import Mathlib import OAI.Geometry.CubicTorus.Estimates.SurfaceJet import OAI.Geometry.CubicTorus.Reflection.CoordinateReflection namespace OAI section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma mem_coordinateHalf_quotient (i : Fin 3) (a : ℝ) (x : Cover) : quotientMap x∈coordinateHalf i a ↔ Real.sin (2*Real.pi*(x i-a))<0 := by let q : Cover → ℝ := fun y => coordinateSine i (y-a • EuclideanSpace.single i 1) have hp : ∀ n y,q (y+latticeVector n)=q y := by intro n y dsimp [q] rw [show y+latticeVector n-a • EuclideanSpace.single i 1= (y-a • EuclideanSpace.single i 1)+latticeVector n by abel,coordinateSine_periodic] have he := periodic_representative hp x simp only [q,coordinateSine,PiLp.sub_apply,PiLp.smul_apply,PiLp.single_apply,ite_true, smul_eq_mul,mul_one] at he change Real.sin (2*Real.pi*((representative (quotientMap x)) i-a))<0 ↔ _ rw [he] lemma coordinateHalf_zero (i : Fin 3) : coordinateHalf i 0=sublevelRegion (coordinateSine i) 0 := by ext z simp only [coordinateHalf,sublevelRegion,coordinateSine,sub_zero,mem_ofPred_eq] lemma coordinateHalf_one_half (i : Fin 3) : coordinateHalf i (1/2)=sublevelRegion (fun x => -coordinateSine i x) 0 := by ext z rw [show (1/2 : ℝ)=0+1/2 by ring,coordinateHalf_halfturn] simp [sublevelRegion,coordinateSine] lemma coordinateHalf_translation (i : Fin 3) (a : ℝ) (z : Torus) : z+quotientMap (a • EuclideanSpace.single i 1)∈coordinateHalf i a ↔ z∈coordinateHalf i 0 := by obtain ⟨x,rfl⟩ := quotientMap_surjective z rw [← quotientMap_add,mem_coordinateHalf_quotient,mem_coordinateHalf_quotient] simp lemma volume_translated_half (i : Fin 3) (a : ℝ) {E : Set Torus} (hE : MeasurableSet E) : torusVolume ((fun z => z+quotientMap (a • EuclideanSpace.single i 1)) ⁻¹' E∩coordinateHalf i 0)= torusVolume (E∩coordinateHalf i a) := by have he : (fun z => z+quotientMap (a • EuclideanSpace.single i 1)) ⁻¹' E∩coordinateHalf i 0= (fun z => z+quotientMap (a • EuclideanSpace.single i 1)) ⁻¹' (E∩coordinateHalf i a) := by ext z simp only [mem_inter_iff,mem_preimage,coordinateHalf_translation] rw [he,volume_translation (hE.inter (coordinateHalf_measurable i a)).nullMeasurableSet] theorem isMinimizerAt_exists_bisecting_reflection_measurable {V : ℝ} (hV : V∈Ioo 0 1) {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) (i : Fin 3) : ∃ a∈Icc (0 : ℝ) (1/2),let F := (fun z => z+quotientMap (a • EuclideanSpace.single i 1)) ⁻¹' E MeasurableSet F ∧ IsMinimizerAt V F ∧ ∃ μ : FiniteMeasure NormalBundle,IsNormalMeasure F μ ∧ IsMinimizerAt V (reflectionCompetitor i F (sublevelRegion (coordinateSine i) 0)) ∧ IsMinimizerAt V (reflectionCompetitor i F (sublevelRegion (fun x => -coordinateSine i x) 0)) ∧ (μ : Measure NormalBundle) {z | coordinateSine i z.1=0}=0 := by obtain ⟨a,ha,hbis⟩ := exists_coordinateHalf_bisects i hE let F := (fun z : Torus => z+quotientMap (a • EuclideanSpace.single i 1)) ⁻¹' E have hF : MeasurableSet F := (continuous_add_const _).measurable hE have hmF : IsMinimizerAt V F := isMinimizerAt_translation hmin _ have hv : torusVolume.real E=V := by rw [Measure.real_def,hmin.2.1,ENNReal.toReal_ofReal hV.1.le] have hvF : torusVolume.real F=V := by rw [Measure.real_def,hmF.2.1,ENNReal.toReal_ofReal hV.1.le] have hm : torusVolume.real (F∩coordinateHalf i 0)=V/2 := by rw [Measure.real_def,volume_translated_half i a