : ℝ)*dist (C.param ⟨y,hss hy⟩) (C.param ⟨z,hss hz⟩) := by | |
| intro i y hy z hz | |
| rw [←map_sub] | |
| apply (abs_matrixRow_sqrt_le Q hQ.posSemidef i (y-z)).trans | |
| have hh := (hnear y hy z hz).1 | |
| simp only [paramExtended,dite_eq_left (hss hy),dite_eq_left (hss hz)] at hh | |
| change Real.sqrt ((WithLp.ofLp (y-z)) ⬝ᵥ Q *ᵥ WithLp.ofLp (y-z)) ≤ _ | |
| rw [show Q = polarizationMatrix p b from rfl,sqrt_polarization_quadratic p hp hpara,WithLp.toLp_ofLp] | |
| change p (y-z) ≤ (1-ε)⁻¹*_ | |
| rw [←div_eq_inv_mul] | |
| apply (le_div_iff₀ (sub_pos.mpr hε1)).mpr | |
| simpa only [mul_comm] using hh | |
| have hm : (Matrix.of fun i j => ℓ i (EuclideanSpace.single j 1)) = CFC.sqrt Q := by | |
| ext i j | |
| simp only [ℓ,Matrix.of_apply,matrixRowFunctional_single] | |
| have hdetQ : |(CFC.sqrt Q).det| = J := by | |
| rw [show (CFC.sqrt Q).det = Real.sqrt Q.det by simpa only [RCLike.sqrt_real] using hQ.posSemidef.det_sqrt] | |
| exact abs_of_nonneg (Real.sqrt_nonneg _) | |
| have hpiece : densityPush (volume.restrict s) C.paramExtended | |
| (fun z => |(C.multiplicity z : ℝ)| * J) ≤ ((K^k) • μ).restrict (C.paramExtended '' s) := by | |
| simpa only [hm,hdetQ] using C.linear_piece_mass_le hC hμ hs hss ℓ K hrow | |
| have hconst : ENNReal.ofReal J • Measure.map C.paramExtended (volume.restrict s) ≤ (K^k) • μ := by | |
| calc | |
| _ = densityPush (volume.restrict s) C.paramExtended (fun _ => J) := by | |
| rw [densityPush,withDensity_const,Measure.map_smul _ C.measurable_paramExtended.aemeasurable] | |
| _ ≤ densityPush (volume.restrict s) C.paramExtended (fun z => |(C.multiplicity z : ℝ)| * J) := by | |
| apply densityPush_mono_ae C.measurable_paramExtended | |
| filter_upwards [hθ] with z hz | |
| exact (le_mul_of_one_le_left hJ.le hz) | |
| _ ≤ _ := hpiece.trans Measure.restrict_le_self | |
| refine ⟨e '' s,C.paramExtended ∘ e.symm, | |
| e.toHomeomorph.measurableEmbedding.measurableSet_image.mpr hs,?_, | |
| C.measurable_paramExtended.comp e.symm.continuous.measurable,?_,?_⟩ | |
| · have hh := Measure.addHaar_image_linearMap volume e.toLinearEquiv.toLinearMap s | |
| change volume (e '' s) = _ at hh | |
| rw [hh,hdet] | |
| exact mul_ne_zero (ENNReal.ofReal_ne_zero_iff.mpr hJ) hs0 | |
| · rintro _ ⟨y,hy,rfl⟩ _ ⟨z,hz,rfl⟩ | |
| simp only [Function.comp_apply,e.symm_apply_apply] | |
| apply ((hnear y hy z hz).2).trans | |
| apply mul_le_mul_of_nonneg_left _ (by linarith only [hε]) | |
| simp only [dist_eq_norm,←map_sub,he,le_refl] | |
| · rw [normalized_linear_image_push volume e hs C.measurable_paramExtended,hdet] | |
| convert hconst using 1 | |
| change (ENNReal.ofReal ((K:ℝ)^k) • μ) = ((K^k) • μ) | |
| simp only [ENNReal.ofReal_pow K.coe_nonneg,ENNReal.ofReal_coe_nnreal] | |
| rfl | |
| end IntegerChart | |
| namespace ChartGeometry | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [CompactSpace X] [Nonempty X] | |
| variable {T : Functional X 2} {hT : IsMetricCurrent T} (q : ChartGeometry hT) | |
| include q in | |
| lemma exists_almost_euclidean_density (hX : IsCAT0 X) (hM : 0 < mass T) | |
| {ε : ℝ} (hε : 0 < ε) (hε1 : ε < 1) : | |
| ∃ o : X, ∀ v : ℝ, 1+ε < v → ∀ d : ℝ, d < Real.pi → | |
| ∀ᶠ r in 𝓝[>] (0:ℝ), d*(r/v)^2 ≤ (1-ε)⁻¹^2 * | |
| (MassMeasure.currentMassMeasure hT).real (Metric.ball o r) := by | |
| obtain ⟨i,s,p,hs,hss,hs0,hθ,hp,hpara,hnear⟩ := q.exists_positive_near_hilbert_piece hX hM hε | |
| obtain ⟨t,f,ht,ht0,hf,hLip,hdom⟩ := (q.chart i).normalized_piece (q.current i) | |
| (q.chart_controls_total i) hs hss hs0 hθ p hp hpara hε hε1 hnear | |
| exact exists_lipschitz_chart_density ht ht0 hf (sq_pos_of_pos (inv_pos.mpr (sub_pos.mpr hε1))) | |
| (by linarith) hLip hdom | |
| end ChartGeometry | |
| end CAT0Fillings | |
| end | |
| end OAI | |