| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Radial.WholeMass |
| import OAI.Geometry.CAT0Fillings.Radial.SeriesDerivative |
| import OAI.Geometry.CAT0Fillings.Calculus.AbsoluteValue |
| import OAI.Geometry.CAT0Fillings.Calculus.ChainSet |
|
|
| namespace OAI |
|
|
| section |
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open ClosedCalculus |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma nonnegative_mem_closed (hz : IsCycle T) {S : Set q.Sobolev} (hS : IsClosed S) |
| (hmem : ∀ (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), |
| (∀ x, 0 ≤ u x) → q.lipSobolev hu ∈ S) |
| (P : q.Sobolev) (hP : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, 0 ≤ q.inclusion P x) : |
| P ∈ S := by |
| let e (j : ℕ) : ℝ := 1/((j:ℝ)+1) |
| have hepos (j : ℕ) : 0 < e j := by dsimp [e]; positivity |
| have he : Tendsto e atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat |
| have hb (t : ℝ) : abs (t/|t|) ≤ 1 := by |
| rw [abs_div,abs_abs] |
| by_cases ht : t = 0 |
| · simp [ht] |
| · rw [div_self (abs_pos.mpr ht).ne'] |
| obtain ⟨Q,hQS,hQ,_⟩ := q.closed_chain_uniform_in_closed (F := abs) (D := fun t => t/|t|) |
| (show LipschitzWith 1 abs from LipschitzWith.of_dist_le_mul (fun x y => by simpa only [Real.dist_eq,NNReal.coe_one,one_mul] using abs_abs_sub_abs_le_abs_sub x y)) |
| (measurable_id.div measurable_id.abs) (by norm_num : (0:ℝ) ≤ 1) hb |
| (fun j => absApprox (e j)) (fun j => absApproxDeriv (e j)) (fun _ => 1) |
| (fun j => absApprox_lipschitz (hepos j)) (fun j => absApprox_hasDerivAt (hepos j)) |
| (fun j => continuous_absApproxDeriv (hepos j)) (fun j => absApproxDeriv_bound (hepos j)) |
| (absApproxDeriv_tendsto he hepos) e he (fun j => absApprox_error (hepos j).le) P hS |
| (fun j u K hu => hmem _ _ _ (fun x => Real.sqrt_nonneg _)) |
| have hQP : Q = P := q.inclusion_injective hz (Lp.ext (by |
| filter_upwards [hQ,hP] with x hx hp |
| exact hx.trans (abs_of_nonneg hp))) |
| rwa [hQP] at hQS |
|
|
| lemma nonnegative_linear_inequality (hz : IsCycle T) (L : q.Sobolev →L[ℝ] ℝ) |
| (hL : ∀ (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), |
| (∀ x, 0 ≤ u x) → L (q.lipSobolev hu) ≤ 0) |
| (P : q.Sobolev) (hP : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, 0 ≤ q.inclusion P x) : |
| L P ≤ 0 := |
| q.nonnegative_mem_closed hz (isClosed_le L.continuous continuous_const) hL P hP |
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal BigOperators |
|
|
| namespace CAT0Fillings |
| open CurrentOperations |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} |
| namespace ChartGeometry |
| variable {T : Functional X k} {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| noncomputable def spatialMass (o : X) (g : X → ℝ) (t : ℝ) : ℝ := |
| ∑' i, ∫ z, |((q.chart i).multiplicity z : ℝ)| * |
| (q.chart i).radialJacobian (q.gram i) o g t z ∂volume.restrict (q.chart i).domain |
|
|
| noncomputable def radialVariationChart (o : X) (g : X → ℝ) (i : ℕ) : ℝ := |
| ∫ z, q.density i z * ((k:ℝ)*(q.chart i).scalar g z + (q.chart i).scalar (dist o) z * |
| (differentialRow (fderivWithin ℝ ((q.chart i).scalar (dist o)) (q.chart i).domain z) ⬝ᵥ |
| (q.gram i z)⁻¹.mulVec |
| (differentialRow (fderivWithin ℝ ((q.chart i).scalar g) (q.chart i).domain z)))) |
| ∂volume.restrict (q.chart i).domain |
|
|
