| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Radial.WholeMass |
| import OAI.Geometry.CAT0Fillings.Radial.SeriesDerivative |
|
|
| namespace OAI |
|
|
| section |
| section |
| 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 MeasureTheory Set Filter Metric Matrix CurrentOperations |
| open scoped Topology NNReal |
|
|
| variable {X : Type*} [MetricSpace X] [CompactSpace X] |
| [MeasurableSpace X] [BorelSpace X] [Nonempty X] {k : ℕ} |
| theorem exists_whole_radial_first_variation {T : Functional X k} |
| (hX : IsCAT0 X) (hT : IsMetricCurrent T) (hrect : IntegerRectifiable T) |
| (seg : X → X → ℝ → X) |
| (hseg : ∀ o x a b, a ∈ Icc (0:ℝ) 1 → b ∈ Icc (0:ℝ) 1 → |
| dist (seg o x a) (seg o x b) = |a-b| *dist o x) |
| (hcomp : ∀ o x y a b, a ∈ Icc (0:ℝ) 1 → b ∈ Icc (0:ℝ) 1 → |
| dist (seg o x a) (seg o y b)^2 ≤ (a*dist o x-b*dist o y)^2+ |
| a*b*(dist x y^2-(dist o x-dist o y)^2)) |
| (o : X) {g : X → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) : |
| ∃ (C : ℕ → IntegerChart X k) (p : ℕ → Euc k → Seminorm ℝ (Euc k)), |
| Pairwise (fun i j => Disjoint (C i).image (C j).image) ∧ |
| (∀ b π, T b π = ∑' i, (C i).action b π) ∧ |
| (∀ i v, Measurable (fun z => p i z v)) ∧ |
| (∀ i, ∀ᵐ z ∂volume.restrict (C i).domain, |
| (∀ hz : z ∈ (C i).domain, MetricDifferentiation.HasCenteredMetricDifferentialWithin |
| (C i).domain (C i).param (p i z) ⟨z,hz⟩) ∧ |
| (∀ u v, p i z (u+v)^2+p i z (u-v)^2 = 2*p i z u^2+2*p i z v^2) ∧ |
| (∀ v, p i z v = 0 ↔ v = 0)) ∧ |
| mass T = (∑' i, ∫ z, |((C i).multiplicity z : ℝ)| *Real.sqrt |
| (polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det |
| ∂volume.restrict (C i).domain) ∧ |
| (let d : ℕ → ℝ := fun i => ∫ z, |
| (|((C i).multiplicity z : ℝ)| * Real.sqrt |
| (polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det) * |
| -((k:ℝ)*(C i).scalar g z+(C i).scalar (dist o) z * |
| (differentialRow (fderivWithin ℝ ((C i).scalar (dist o)) (C i).domain z) ⬝ᵥ |
| (polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis)⁻¹.mulVec |
| (differentialRow (fderivWithin ℝ ((C i).scalar g) (C i).domain z)))) |
| ∂volume.restrict (C i).domain |
| Summable d ∧ HasDerivWithinAt |
| (fun t => ∑' i, ∫ z, |((C i).multiplicity z : ℝ)| * (C i).radialJacobian |
| (fun z => polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) |
| o g t z ∂volume.restrict (C i).domain) (∑' i, d i) (Ici 0) 0) ∧ |
| ∀ t ∈ Icc (0:ℝ) 1, (∀ y, 1-t*g y ∈ Icc (0:ℝ) 1) → |
| Summable (fun i => ∫ z, |((C i).multiplicity z : ℝ)| *(C i).radialJacobian |
| (fun z => polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) |
| o g t z ∂volume.restrict (C i).domain) ∧ |
| mass (pushCurrent (fun y => seg o y (1-t*g y)) T) ≤ |
| ∑' i, ∫ z, |((C i).multiplicity z : ℝ)| *(C i).radialJacobian |
| (fun z => polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) |
| o g t z ∂volume.restrict (C i).domain := by |
| classical |
| obtain ⟨C,hdis,hC,hsum,heq⟩ := hrect |
| choose P hPm hP hρ hfin hcontrol hmin hfield using |
| fun i => (C i).exists_exact_quadratic_mass hX (hC i) |
| choose p hPeq hpm hp using hfield |
| simp only [hPeq] at hρ hfin hcontrol hmin |
| let J (i : ℕ) (z : Euc k) := Real.sqrt |
| (polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis).det |
| have hmass (i : ℕ) : mass (C i).action = |
| ∫ z, |((C i).multiplicity z : ℝ)| *J i z ∂volume.restrict (C i).domain := |
| (C i).mass_eq_integral_majorant (hC i) (hρ i) |
| (Eventually.of_forall fun _ => Real.sqrt_nonneg _) (hcontrol i) (hmin i) |
| let μ (i : ℕ) : Measure X := (C i).majorantMeasure (J i) |
| have hμfin : ∀ i, IsFiniteMeasure (μ i) := hfin |
| let := hμfin |
| have htot (i : ℕ) : (μ i).real univ = mass (C i).action := by |
| have heqm : MassMeasure.currentMassMeasure (hC i) = μ i := by |
| apply le_antisymm (MassMeasure.currentMassMeasure_le (hC i) (hcontrol i)) |
| exact hmin i _ inferInstance (MassMeasure.currentMassMeasure_controls (hC i)) |
| rw [←heqm] |