hE,← Measure.real_def,hbis,hv] have hp : torusVolume.real (F∩coordinateHalf i (1/2))=V/2 := by have hh := coordinateHalf_volume_sum i 0 hF norm_num only [zero_add] at hh rw [hm,hvF] at hh linarith have hem : 2*torusVolume (F∩sublevelRegion (coordinateSine i) 0)=ENNReal.ofReal V := by apply (ENNReal.toReal_eq_toReal_iff' (by finiteness) ENNReal.ofReal_ne_top).mp rw [ENNReal.toReal_mul,ENNReal.toReal_ofNat,ENNReal.toReal_ofReal hV.1.le, ← coordinateHalf_zero,← Measure.real_def,hm] ring have hep : 2*torusVolume (F∩sublevelRegion (fun x => -coordinateSine i x) 0)=ENNReal.ofReal V := by apply (ENNReal.toReal_eq_toReal_iff' (by finiteness) ENNReal.ofReal_ne_top).mp rw [ENNReal.toReal_mul,ENNReal.toReal_ofNat,ENNReal.toReal_ofReal hV.1.le, ← coordinateHalf_one_half,← Measure.real_def,hp] ring obtain ⟨μ,hμ,hflux⟩ := exists_normalMeasure hF hmF.1.2.ne have hnormal : IsNormalMeasure F μ := ⟨hμ,hflux⟩ exact ⟨a,ha,hF,hmF,μ,hnormal,isMinimizerAt_reflection_competitors hF hmF hnormal i hem hep⟩ theorem isMinimizerAt_exists_bisecting_reflection {V : ℝ} (hV : V∈Ioo 0 1) {E : Set Torus} (hmin : IsMinimizerAt V E) (i : Fin 3) : ∃ a∈Icc (0 : ℝ) (1/2),∃ F : Set Torus, F =ᵐ[torusVolume] ((fun z => z+quotientMap (a • EuclideanSpace.single i 1)) ⁻¹' E) ∧ MeasurableSet F ∧ IsMinimizerAt V F ∧ ∃ μ : FiniteMeasure NormalBundle,IsNormalMeasure F μ ∧ IsMinimizerAt V (reflectionCompetitor i F (sublevelRegion (coordinateSine i) 0)) ∧ IsMinimizerAt V (reflectionCompetitor i F (sublevelRegion (fun x => -coordinateSine i x) 0)) ∧ (μ : Measure NormalBundle) {z | coordinateSine i z.1=0}=0 := by let E' := toMeasurable torusVolume E have hae : E' =ᵐ[torusVolume] E := hmin.1.1.toMeasurable_ae_eq have hE' : MeasurableSet E' := measurableSet_toMeasurable _ _ have hmE' : IsMinimizerAt V E' := (isMinimizerAt_congr_ae hae).mpr hmin obtain ⟨a,ha,hF,hmF,hrest⟩ := isMinimizerAt_exists_bisecting_reflection_measurable hV hE' hmE' i refine ⟨a,ha,_,?_,hF,hmF,hrest⟩ exact (measurePreserving_add_right torusVolume _).quasiMeasurePreserving.preimage_ae_eq hae end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma coordinateHalf_open (i : Fin 3) : IsOpen (coordinateHalf i 0) := by rw [coordinateHalf_zero] change IsOpen {z : Torus | coordinateSine i (representative z)<0} exact isOpen_lt (continuous_periodic_descent (coordinateSine_contDiff i).continuous (coordinateSine_periodic i)) continuous_const lemma coordinateHalf_eq_slab_image (i : Fin 3) : coordinateHalf i 0= quotientMap '' {x : Cover | (1/2 : ℝ) ⟨Or.inl hz,hz.2⟩ · rintro ⟨hz,hzB⟩ rcases hz with hh|hh · exact hh · exact False.elim (disjoint_left.mp hd hzB hh.2) lemma reflectionCompetitor_invariant (i : Fin 3) (E B : Set Torus) : coordinateReflection i ⁻¹' reflectionCompetitor i E B=reflectionCompetitor i E B := by ext z simp only [reflectionCompetitor,mem_preimage,mem_union,mem_inter_iff,coordinateReflection_involutive] tauto end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus theorem cutoff_graph_patch_continuous_mass {A O U : Set Cover} (hA : MeasurableSet A) (a : Cover) {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) {ν : FiniteMeasure BlowupNormalBundle} (hsat : HasCutoffSaturation ν O) (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) (f : BoundedContinuousFunction Cover ℝ) : (∫ z in {z : BlowupNormalBundle | z.1 ∈ O}, φ z.1*f z.1 ∂(ν : Measure BlowupNormalBundle)) = ∫ y : Plane, φ (a+normalGraph n g y)*f (a+normalGraph n g y)*‖graphAreaVector n g y‖ := by let M : ℝ := ‖f‖ + 1 have hM : 0 < M := by dsimp [M]; positivity let f₀ : Cover → ℝ := fun x => f x / M have hfc : Continuous f₀ := f.continuous.div_const M have hfb (x : Cover) : |f₀ x| ≤ 1 := by rw [show f₀ x = f x / M from rfl, abs_div, abs_of_pos hM] apply (div_le_one hM).mpr exact (f.norm_coe_le_norm x).trans (by dsimp [M]; linarith) have he := cutoff_graph_patch_limit_mass hA a hn hg hc hgraph hsat hflux hφ hsφ hφU hφO hφR hφb (fun i => smoothApprox_contDiff hfc.measurable hfb i) (fun i x => smoothApprox_bound hfb i x) (fun x _ => ContDiffBump.convolution_tendsto_right_of_continuous mollifier_rOut_tendsto hfc x) have he' := congrArg (fun t : ℝ => M * t) he rw [← integral_const_mul, ← integral_const_mul] at he' convert he' using 1 <;> congr 1 <;> funext x <;> dsimp only [f₀] <;> field_simp [ne_of_gt hM] theorem cutoff_graph_patch_measure {A O U : Set Cover} (hA : MeasurableSet A) (a : Cover) {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) {ν : FiniteMeasure BlowupNormalBundle} (hsat : HasCutoffSaturation ν O) (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) : Measure.map Prod.fst (((ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}).withDensity (fun z => ENNReal.ofReal (φ z.1))) = Measure.map (fun y : Plane => a + normalGraph n g y) (volume.withDensity (fun y => ENNReal.ofReal (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖))) := by let μ := (ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O} have hI : Integrable (fun z : BlowupNormalBundle => φ z.1) μ := Integrable.of_bound (hφ.continuous.comp continuous_fst).aestronglyMeasurable 1 (ae_of_all μ fun z => by rw [Real.norm_eq_abs, abs_of_nonneg (hφb _).1]; exact (hφb _).2) have hG := graph_weighted_integrable a hn hg.continuous hφ.continuous hsφ hφR (fun y hy => (graphAreaVector_continuousAt n (hc y hy)).norm) have hGc := graph_weighted_continuous a hn hg.continuous hφ.continuous hφR (fun y hy => (graphAreaVector_continuousAt n (hc y hy)).norm) let := isFiniteMeasure_withDensity_ofReal hI.hasFiniteIntegral let := isFiniteMeasure_withDensity_ofReal hG.hasFiniteIntegral have hchart : Continuous (fun y : Plane => a + normalGraph n g y) := by simpa only [id_eq] using graph_trace_continuous a n hg.continuous continuous_id apply ext_of_forall_integral_eq_of_IsFiniteMeasure intro f rw [integral_map_of_stronglyMeasurable measurable_fst f.continuous.stronglyMeasurable, integral_map_of_stronglyMeasurable hchart.measurable f.continuous.stronglyMeasurable] rw [integral_withDensity_eq_integral_toReal_smul (f := fun z : BlowupNormalBundle => ENNReal.ofReal (φ z.1)) (hφ.continuous.comp continuous_fst).measurable.ennreal_ofReal (ae_of_all μ fun _ => ENNReal.ofReal_lt_top) _, integral_withDensity_eq_integral_toReal_smul hGc.measurable.ennreal_ofReal (ae_of_all volume fun _ => ENNReal.ofReal_lt_top) _] simp_rw [ENNReal.toReal_ofReal (hφb _).1, smul_eq_mul, ENNReal.toReal_ofReal (mul_nonneg (hφb _).1 (norm_nonneg _))] convert cutoff_graph_patch_continuous_mass hA a hn hg hc hgraph hsat hflux hφ hsφ hφU hφO hφR hφb f using 1 congr 1 funext y ring end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma graphUnitNormalExtension_measurable (a : Cover) {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) : Measurable (graphUnitNormalExtension a n g) := by have hm : Measurable (gradient g) := (InnerProductSpace.toDual ℝ Plane).symm.continuous.measurable.comp (measurable_fderiv ℝ g) exact (graphSlopeNormal_contDiff hn).continuous.measurable.comp (hm.comp ((normalPlaneCoordinates n).continuous.measurable.comp (measurable_id.sub measurable_const))) def graphNormalSection (a : Cover) {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) (x : Cover) : BlowupNormalBundle := (x, ⟨graphUnitNormalExtension a n g x, graphUnitNormalExtension_unit a hn g x⟩) lemma graphNormalSection_measurable (a : Cover) {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) : Measurable (graphNormalSection a hn g) := measurable_id.prodMk (graphUnitNormalExtension_measurable a hn g).subtype_mk lemma weighted_bundle_section {μ : Measure BlowupNormalBundle} {φ : Cover → ℝ} (hφ : Measurable φ) {s : Cover → BlowupNormalBundle} (hs : Measurable s) (hae : ∀ᵐ z ∂μ, 0 < φ z.1 → s z.1 = z) : μ.withDensity (fun z => ENNReal.ofReal (φ z.1)) = Measure.map s (Measure.map Prod.fst (μ.withDensity (fun z => ENNReal.ofReal (φ z.1)))) := by rw [Measure.map_map hs measurable_fst] symm calc _ = Measure.map id (μ.withDensity (fun z => ENNReal.ofReal (φ z.1))) := by apply Measure.map_congr apply (ae_withDensity_iff (hφ.comp measurable_fst).ennreal_ofReal).mpr filter_upwards [hae] with z hz hpos exact hz (ENNReal.ofReal_ne_zero_iff.mp hpos) _ = _ := Measure.map_id theorem cutoff_graph_patch_bundle_measure {A O U : Set Cover} (hA : MeasurableSet A) (a : Cover) {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (x ∈ A ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) {ν : FiniteMeasure BlowupNormalBundle} (hsat : HasCutoffSaturation ν O) (hflux : ∀ F : Cover → Cover, ContDiff ℝ (⊤ : ℕ∞) F → HasCompactSupport F → tsupport F ⊆ O → (∀ x, ‖F x‖ ≤ 1) → (∫ z, ⟪(z.2 : Cover),F z.1⟫ ∂(ν : Measure BlowupNormalBundle)) = ∫ x in A, divergence F x) {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ O) (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) : (((ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}).withDensity (fun z => ENNReal.ofReal (φ z.1))) = Measure.map (fun y : Plane => graphNormalSection a hn g (a + normalGraph n g y)) (volume.withDensity (fun y => ENNReal.ofReal (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖))) := by have ha := cutoff_graph_patch_normal_ae hA a hn hg hc hgraph hsat hflux hφ hsφ hφU hφO hφR hφb have hs := weighted_bundle_section hφ.continuous.measurable (graphNormalSection_measurable a hn g) (show ∀ᵐ z ∂(ν : Measure BlowupNormalBundle).restrict {z | z.1 ∈ O}, 0 < φ z.1 → graphNormalSection a hn g z.1 = z from by filter_upwards [ha] with z hz hpos refine Prod.ext ?_ ?_ · rfl · exact Subtype.ext (hz hpos).symm) rw [hs, cutoff_graph_patch_measure hA a hn hg hc hgraph hsat hflux hφ hsφ hφU hφO hφR hφb] have hchart : Measurable (fun y : Plane => a + normalGraph n g y) := by simpa only [id_eq] using (graph_trace_continuous a n hg.continuous continuous_id).measurable exact Measure.map_map (graphNormalSection_measurable a hn g) hchart end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus def clampCell (x : Cover) : CompactCell := ⟨WithLp.toLp 2 (fun i => max 0 (min 1 (x i))), ⟨fun i => max 0 (min 1 (x i)), ⟨fun _ => le_max_left _ _, fun _ => max_le zero_le_one (min_le_left _ _)⟩, rfl⟩⟩ lemma continuous_clampCell : Continuous clampCell := by apply Continuous.subtype_mk exact (EuclideanSpace.equiv (Fin 3) ℝ).symm.continuous.comp (continuous_pi (fun i => continuous_const.max (continuous_const.min (by fun_prop)))) lemma clampCell_coe (x : CompactCell) : clampCell (x : Cover) = x := by apply Subtype.ext ext i change max 0 (min 1 ((x : Cover) i)) = (x : Cover) i rw [min_eq_right (compactCell_coordinates x i).2, max_eq_right (compactCell_coordinates x i).1] def normalCellProjection (z : BlowupNormalBundle) : NormalBundle := (clampCell z.1, z.2) lemma continuous_normalCellProjection : Continuous normalCellProjection := (continuous_clampCell.comp continuous_fst).prodMk continuous_snd lemma normalCellProjection_cover (z : NormalBundle) : normalCellProjection (normalCoverMap z) = z := by change (clampCell (z.1 : Cover), z.2) = z rw [clampCell_coe] lemma map_withDensity_comp {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] (μ : Measure α) {f : α → β} (hf : Measurable f) {w : β → ℝ≥0∞} (hw : Measurable w) : (Measure.map f μ).withDensity w = Measure.map f (μ.withDensity (w ∘ f)) := by apply Measure.ext intro s hs rw [withDensity_apply _ hs, Measure.map_apply hf hs, withDensity_apply _ (hs.preimage hf), setLIntegral_map hs hw hf] rfl lemma withDensity_restrict_of_support {α : Type*} [MeasurableSpace α] {μ : Measure α} {w : α → ℝ≥0∞} {S : Set α} (hS : MeasurableSet S) (hs : ∀ x, x ∉ S → w x = 0) : (μ.restrict S).withDensity w = μ.withDensity w := by rw [← withDensity_indicator hS] congr 1 apply Set.indicator_eq_self.2 intro x hx by_contra hn exact hx (hs x hn) def actualNormalGraphChart (a : Cover) {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) (y : Plane) : NormalBundle := normalCellProjection (graphNormalSection a hn g (a + normalGraph n g y)) lemma actualNormalGraphChart_measurable (a : Cover) {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} (hg : Continuous g) : Measurable (actualNormalGraphChart a hn g) := by apply continuous_normalCellProjection.measurable.comp apply (graphNormalSection_measurable a hn g).comp simpa only [id_eq] using (graph_trace_continuous a n hg continuous_id).measurable theorem normalMeasure_graph_area_measure {E : Set Torus} (hE : MeasurableSet E) {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {U : Set Cover} (a : Cover) {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {R : ℝ} (hc : ∀ y ∈ Metric.closedBall (0 : Plane) R, ContinuousAt (fderiv ℝ g) y) (hgraph : ∀ᵐ x : Cover ∂volume, x ∈ U → (quotientMap x ∈ E ↔ ⟪n,x-a⟫ < graphHeightExtension a n g x)) {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) (hφU : tsupport φ ⊆ U) (hφO : tsupport φ ⊆ {x | ∀ i, 0 < x i ∧ x i < 1}) (hφR : tsupport φ ⊆ Metric.closedBall a R) (hφb : ∀ x, φ x ∈ Icc 0 1) : (μ : Measure NormalBundle).withDensity (fun z => ENNReal.ofReal (φ z.1)) = Measure.map (actualNormalGraphChart a hn g) (volume.withDensity (fun y => ENNReal.ofReal (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖))) := by let O : Set Cover := {x | ∀ i, 0 < x i ∧ x i < 1} have hmO : MeasurableSet {z : BlowupNormalBundle | z.1 ∈ O} := by change MeasurableSet {z : BlowupNormalBundle | ∀ i, 0 < z.1 i ∧ z.1 i < 1} simp only [Set.ofPred_forall] apply MeasurableSet.iInter intro i have hm : Measurable (fun z : BlowupNormalBundle => z.1 i) := by fun_prop exact (measurableSet_lt measurable_const hm).inter (measurableSet_lt hm measurable_const) have hf (F : Cover → Cover) (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hs : HasCompactSupport F) (hO : tsupport F ⊆ O) (_hb : ∀ x, ‖F x‖ ≤ 1) := normalCoverMeasure_local_flux hE hμ hF hs hO have he := cutoff_graph_patch_bundle_measure (hE.preimage measurable_quotientMap) a hn hg hc hgraph (normalMeasure_cutoff_saturation hμ O) hf hφ hsφ hφU hφO hφR hφb rw [withDensity_restrict_of_support hmO (fun z hz => by rw [image_eq_zero_of_notMem_tsupport (fun h => hz (hφO h)), ENNReal.ofReal_zero])] at he have hcmap : (normalCoverMeasure μ : Measure BlowupNormalBundle) = Measure.map normalCoverMap (μ : Measure NormalBundle) := FiniteMeasure.toMeasure_map μ normalCoverMap rw [hcmap, map_withDensity_comp _ continuous_normalCoverMap.measurable (w := fun z : BlowupNormalBundle => ENNReal.ofReal (φ z.1)) ((hφ.continuous.comp continuous_fst).measurable.ennreal_ofReal)] at he have hp := congrArg (Measure.map normalCellProjection) he rw [Measure.map_map continuous_normalCellProjection.measurable continuous_normalCoverMap.measurable] at hp have hid : normalCellProjection ∘ normalCoverMap = id := funext normalCellProjection_cover rw [hid, Measure.map_id] at hp have hchart : Measurable (fun y : Plane => graphNormalSection a hn g (a+normalGraph n g y)) := by apply (graphNormalSection_measurable a hn g).comp simpa only [id_eq] using (graph_trace_continuous a n hg.continuous continuous_id).measurable rw [Measure.map_map continuous_normalCellProjection.measurable hchart] at hp exact hp end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus def lpPullbackLE {α β F : Type*} [MeasurableSpace α] [MeasurableSpace β] [NormedAddCommGroup F] [NormedSpace ℝ F] {ν : Measure α} {μ : Measure β} {f : α → β} (hf : Measurable f) (hle : Measure.map f ν ≤ μ) : Lp F 2 μ →L[ℝ] Lp F 2 ν := (Lp.compMeasurePreservingₗᵢ ℝ f ⟨hf,rfl⟩).toContinuousLinearMap.comp (Lp.LpToLpOfMeasureLeSMul (c := 1) (by simp) (by simpa using hle)) lemma lpPullbackLE_ae {α β F : Type*} [MeasurableSpace α] [MeasurableSpace β] [NormedAddCommGroup F] [NormedSpace ℝ F] {ν : Measure α} {μ : Measure β} {f : α → β} (hf : Measurable f) (hle : Measure.map f ν ≤ μ) (u : Lp F 2 μ) : lpPullbackLE hf hle u =ᵐ[ν] u ∘ f := by have hmp : MeasurePreserving f ν (Measure.map f ν) := ⟨hf,rfl⟩ have hr := Lp.coeFn_LpToLpOfMeasureLeSMul (c := 1) (by simp : (1 : ℝ≥0∞) ≠ ∞) (by simpa using hle) u exact (Lp.coeFn_compMeasurePreserving _ hmp).trans (hmp.quasiMeasurePreserving.ae_eq_comp hr) lemma lpPullbackLE_congr_ae {α β F : Type*} [MeasurableSpace α] [MeasurableSpace β] [NormedAddCommGroup F] [NormedSpace ℝ F] {ν : Measure α} {μ : Measure β} {f : α → β} (hf : Measurable f) (hle : Measure.map f ν ≤ μ) {u v : β → F} (huv : u =ᵐ[μ] v) : u ∘ f =ᵐ[ν] v ∘ f := by exact (show Measure.QuasiMeasurePreserving f ν μ from ⟨hf,Measure.absolutelyContinuous_of_le hle⟩).ae_eq_comp huv lemma graphAreaVector_norm_ge_one {n : Cover} (hn : ‖n‖ = 1) (g : Plane → ℝ) (y : Plane) : 1 ≤ ‖graphAreaVector n g y‖ := by have hs := graphAreaVector_norm_sq hn g y nlinarith [norm_nonneg (graphAreaVector n g