| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.ExactMass |
| import OAI.Geometry.IntegralFillings.Currents.CanonicalMass |
| import OAI.Geometry.IntegralFillings.Currents.ComponentMass |
|
|
| namespace OAI |
|
|
| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Set MeasureTheory Filter |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
| open Filter Set |
| open scoped Topology NNReal |
| open Set Filter MeasureTheory TopologicalSpace |
| open scoped Topology ENNReal |
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
| open Matrix |
| open scoped RealInnerProductSpace MatrixOrder |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| namespace IntegerChart |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] [Nonempty X] {k : ℕ} (C : IntegerChart X k) |
| lemma measurableSet_image : MeasurableSet C.image := by |
| have heq : C.image = C.paramExtended '' C.domain := by |
| ext x |
| constructor |
| · rintro ⟨z,rfl⟩ |
| exact ⟨z,z.property,by simp only [paramExtended,dite_eq_left z.property]⟩ |
| · rintro ⟨z,hz,rfl⟩ |
| exact ⟨⟨z,hz⟩,by simp only [paramExtended,dite_eq_left hz]⟩ |
| rw [heq] |
| exact C.measurableSet_paramExtended_image C.borel (Subset.rfl) |
|
|
| lemma majorantMeasure_ae_image (J : Euc k → ℝ) : |
| ∀ᵐ x ∂C.majorantMeasure J, x ∈ C.image := by |
| rw [majorantMeasure,densityPush_eq_map _ C.measurable_paramExtended] |
| apply (ae_map_iff C.measurable_paramExtended.aemeasurable C.measurableSet_image).mpr |
| have hmem := (withDensity_absolutelyContinuous (volume.restrict C.domain) |
| (fun z => ENNReal.ofReal (|(C.multiplicity z : ℝ)| * J z))).ae_le (ae_restrict_mem C.borel) |
| filter_upwards [hmem] with z hz |
| exact ⟨⟨z,hz⟩,by simp only [paramExtended,dite_eq_left hz]⟩ |
|
|
| end IntegerChart |
|
|
| theorem integerRectifiable_exact_mass |
| {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] [Nonempty X] |
| {k : ℕ} {T : Functional X k} (hX : IsCAT0 X) |
| (hT : IsMetricCurrent T) (hrect : IntegerRectifiable T) : |
| ∃ (C : ℕ → IntegerChart X k) (P : ℕ → Euc k → Matrix (Fin k) (Fin k) ℝ), |
| Pairwise (fun i j => Disjoint (C i).image (C j).image) ∧ |
| (∀ b π, T b π = ∑' i, (C i).action b π) ∧ |
| (∀ i a b, Measurable (fun z => P i z a b)) ∧ |
| (∀ i, ∀ᵐ z ∂volume.restrict (C i).domain, (P i z).PosDef) ∧ |
| (∀ i, Integrable (fun z => |((C i).multiplicity z : ℝ)| * Real.sqrt (P i z).det) |
| (volume.restrict (C i).domain)) ∧ |
| (∀ i, ∃ p : Euc k → Seminorm ℝ (Euc k), |
| P i = (fun z => polarizationMatrix (p z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) ∧ |
| (∀ v, Measurable (fun z => p z v)) ∧ |
| ∀ᵐ z ∂volume.restrict (C i).domain, |
| (∀ hz : z ∈ (C i).domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| (C i).domain (C i).param (p z) ⟨z,hz⟩) ∧ |
| (∀ u v, p z (u+v)^2+p z (u-v)^2 = 2*p z u^2+2*p z v^2) ∧ |
| (∀ v, p z v = 0 ↔ v = 0)) ∧ |
| MassMeasure.currentMassMeasure hT = |
| Measure.sum (fun i => (C i).majorantMeasure (fun z => Real.sqrt (P i z).det)) := by |
| classical |
| let C := hrect.choose |
| have hdis := hrect.choose_spec.1 |
| have hC := hrect.choose_spec.2.1 |
| have hsum := hrect.choose_spec.2.2.1 |
| have heq := hrect.choose_spec.2.2.2 |
| choose P hPm hP hρ hfin hctrl hmin hfield using |
| (fun i => (C i).exists_exact_quadratic_mass hX (hC i)) |
| let μ : ℕ → Measure X := fun i => (C i).majorantMeasure (fun z => Real.sqrt (P i z).det) |
| have hμfin : ∀ i, IsFiniteMeasure (μ i) := hfin |
| let := hμfin |
| have hmass (i) : MassMeasure.currentMassMeasure (hC i) = μ i := by |
| apply le_antisymm (MassMeasure.currentMassMeasure_le (hC i) (hctrl i)) |
| exact hmin i _ inferInstance (MassMeasure.currentMassMeasure_controls (hC i)) |
| have htot (i) : (μ i).real univ = mass (C i).action := by |
| rw [←hmass i] |
| exact MassMeasure.currentMassMeasure_total _ |
| have hμsum : Summable (fun i => (μ i).real univ) := by |
| simpa only [htot] using hsum |
| let := finite_sum_measure_of_summable_total μ hμsum |
| refine ⟨C,P,hdis,heq,hPm,hP,hρ,hfield,?_⟩ |
| apply le_antisymm |
| · exact MassMeasure.currentMassMeasure_le hT (controls_sum hctrl heq) |
| · exact sum_measure_le_of_component_minimal hC hctrl hmin heq |
| (fun i => (C i).measurableSet_image) hdis |
| (fun i => (C i).majorantMeasure_ae_image _) (MassMeasure.currentMassMeasure_controls hT) |
| |
| end SharpIntegralFillings |
| |
| namespace SharpIntegralFillings |
| attribute [local instance] Classical.propDecidable |
| end SharpIntegralFillings |
| end |
| |
| end OAI |
| |
