| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Density.RadialEstimate |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal BigOperators |
|
|
| namespace CAT0Fillings |
| namespace ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] |
| variable {T : Functional X 2} {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma cutoff_mass_upper (o : X) {ε ell : ℝ} (hε : 0 < ε) (hell : 0 < ell) |
| (hμ : MassMeasure.currentMassMeasure hT {x | dist o x = ε} = 0) |
| (hrad : ell*q.radialVariation o (fun x => inverseSquareCutoff ε (dist o x)) ≤ |
| q.sweptMass o (fun x => inverseSquareCutoff ε (dist o x))) : |
| (2*ell-ε)*(MassMeasure.currentMassMeasure hT).real (Metric.ball o ε) ≤ |
| ε^2/(8*ell)*mass T := by |
| classical |
| let g : X → ℝ := fun x => inverseSquareCutoff ε (dist o x) |
| have hg : LipschitzWith ((2/ε).toNNReal*1) g := |
| (inverseSquareCutoff_lipschitz hε).comp (LipschitzWith.dist_right o) |
| have hg0 : ∀ x, 0 ≤ g x := fun x => inverseSquareCutoff_nonneg _ |
| let w : X → ℝ := fun x => (if dist o x < ε then 2*ell-ε else 0)-ε^2/(8*ell) |
| have hw : Measurable w := |
| ((measurable_const.piecewise (measurableSet_lt (continuous_const.dist continuous_id).measurable |
| measurable_const) measurable_const).sub measurable_const) |
| have hwb : ∃ B : ℝ, ∀ x, |w x| ≤ B := by |
| refine ⟨|2*ell-ε|+|ε^2/(8*ell)|,fun x => ?_⟩ |
| dsimp only [w] |
| split_ifs |
| · exact abs_sub _ _ |
| · simpa only [zero_sub,abs_neg] using le_add_of_nonneg_left (abs_nonneg (2*ell-ε)) |
| let V (i : ℕ) := q.radialVariationChart o g i |
| let W (i : ℕ) := ∫ z, |((q.chart i).multiplicity z : ℝ)| * |
| (q.chart i).sweptWeight (q.field i) o g z ∂volume.restrict (q.chart i).domain |
| have hW := q.swept_integrable_summable o hg hg0 |
| have hV := (q.spatialMass_first_variation o hg).1 |
| have hchart (i : ℕ) : (∫ z, q.density i z*w ((q.chart i).paramExtended z) |
| ∂volume.restrict (q.chart i).domain) ≤ ell*V i-W i := by |
| dsimp only [V,W] |
| rw [q.radialVariationChart_cutoff o i hε hμ,←integral_const_mul,←integral_sub |
| ((q.cutoff_weight_integrable o i hε).const_mul ell) (hW.1 i)] |
| apply integral_mono_ae (q.weighted_density_integrable_borel i hw hwb) |
| (((q.cutoff_weight_integrable o i hε).const_mul ell).sub (hW.1 i)) |
| filter_upwards [ae_restrict_mem (q.chart i).borel,q.ae_radialNormSq_bounds o i] with z hz ha |
| have hin := cutoff_pointwise_estimate hε hell (dist_nonneg (x := o) |
| (y := (q.chart i).paramExtended z)) ha |
| have h := mul_le_mul_of_nonneg_left hin (q.density_nonneg i z) |
| have hs (f : X → ℝ) : (q.chart i).scalar f z = f ((q.chart i).paramExtended z) := by |
| rw [IntegerChart.scalar_eq _ hz] |
| simp only [IntegerChart.paramExtended,dite_eq_left hz] |
| dsimp only [w,g] at h ⊢ |
| simp only [Pi.sub_apply,IntegerChart.sweptWeight,hs,density,gram,radialNormSq] at h ⊢ |
| convert h using 1 |
| ring |
| have hint : (∫ x, w x ∂MassMeasure.currentMassMeasure hT) ≤ 0 := by |
| rw [q.integral_massMeasure_borel hw hwb] |
| calc |
| _ ≤ ∑' i, (ell*V i-W i) := (q.weighted_density_summable_borel hw hwb).tsum_le_tsum |
| hchart ((hV.mul_left ell).sub hW.2) |
| _ = ell*q.radialVariation o g-q.sweptMass o g := by |
| rw [(hV.mul_left ell).tsum_sub hW.2,tsum_mul_left] |
| rfl |
| _ ≤ 0 := sub_nonpos.mpr hrad |
| have heq : w = fun x => (Metric.ball o ε).indicator (fun _ => 2*ell-ε) x-ε^2/(8*ell) := by |
| funext x |
| simp only [w,indicator,Metric.mem_ball,dist_comm] |
| rw [heq,integral_sub ((integrable_const (2*ell-ε)).indicator measurableSet_ball) |
| (integrable_const _),integral_indicator measurableSet_ball,integral_const,integral_const] at hint |
| simp only [smul_eq_mul,measureReal_def,Measure.restrict_apply_univ] at hint |
