| import Mathlib |
| import OAI.Geometry.IntegralFillings.Charts.MultiplicityMass |
| import OAI.Geometry.IntegralFillings.Charts.Transitions |
| import OAI.Geometry.IntegralFillings.Charts.ContainingCharts |
| import OAI.Geometry.IntegralFillings.Charts.IntegerSeries |
|
|
| 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 Filter MeasureTheory |
| open scoped Topology |
|
|
| namespace IntegerChart |
| variable {X : Type*} [MetricSpace X] {k : ℕ} (C : IntegerChart X k) |
| lemma actionWeight_bound {b : X → ℝ} {π : Fin k → X → ℝ} (hab : Admissible b π) : |
| ∃ M : ℝ, ∀ᵐ x ∂volume.restrict C.domain, ‖C.scalar b x * C.jacobian π x‖ ≤ M := by |
| obtain ⟨M,hM⟩ := hab.1.2 |
| obtain ⟨B,hB⟩ := C.exists_jacobian_bound hab.2 |
| refine ⟨max M 0 * max B 0,?_⟩ |
| filter_upwards [hB,ae_restrict_mem C.borel] with x hx hxs |
| rw [norm_mul] |
| apply mul_le_mul |
| · rw [C.scalar_eq hxs,Real.norm_eq_abs] |
| exact (hM _).trans (le_max_left _ _) |
| · exact hx.trans (le_max_left _ _) |
| · exact norm_nonneg _ |
| · exact le_max_right _ _ |
|
|
| noncomputable def sumMultiplicities (θ : ℕ → Euc k → ℤ) |
| (hθ : ∀ i, Integrable (fun x => (θ i x : ℝ)) (volume.restrict C.domain)) |
| (hS : Summable (fun i => ∫ x in C.domain, |(θ i x : ℝ)|)) : IntegerChart X k := |
| C.withMultiplicity (summedMultiplicity θ) (summedMultiplicity_integrable hθ hS) |
|
|
| lemma sumMultiplicities_action (θ : ℕ → Euc k → ℤ) |
| (hθ : ∀ i, Integrable (fun x => (θ i x : ℝ)) (volume.restrict C.domain)) |
| (hS : Summable (fun i => ∫ x in C.domain, |(θ i x : ℝ)|)) (b : X → ℝ) (π : Fin k → X → ℝ) : |
| (C.sumMultiplicities θ hθ hS).action b π = |
| ∑' i, (C.withMultiplicity (θ i) (hθ i)).action b π := by |
| by_cases hab : Admissible b π |
| · simp only [action,ite_eq_left hab] |
| change (∫ x in C.domain, (summedMultiplicity θ x : ℝ) * C.scalar b x * C.jacobian π x) = |
| ∑' i, ∫ x in C.domain, (θ i x : ℝ) * C.scalar b x * C.jacobian π x |
| simp_rw [mul_assoc] |
| obtain ⟨M,hM⟩ := C.actionWeight_bound hab |
| obtain ⟨K,hK⟩ := hab.1.1 |
| exact summedMultiplicity_integral_weight hθ hS |
| ((C.measurable_scalar hK).aestronglyMeasurable.mul (C.jacobian_aestronglyMeasurable hab.2)) hM |
| · simp only [action,ite_eq_right hab,tsum_zero] |
|
|
| end IntegerChart |
| end SharpIntegralFillings |
|
|
| namespace SharpIntegralFillings |
| open Set MeasureTheory |
| open scoped NNReal |
|
|
| namespace IntegerChart |
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] [Nonempty X] {k : ℕ} |
|
|
| theorem exists_overlap_sum (D : IntegerChart X k) (C : ℕ → IntegerChart X k) |
| (hCD : ∀ i, (C i).image ⊆ D.image) (hC : ∀ i, IsMetricCurrent (C i).action) |
| (hS : Summable (fun i => mass (C i).action)) : |
| ∃ E : IntegerChart X k, E.image = D.image ∧ |
| ∀ b π, E.action b π = ∑' i, (C i).action b π := by |
| choose F hFD hparam hact him using fun i => (C i).exists_inCoordinates D (hCD i) |
| let G (i : ℕ) : IntegerChart X k := (F i).extendOn D (hFD i) |
| have hG i : (G i).action = (C i).action := |
| ((F i).extendOn_action D (hFD i) (hparam i)).trans (hact i) |
| have hGcurrent i : IsMetricCurrent (G i).action := by rw [hG]; exact hC i |
| obtain ⟨L,U,_,hU⟩ := D.bilipschitz |
| let θ (i : ℕ) := (G i).multiplicity |
| have hθ i : Integrable (fun x => (θ i x : ℝ)) (volume.restrict D.domain) := (G i).integrable |
| have hl1 : Summable (fun i => ∫ x in D.domain, |(θ i x : ℝ)|) := by |
| apply (hS.mul_left ((U : ℝ)^k)).of_nonneg_of_le |
| · intro i |
| exact integral_nonneg (fun _ => abs_nonneg _) |
| · intro i |
| have hbound := (G i).multiplicity_integral_le_mass (hGcurrent i) hU |
| change (∫ x in D.domain, |(θ i x : ℝ)|) ≤ (U : ℝ)^k * mass (G i).action at hbound |
| simpa only [hG] using hbound |
| refine ⟨D.sumMultiplicities θ hθ hl1,rfl,?_⟩ |
| intro b π |
| rw [D.sumMultiplicities_action θ hθ hl1] |
| apply tsum_congr |
| intro i |
| change (G i).action b π = _ |
| rw [hG] |
| |
| end IntegerChart |
| end SharpIntegralFillings |
| |
| end |
| |
| end OAI |
| |
