| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.CyclePairing |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology ENNReal NNReal BigOperators |
|
|
| namespace CAT0Fillings.ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma exists_inverse_coordinates (i : ℕ) : |
| ∃ (F : Fin n → X → ℝ) (K : ℝ≥0), (∀ a, LipschitzWith K (F a)) ∧ |
| ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| ∀ a, q.covector i (F a) z = (1 : Matrix (Fin n) (Fin n) ℝ) a := by |
| obtain ⟨K₁,K₂,hK₁,hK₂⟩ := (q.chart i).bilipschitz |
| have hrow : ∀ a y (hy : y ∈ (q.chart i).domain) z (hz : z ∈ (q.chart i).domain), |
| |EuclideanSpace.proj a y-EuclideanSpace.proj a z| ≤ |
| (K₂ : ℝ)*dist ((q.chart i).param ⟨y,hy⟩) ((q.chart i).param ⟨z,hz⟩) := by |
| intro a y hy z hz |
| have h := PiLp.norm_apply_le (p := (2 : ℝ≥0∞)) (y-z) a |
| have he : |EuclideanSpace.proj a y-EuclideanSpace.proj a z| ≤ dist y z := by |
| change |y.ofLp a-z.ofLp a| ≤ ‖y-z‖ |
| simpa only [Real.norm_eq_abs,PiLp.sub_apply] using h |
| exact he.trans (hK₂.le_mul_dist ⟨y,hy⟩ ⟨z,hz⟩) |
| obtain ⟨F,hF,hFeq,_⟩ := (q.chart i).exists_inverse_coordinate_tests |
| (q.chart i).borel subset_rfl (fun a => EuclideanSpace.proj a) K₂ hrow |
| refine ⟨F,K₂,hF,?_⟩ |
| rw [eventually_all] |
| intro a |
| have h := (q.chart i).ae_fderivWithin_scalar_eq_linear (q.chart i).borel subset_rfl |
| (hF a) (EuclideanSpace.proj a) (hFeq a) |
| filter_upwards [h] with z hz |
| ext b |
| unfold covector |
| rw [hz] |
| change (EuclideanSpace.single b (1 : ℝ)).ofLp a = (1 : Matrix (Fin n) (Fin n) ℝ) a b |
| simp [Matrix.one_apply] |
|
|
| lemma ae_coordinate_density_pos (i : ℕ) : |
| ∀ᵐ z ∂q.coordinateMeasure i, 0 < q.density i z := by |
| apply (ae_withDensity_iff' (q.density_integrable i).aestronglyMeasurable.aemeasurable.ennreal_ofReal).mpr |
| exact Eventually.of_forall fun z hz => lt_of_not_ge (fun h => hz (ENNReal.ofReal_eq_zero.mpr h)) |
|
|
| lemma ae_multiplicity_ne_zero (i : ℕ) : |
| ∀ᵐ z ∂q.coordinateMeasure i, ((q.chart i).multiplicity z : ℝ) ≠ 0 := by |
| filter_upwards [q.ae_coordinate_density_pos i] with z hz |
| intro he |
| simp only [density,he,abs_zero,zero_mul] at hz |
| exact lt_irrefl 0 hz |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology ENNReal NNReal BigOperators |
|
|
| namespace CAT0Fillings.ChartGeometry |
| 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 eq_zero_of_pairing_null (L : Lp (Euc (k+1)) 2 q.atlasMeasure) |
| (hnull : ∀ (π : Fin k → X → ℝ) (K : Fin k → ℝ≥0), |
| (∀ j, LipschitzWith (K j) (π j)) → ∀ b, BoundedLip b → q.pairing π b L = 0) : |
| L = 0 := by |
| apply Lp.ext |
| apply EventuallyEq.trans _ (Lp.coeFn_zero (Euc (k+1)) 2 q.atlasMeasure).symm |
| apply Measure.ae_sum_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr |
| obtain ⟨F,K,hF,hrows⟩ := q.exists_inverse_coordinates i |
| have hzero : ∀ᵐ z ∂q.coordinateMeasure i, ∀ a : Fin (k+1), |
| q.testWeight (fun j => F (Equiv.swap 0 a j.succ)) L (i,z) = 0 := by |
| rw [eventually_all] |
| intro a |
| have hh := q.testWeight_eq_zero_of_pairings (fun j => F (Equiv.swap 0 a j.succ)) |