| change (MassMeasure.currentMassMeasure hT).real (Metric.ball o ε)*(2*ell-ε)- |
| (MassMeasure.currentMassMeasure hT).real univ*(ε^2/(8*ell)) ≤ 0 at hint |
| rw [MassMeasure.currentMassMeasure_total hT] at hint |
| nlinarith |
| |
| end ChartGeometry |
| end CAT0Fillings |
| end |
| |
| section |
| |
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
| |
| namespace CAT0Fillings |
| |
| lemma exists_euclidean_density_point_two {s : Set (Euc 2)} (hs : MeasurableSet s) |
| (hs0 : volume s ≠ 0) : |
| ∃ u ∈ s, Tendsto (fun r : ℝ => volume.real (s ∩ closedBall u r)/r^2) |
| (𝓝[>] 0) (𝓝 Real.pi) := by |
| obtain ⟨u,hu,hd⟩ := Measure.exists_mem_of_measure_ne_zero_of_ae hs0 |
| (ae_restrict_of_ae (Besicovitch.ae_tendsto_measure_inter_div_of_measurableSet volume hs)) |
| simp only [indicator_of_mem hu,Pi.one_apply] at hd |
| refine ⟨u,hu,?_⟩ |
| have hreal := (ENNReal.continuousAt_toReal (by norm_num : (1:ℝ≥0∞) ≠ ∞)).tendsto.comp hd |
| simp only [ENNReal.toReal_one] at hreal |
| have hlim := hreal.mul_const Real.pi |
| simp only [one_mul] at hlim |
| apply hlim.congr' |
| filter_upwards [self_mem_nhdsWithin] with r hr |
| simp only [Function.comp_apply,ENNReal.toReal_div,measureReal_def,EuclideanSpace.volume_closedBall_fin_two,ENNReal.toReal_mul, |
| ENNReal.toReal_pow,ENNReal.toReal_ofReal hr.le,ENNReal.toReal_ofReal Real.pi_pos.le] |
| field_simp [ne_of_gt (show (0:ℝ) < r from hr),Real.pi_ne_zero] |
|
|
| lemma exists_lipschitz_chart_density {X : Type*} [MetricSpace X] [MeasurableSpace X] |
| [BorelSpace X] {μ : Measure X} [IsFiniteMeasure μ] |
| {s : Set (Euc 2)} (hs : MeasurableSet s) (hs0 : volume s ≠ 0) |
| {f : Euc 2 → X} (hf : Measurable f) {a b : ℝ} (ha : 0 < a) (hb : 0 ≤ b) |
| (hLip : ∀ y ∈ s, ∀ z ∈ s, dist (f y) (f z) ≤ b*dist y z) |
| (hdom : Measure.map f (volume.restrict s) ≤ ENNReal.ofReal a • μ) : |
| ∃ o : X, ∀ v : ℝ, b < v → ∀ d : ℝ, d < Real.pi → |
| ∀ᶠ r in 𝓝[>] (0:ℝ), d*(r/v)^2 ≤ a*μ.real (ball o r) := by |
| obtain ⟨u,hu,hd⟩ := exists_euclidean_density_point_two hs hs0 |
| refine ⟨f u,fun v hv d hdπ => ?_⟩ |
| have hv0 : 0 < v := hb.trans_lt hv |
| have ht : Tendsto (fun r : ℝ => r/v) (𝓝[>] 0) (𝓝[>] 0) := by |
| apply tendsto_nhdsWithin_iff.mpr |
| refine ⟨by simpa using ((tendsto_id.div_const v : Tendsto (fun r : ℝ => r/v) (𝓝 0) (𝓝 (0/v))).mono_left nhdsWithin_le_nhds),?_⟩ |
| filter_upwards [self_mem_nhdsWithin] with r hr using div_pos hr hv0 |
| have hlimit := hd.comp ht |
| have hevent := hlimit.eventually (eventually_gt_nhds hdπ) |
| filter_upwards [self_mem_nhdsWithin,hevent] with r hr hrd |
| have hr0 : 0 < r/v := div_pos hr hv0 |
| have hincl : s ∩ closedBall u (r/v) ⊆ f ⁻¹' ball (f u) r ∩ s := by |
| intro z hz |
| refine ⟨?_,hz.1⟩ |
| change dist (f z) (f u) < r |
| have hh := (hLip z hz.1 u hu).trans (mul_le_mul_of_nonneg_left hz.2 hb) |
| exact hh.trans_lt (by have := mul_lt_mul_of_pos_right hv hr0; nlinarith [div_mul_cancel₀ r hv0.ne']) |
| have hbnd : volume (s ∩ closedBall u (r/v)) ≤ ENNReal.ofReal a*μ (ball (f u) r) := by |
| calc |
| _ ≤ (volume.restrict s) (f ⁻¹' ball (f u) r) := by |
| rw [Measure.restrict_apply (hf measurableSet_ball)] |
| exact measure_mono hincl |
| _ = (Measure.map f (volume.restrict s)) (ball (f u) r) := |
| (Measure.map_apply hf measurableSet_ball).symm |
| _ ≤ _ := hdom _ |
| have hfin : ENNReal.ofReal a*μ (ball (f u) r) ≠ ∞ := |
| ENNReal.mul_ne_top ENNReal.ofReal_ne_top (measure_ne_top _ _) |
| have hrb := ENNReal.toReal_mono hfin hbnd |
| simp only [ENNReal.toReal_mul,ENNReal.toReal_ofReal ha.le] at hrb |
| change volume.real (s ∩ closedBall u (r/v)) ≤ a*μ.real (ball (f u) r) at hrb |
| exact ((lt_div_iff₀ (sq_pos_of_pos hr0)).mp hrd).le.trans hrb |
| |
| end CAT0Fillings |
| end |
| |
| end OAI |
| |
