| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Charts.DensityPush |
| import OAI.Geometry.CAT0Fillings.Currents.Pushforward |
| import OAI.Geometry.CAT0Fillings.Charts.Extensions |
| import OAI.Geometry.CAT0Fillings.Fillings.ConeBound |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open Set MeasureTheory Filter |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
| 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 CAT0Fillings |
| open Set MeasureTheory CurrentOperations |
| open scoped NNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
| theorem IsCAT0.exists_integral_filling_diam (hX : IsCAT0 X) {k : ℕ} |
| {T : Functional X (k+1)} (hT : IsIntegral (k+1) T) (hz : boundarySucc T = 0) : |
| ∃ S : Functional X (k+2), IsIntegral (k+2) S ∧ boundarySucc S = T ∧ |
| mass S ≤ (k+2:ℝ)*Metric.diam (univ : Set X)*mass T := by |
| let o : X := Classical.choice inferInstance |
| let R : ℝ≥0 := ⟨Metric.diam (univ : Set X),Metric.diam_nonneg⟩ |
| exact hX.exists_integral_filling_mass_le hT hz o R (fun x => |
| Metric.dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ o) (mem_univ x)) |
|
|
| end CAT0Fillings |
|
|
| end |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| universe u |
|
|
| end CAT0Fillings |
|
|
| 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 CAT0Fillings |
| attribute [local instance] Classical.propDecidable |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| attribute [local instance] Classical.propDecidable |
|
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|
|
| attribute [local instance] Classical.propDecidable |
| open Set Metric |
| open scoped NNReal |
|
|
| end CAT0Fillings |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| universe u |
|
|
| end CAT0Fillings |
|
|
| 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 |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| universe u |
|
|
| end CAT0Fillings |
|
|
| open MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
|
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| attribute [local instance] Classical.propDecidable |
|
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| universe u |
|
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| end CAT0Fillings |
| namespace CAT0Fillings |
| open MeasureTheory Set Filter CurrentOperations |
|
|
| namespace IntegerChart |
| variable {X Y : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [Nonempty X] [MetricSpace Y] [MeasurableSpace Y] [BorelSpace Y] |
| {k : ℕ} (C : IntegerChart X k) |
|
|
| theorem push_majorant_controls {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) |
| {J : Euc k → ℝ} |
| (hρ : Integrable (fun z => |(C.multiplicity z : ℝ)| *J z) (volume.restrict C.domain)) |
| (hJ : 0 ≤ᵐ[volume.restrict C.domain] J) |
| (hdet : ∀ π : Fin k → Y → ℝ, (∀ i, LipschitzWith 1 (π i)) → |
| ∀ᵐ z ∂volume.restrict C.domain, |C.jacobian (fun i => π i ∘ f) z| ≤ J z) : |
| Controls (pushCurrent f C.action) |
| (densityPush (volume.restrict C.domain) (f ∘ C.paramExtended) |
| (fun z => |(C.multiplicity z : ℝ)| *J z)) := by |
| intro b π hb hπ |
| let μ := volume.restrict C.domain |
| have hφ := hf.continuous.measurable.comp C.measurable_paramExtended |
| have hbmeas := hb.continuous.measurable.comp hφ |
| obtain ⟨M,hM⟩ := hb.2 |
| have hupper : Integrable |
| (fun z => (|(C.multiplicity z : ℝ)| *J z)*|b (f (C.paramExtended z))|) μ := |
| hρ.mul_bdd hbmeas.abs.aestronglyMeasurable |
| (Eventually.of_forall fun z => by simpa only [Real.norm_eq_abs,abs_abs] using hM _) |
| have hρ0 : 0 ≤ᵐ[μ] fun z => |(C.multiplicity z : ℝ)| *J z := |
| hJ.mono fun z hz => mul_nonneg (abs_nonneg _) hz |
| rw [integral_densityPush μ hφ hρ hρ0 hb.continuous.abs] |
| rw [pushCurrent_apply f C.action ⟨hb,fun i => ⟨1,hπ i⟩⟩, |
| action,ite_eq_left (admissible_comp ⟨hb,fun i => ⟨1,hπ i⟩⟩ hf)] |
| rw [←Real.norm_eq_abs] |
| apply le_trans (norm_integral_le_integral_norm _) ?_ |
| change (∫ z, |(C.multiplicity z : ℝ)*C.scalar (b ∘ f) z* |
| C.jacobian (fun i => π i ∘ f) z| ∂μ) ≤ _ |
| apply integral_mono_ae |
| (C.integrable_action_integrand (boundedLip_comp hb hf) |
| (admissible_comp ⟨hb,fun i => ⟨1,hπ i⟩⟩ hf).2).abs hupper |
| filter_upwards [hdet π hπ,C.scalar_ae_paramExtended (b ∘ f)] with z hz hbz |
| rw [abs_mul,abs_mul,hbz] |
| change |(C.multiplicity z : ℝ)| *|b (f (C.paramExtended z))| * |
| |C.jacobian (fun i => π i ∘ f) z| ≤ _ |
| calc _ ≤ (|(C.multiplicity z : ℝ)| *|b (f (C.paramExtended z))|)*J z := |
| mul_le_mul_of_nonneg_left hz (mul_nonneg (abs_nonneg _) (abs_nonneg _)) |
| _ = _ := by ring |
|
|
| theorem push_mass_le_majorant {f : X → Y} {K : ℝ≥0} (hf : LipschitzWith K f) |
| {J : Euc k → ℝ} |
| (hρ : Integrable (fun z => |(C.multiplicity z : ℝ)| *J z) (volume.restrict C.domain)) |
| (hJ : 0 ≤ᵐ[volume.restrict C.domain] J) |
| (hdet : ∀ π : Fin k → Y → ℝ, (∀ i, LipschitzWith 1 (π i)) → |
| ∀ᵐ z ∂volume.restrict C.domain, |C.jacobian (fun i => π i ∘ f) z| ≤ J z) : |
| mass (pushCurrent f C.action) ≤ |
| ∫ z, |(C.multiplicity z : ℝ)| *J z ∂volume.restrict C.domain := by |
| let ν := densityPush (volume.restrict C.domain) (f ∘ C.paramExtended) |
| (fun z => |(C.multiplicity z : ℝ)| *J z) |
| have hfin : IsFiniteMeasure ν := densityPush_finite _ _ hρ |
| have hc := C.push_majorant_controls hf hρ hJ hdet |
| have hu : ν.real univ = ∫ z, |(C.multiplicity z : ℝ)| *J z ∂volume.restrict C.domain := by |
| have hi := integral_densityPush (volume.restrict C.domain) |
| (hf.continuous.measurable.comp C.measurable_paramExtended) hρ |
| (hJ.mono fun z hz => mul_nonneg (abs_nonneg _) hz) continuous_const (b := fun _ => (1:ℝ)) |
| simpa only [integral_const,smul_eq_mul,mul_one] using hi |
| exact (mass_le_measure hfin hc).trans_eq hu |
| end IntegerChart |
| end CAT0Fillings |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings |
|
|
| attribute [local instance] Classical.propDecidable |
|
|
| universe u |
|
|
| end CAT0Fillings |
|
|
| 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 |
|
|
| end |
| end |
|
|
| end OAI |
|
|