| (fun _ => K) (fun j => hF _) L (hnull _ _ (fun j => hF _)) |
| exact (measurableEmbedding_prodMk_left i).ae_map_iff.mp (Measure.ae_sum_iff.mp hh i) |
| have hrows' := (withDensity_absolutelyContinuous _ (fun z => ENNReal.ofReal (q.density i z))).ae_le hrows |
| have hdiff := (withDensity_absolutelyContinuous _ (fun z => ENNReal.ofReal (q.density i z))).ae_le (q.differential i) |
| filter_upwards [hzero,hrows',hdiff,q.ae_multiplicity_ne_zero i] with z hz hr hd hθ |
| have hG : (q.gram i z).PosDef := polarizationMatrix_posDef _ _ hd.2.2 hd.2.1 |
| have hJ : Real.sqrt (q.gram i z).det ≠ 0 := (Real.sqrt_pos.mpr hG.det_pos).ne' |
| have hraw : rawCovector (q.gram i z) (L (i,z)) = 0 := by |
| funext a |
| apply component_zero_of_omitted_det _ a |
| have hh := hz a |
| change ((_ : ℝ)/abs _) * normalizedDet _ _ _ = 0 at hh |
| have hh' := (mul_eq_zero.mp hh).resolve_left (div_ne_zero hθ (abs_ne_zero.mpr hθ)) |
| unfold normalizedDet at hh' |
| have he : (fun j : Fin k => q.covector i (F (Equiv.swap 0 a j.succ)) z) = |
| fun j => (1 : Matrix (Fin (k+1)) (Fin (k+1)) ℝ) (Equiv.swap 0 a j.succ) := by |
| funext j |
| exact hr _ |
| rw [he] at hh' |
| exact (div_eq_zero_iff.mp hh').resolve_right hJ |
| have he := normalize_rawCovector (q.gram i z) hG (L (i,z)) |
| rw [hraw,Matrix.mulVec_zero] at he |
| apply (WithLp.equiv 2 (Fin (k+1) → ℝ)).injective |
| exact he.symm |
|
|
| theorem gradient_closable (hz : IsCycle T) |
| (u : ℕ → X → ℝ) (Ku : ℕ → ℝ≥0) (hu : ∀ j, LipschitzWith (Ku j) (u j)) |
| (hu0 : Tendsto (fun j => lpNorm (u j) 2 (MassMeasure.currentMassMeasure hT)) atTop (𝓝 0)) |
| (L : Lp (Euc (k+1)) 2 q.atlasMeasure) |
| (hL : Tendsto (fun j => q.gradient (hu j)) atTop (𝓝 L)) : L = 0 := by |
| apply q.eq_zero_of_pairing_null L |
| intro π K hπ b hb |
| exact q.pairing_eq_zero_of_gradient_limit hz u Ku hu hu0 L hL π K hπ hb |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma gradient_add {u v : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hv : LipschitzWith J v) : |
| q.gradient (hu.add hv) = q.gradient hu + q.gradient hv := by |
| apply Lp.ext |
| apply EventuallyEq.trans _ (Lp.coeFn_add _ _).symm |
| have hea := (q.memLp_gradientFunction (hu.add hv)).coeFn_toLp |
| have heb := (q.memLp_gradientFunction hu).coeFn_toLp |
| have hec := (q.memLp_gradientFunction hv).coeFn_toLp |
| have he : q.gradientFunction (fun x => u x+v x) =ᵐ[q.atlasMeasure] |
| fun w => q.gradientFunction u w+q.gradientFunction v w := by |
| apply Measure.ae_sum_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr |
| apply (withDensity_absolutelyContinuous _ _).ae_le |
| filter_upwards [ae_uniqueDiffWithinAt volume (q.chart i).domain, |
| (q.chart i).ae_differentiableWithinAt_scalar hu, |
| (q.chart i).ae_differentiableWithinAt_scalar hv] with z hz hdu hdv |
| have hs : (q.chart i).scalar (fun x => u x+v x) = |
| (q.chart i).scalar u+(q.chart i).scalar v := by |
| funext z |
| by_cases hz : z ∈ (q.chart i).domain <;> simp [IntegerChart.scalar,hz] |
| change q.normalizedCovector i (fun x => u x+v x) z = |
| q.normalizedCovector i u z+q.normalizedCovector i v z |
| unfold normalizedCovector covector |