| 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 |
| have hmeasure : MassMeasure.currentMassMeasure hT = Measure.sum μ := by |
| apply le_antisymm |
| · exact MassMeasure.currentMassMeasure_le hT (controls_sum_general hcontrol heq) |
| · exact sum_measure_le_of_component_minimal hC hcontrol hmin heq |
| (fun i => (C i).measurableSet_image) hdis |
| (fun i => (C i).majorantMeasure_ae_image _) (MassMeasure.currentMassMeasure_controls hT) |
| have htotal : mass T = ∑' i, mass (C i).action := by |
| rw [←MassMeasure.currentMassMeasure_total hT,hmeasure] |
| rw [measureReal_def,Measure.sum_apply _ MeasurableSet.univ, |
| ENNReal.tsum_toReal_eq (fun i => measure_ne_top (μ i) univ)] |
| exact tsum_congr htot |
| refine ⟨C,p,hdis,heq,hpm,hp,?_,?_,?_⟩ |
| · simpa only [hmass,J] using htotal |
| · exact countable_chart_radial_first_variation C p hpm hp hρ |
| (by simpa only [hmass,J] using hsum) o hg |
| obtain ⟨M,hM⟩ := isCompact_univ.exists_bound_of_continuousOn hg.continuous.continuousOn |
| let B : ℝ := max 1 (max (K:ℝ) (max M (Metric.diam (univ : Set X)))) |
| have hB : 1 ≤ B := le_max_left .. |
| have hK : (K:ℝ) ≤ B := (le_max_left ..).trans (le_max_right ..) |
| have hgb : ∀ x, |g x| ≤ B := by |
| intro x |
| have hm : |g x| ≤ M := by simpa only [Real.norm_eq_abs] using hM x (mem_univ x) |
| exact hm.trans ((le_max_left ..).trans ((le_max_right ..).trans (le_max_right ..))) |
| have hrb : ∀ x, dist o x ≤ B := by |
| intro x |
| exact (Metric.dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ o) (mem_univ x)).trans |
| ((le_max_right ..).trans ((le_max_right ..).trans (le_max_right ..))) |
| obtain ⟨A,hA,hbound⟩ := uniform_chart_radialJacobian_majorant (k := k) o hg hB hK hgb hrb |
| intro t ht htg |
| let R (i : ℕ) (z : Euc k) := (C i).radialJacobian |
| (fun z => polarizationMatrix (p i z) (EuclideanSpace.basisFun (Fin k) ℝ).toBasis) o g t z |
| let I (i : ℕ) : ℝ := ∫ z, |((C i).multiplicity z : ℝ)| *R i z |
| ∂volume.restrict (C i).domain |
| have hrad (i : ℕ) : Integrable (fun z => |((C i).multiplicity z : ℝ)| *R i z) |
| (volume.restrict (C i).domain) := |
| (C i).integrable_radialJacobian (p i) (hpm i) (hp i) (hρ i) o hg hB hK hgb hrb ht |
| have hI0 (i : ℕ) : 0 ≤ I i := integral_nonneg fun z => |
| mul_nonneg (abs_nonneg _) (Real.sqrt_nonneg _) |
| have hIle (i : ℕ) : I i ≤ A*mass (C i).action := by |
| rw [hmass,←integral_const_mul] |
| apply integral_mono_ae (hrad i) ((hρ i).const_mul A) |
| filter_upwards [hbound (C i) (p i) (hp i) t ht] with z hz |
| calc |
| _ ≤ |((C i).multiplicity z : ℝ)| *(A*J i z) := |
| mul_le_mul_of_nonneg_left hz (abs_nonneg _) |
| _ = _ := by ring |
| have hIsum : Summable I := (hsum.mul_left A).of_nonneg_of_le hI0 hIle |
| obtain ⟨L,hf⟩ := radial_lipschitzWith seg hseg hcomp o hg t htg |
| let f : X → X := fun y => seg o y (1-t*g y) |
| have hpush (i : ℕ) : IsMetricCurrent (pushCurrent f (C i).action) := |
| pushCurrent_isMetricCurrent (hC i) hf |
| have hpushle (i : ℕ) : mass (pushCurrent f (C i).action) ≤ I i := |
| (C i).radial_push_mass_le seg hcomp o g hg t htg hf (p i) (hp i) (hrad i) |
| have hpushsum : Summable (fun i => mass (pushCurrent f (C i).action)) := |
| hIsum.of_nonneg_of_le (fun i => mass_nonneg _) hpushle |
| have hpusheq : pushCurrent f T = fun b π => ∑' i, pushCurrent f (C i).action b π := by |
| funext b π |
| by_cases hab : Admissible b π |
| · simp only [pushCurrent_apply _ _ hab] |
| exact heq _ _ |
| · simp only [pushCurrent,ite_eq_right hab,tsum_zero] |
| refine ⟨hIsum,?_⟩ |
| change mass (pushCurrent f T) ≤ ∑' i, I i |
| rw [hpusheq] |
| exact (mass_tsum_le hpush hpushsum).trans (hpushsum.tsum_le_tsum hpushle hIsum) |
| end CAT0Fillings |
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| end |
|
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| end |
|
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| end OAI |
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