| noncomputable def radialVariation (o : X) (g : X → ℝ) : ℝ := ∑' i, q.radialVariationChart o g i |
|
|
| lemma spatialMass_zero (o : X) (g : X → ℝ) : q.spatialMass o g 0 = mass T := by |
| rw [q.density_mass] |
| apply tsum_congr |
| intro i |
| simp only [density,IntegerChart.radialJacobian,metricSpatialRadialForm] |
| simp |
|
|
| lemma spatialMass_nonneg (o : X) (g : X → ℝ) (t : ℝ) : 0 ≤ q.spatialMass o g t := by |
| apply tsum_nonneg |
| intro i |
| exact integral_nonneg fun z => mul_nonneg (abs_nonneg _) (Real.sqrt_nonneg _) |
|
|
| lemma spatialMass_first_variation (o : X) {g : X → ℝ} {K : ℝ≥0} |
| (hg : LipschitzWith K g) : |
| Summable (q.radialVariationChart o g) ∧ |
| HasDerivWithinAt (q.spatialMass o g) (-q.radialVariation o g) (Ici 0) 0 := by |
| have hh := countable_chart_radial_first_variation q.chart q.field q.measurable q.differential |
| q.integrable q.density_summable o hg |
| dsimp only at hh |
| simp_rw [mul_neg,integral_neg,tsum_neg] at hh |
| refine ⟨?_,hh.2⟩ |
| have hs := hh.1.neg |
| change Summable (fun i => -(-q.radialVariationChart o g i)) at hs |
| simpa only [neg_neg] using hs |
|
|
| lemma spatial_push_mass_le |
| (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) |
| {t : ℝ} (ht : t ∈ Icc (0:ℝ) 1) (htg : ∀ y, 1-t*g y ∈ Icc (0:ℝ) 1) : |
| mass (pushCurrent (fun y => seg o y (1-t*g y)) T) ≤ q.spatialMass o g t := by |
| 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 |
| let I (i : ℕ) : ℝ := ∫ z, |((q.chart i).multiplicity z : ℝ)| * |
| (q.chart i).radialJacobian (q.gram i) o g t z ∂volume.restrict (q.chart i).domain |
| have hrad (i : ℕ) : Integrable (fun z => |((q.chart i).multiplicity z : ℝ)| * |
| (q.chart i).radialJacobian (q.gram i) o g t z) (volume.restrict (q.chart i).domain) := |
| (q.chart i).integrable_radialJacobian (q.field i) (q.measurable i) (q.differential i) |
| (q.integrable 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 (q.chart i).action := by |
| rw [q.chart_mass,←integral_const_mul] |
| apply integral_mono_ae (hrad i) ((q.integrable i).const_mul A) |
| filter_upwards [hbound (q.chart i) (q.field i) (q.differential i) t ht] with z hz |
| calc |
| _ ≤ |((q.chart i).multiplicity z : ℝ)| *(A*Real.sqrt (q.gram i z).det) := |
| mul_le_mul_of_nonneg_left hz (abs_nonneg _) |
| _ = _ := by dsimp only [gram]; ring |
| have hIsum : Summable I := (q.summable.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 (q.chart i).action) := |
| pushCurrent_isMetricCurrent (q.chart i).action_isMetricCurrent hf |
| have hpushle (i : ℕ) : mass (pushCurrent f (q.chart i).action) ≤ I i := |
| (q.chart i).radial_push_mass_le seg hcomp o g hg t htg hf (q.field i) (q.differential i) (hrad i) |
| have hpushsum : Summable (fun i => mass (pushCurrent f (q.chart i).action)) := |
| hIsum.of_nonneg_of_le (fun i => mass_nonneg _) hpushle |
| have hpusheq : pushCurrent f T = fun b π => ∑' i, pushCurrent f (q.chart i).action b π := by |
| funext b π |
| by_cases hab : Admissible b π |
| · simp only [pushCurrent_apply _ _ hab] |
| exact q.action _ _ |
| · simp only [pushCurrent,ite_eq_right hab,tsum_zero] |
| change mass (pushCurrent f T) ≤ ∑' i, I i |
| rw [hpusheq] |
| exact (mass_tsum_le hpush hpushsum).trans (hpushsum.tsum_le_tsum hpushle hIsum) |
|
|
| end ChartGeometry |
| end CAT0Fillings |
| end |
|
|
| end OAI |
|
|