| rw [hs,fderivWithin_add hz hdu hdv] |
| have he : differentialRow (fderivWithin ℝ ((q.chart i).scalar u) (q.chart i).domain z + |
| fderivWithin ℝ ((q.chart i).scalar v) (q.chart i).domain z) = |
| differentialRow (fderivWithin ℝ ((q.chart i).scalar u) (q.chart i).domain z)+ |
| differentialRow (fderivWithin ℝ ((q.chart i).scalar v) (q.chart i).domain z) := rfl |
| rw [he,Matrix.mulVec_add,WithLp.toLp_add] |
| filter_upwards [hea,heb,hec,he] with w ha hb hc he |
| change (q.gradient (hu.add hv)) w = (q.gradient hu) w+(q.gradient hv) w |
| exact ha.trans (he.trans (congrArg₂ (·+·) hb.symm hc.symm)) |
|
|
| lemma gradient_smul {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) (c : ℝ) : |
| q.gradient ((lipschitzWith_smul c).comp hu) = c • q.gradient hu := by |
| apply Lp.ext |
| apply EventuallyEq.trans _ (Lp.coeFn_smul c _).symm |
| have hea := (q.memLp_gradientFunction ((lipschitzWith_smul c).comp hu)).coeFn_toLp |
| have heb := (q.memLp_gradientFunction hu).coeFn_toLp |
| have he : q.gradientFunction (fun x => c • u x) =ᵐ[q.atlasMeasure] |
| fun w => c • q.gradientFunction u w := by |
| apply Measure.ae_sum_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr |
| apply (withDensity_absolutelyContinuous _ _).ae_le |
| filter_upwards [ae_uniqueDiffWithinAt volume (q.chart i).domain, |
| (q.chart i).ae_differentiableWithinAt_scalar hu] with z hz hdu |
| have hs : (q.chart i).scalar (fun x => c • u x) = c • (q.chart i).scalar u := by |
| funext z |
| by_cases hz : z ∈ (q.chart i).domain <;> simp [IntegerChart.scalar,hz] |
| change q.normalizedCovector i (fun x => c • u x) z = c • q.normalizedCovector i u z |
| unfold normalizedCovector covector |
| rw [hs,fderivWithin_const_smul hz hdu] |
| have he : differentialRow (c • fderivWithin ℝ ((q.chart i).scalar u) (q.chart i).domain z) = |
| c • differentialRow (fderivWithin ℝ ((q.chart i).scalar u) (q.chart i).domain z) := rfl |
| rw [he,Matrix.mulVec_smul,WithLp.toLp_smul] |
| filter_upwards [hea,heb,he] with w ha hb he |
| change (q.gradient ((lipschitzWith_smul c).comp hu)) w = c • (q.gradient hu) w |
| exact ha.trans (he.trans (congrArg (c • ·) hb.symm)) |
|
|
| noncomputable def value {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| Lp ℝ 2 (MassMeasure.currentMassMeasure hT) := |
| ((Foundations.boundedLip_of_lipschitz hu).memLp 2).toLp u |
|
|
| lemma value_add {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} {u v : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hv : LipschitzWith J v) : |
| value (hT := hT) (hu.add hv) = value hu + value hv := |
| MemLp.toLp_add _ _ |
|
|
| lemma value_smul {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) (c : ℝ) : |
| value (hT := hT) ((lipschitzWith_smul c).comp hu) = c • value hu := |
| MemLp.toLp_const_smul c ((Foundations.boundedLip_of_lipschitz hu).memLp 2) |
|
|
| lemma norm_value {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| ‖value (hT := hT) hu‖ = lpNorm u 2 (MassMeasure.currentMassMeasure hT) := by |
| rw [value,Lp.norm_toLp,toReal_eLpNorm] |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