y), sq_nonneg ‖gradient g y‖] lemma graph_chart_measure_le {μ : Measure NormalBundle} (a : Cover) {n : Cover} (hn : ‖n‖ = 1) {g : Plane → ℝ} (hg : Continuous g) {φ : Cover → ℝ} (hφ : ∀ x, φ x ∈ Icc 0 1) (hm : μ.withDensity (fun z => ENNReal.ofReal (φ z.1)) = Measure.map (actualNormalGraphChart a hn g) (volume.withDensity (fun y => ENNReal.ofReal (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)))) {S : Set Plane} (hS : MeasurableSet S) (hunit : ∀ y ∈ S, φ (a+normalGraph n g y)=1) : Measure.map (actualNormalGraphChart a hn g) (volume.restrict S) ≤ μ := by have hd : volume.restrict S ≤ volume.withDensity (fun y => ENNReal.ofReal (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)) := by rw [← withDensity_indicator_one hS] apply withDensity_mono filter_upwards [] with y by_cases hy : y∈S · simp only [Set.indicator_of_mem hy, Pi.one_apply, hunit y hy, one_mul] simpa using ENNReal.ofReal_le_ofReal (graphAreaVector_norm_ge_one hn g y) · simp [hy] have hcut : μ.withDensity (fun z => ENNReal.ofReal (φ z.1)) ≤ μ := by conv_rhs => rw [← withDensity_one (μ := μ)] apply withDensity_mono exact ae_of_all _ (fun z => by simpa using ENNReal.ofReal_le_ofReal (hφ z.1).2) exact ((Measure.map_mono hd (actualNormalGraphChart_measurable a hn hg)).trans_eq hm.symm).trans hcut def graphCovector (n : Cover) (g : Plane → ℝ) (y : Plane) : Cover →L[ℝ] Plane := normalPlaneCoordinates n + (innerSL ℝ n).smulRight (gradient g y) lemma graphCovector_apply (n : Cover) (g : Plane → ℝ) (y : Plane) (v : Cover) : graphCovector n g y v = normalPlaneCoordinates n v + ⟪n,v⟫ • gradient g y := rfl lemma measurable_graphCovector (n : Cover) (g : Plane → ℝ) : Measurable (graphCovector n g) := by have hm : Measurable (gradient g) := (InnerProductSpace.toDual ℝ Plane).symm.continuous.measurable.comp (measurable_fderiv ℝ g) exact measurable_const.add ((ContinuousLinearMap.smulRightL ℝ Cover Plane (innerSL ℝ n)).continuous.measurable.comp hm) lemma graphCovector_norm_le {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) (y : Plane) : ‖graphCovector n g y‖ ≤ 1+K := by calc _ ≤ ‖normalPlaneCoordinates n‖+‖(innerSL ℝ n).smulRight (gradient g y)‖ := norm_add_le _ _ _ ≤ 1+‖innerSL ℝ n‖*‖gradient g y‖ := add_le_add (norm_normalPlaneCoordinates_le n) (ContinuousLinearMap.norm_smulRight_apply _ _).le _ ≤ 1+K := by rw [innerSL_apply_norm,hn,one_mul] apply add_le_add le_rfl simpa only [gradient, LinearIsometryEquiv.norm_map] using norm_fderiv_le_of_lipschitz ℝ hg (x₀ := y) lemma graphCovector_memLp {ν : Measure Plane} {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) : MemLp (graphCovector n g) ∞ ν := memLp_top_of_bound (measurable_graphCovector n g).aestronglyMeasurable (1+K) (ae_of_all _ (graphCovector_norm_le hn hg)) def graphCovectorLp {ν : Measure Plane} {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) : Lp Cover 2 ν →L[ℝ] Lp Plane 2 ν := (ContinuousLinearMap.id ℝ (Cover →L[ℝ] Plane)).holderL ν ∞ 2 2 ((graphCovector_memLp (ν := ν) hn hg).toLp (graphCovector n g)) lemma graphCovectorLp_ae {ν : Measure Plane} {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) (u : Lp Cover 2 ν) : graphCovectorLp hn hg u =ᵐ[ν] fun y => graphCovector n g y (u y) := by have ha := (graphCovector_memLp (ν := ν) hn hg).coeFn_toLp have hb := (ContinuousLinearMap.id ℝ (Cover →L[ℝ] Plane)).coeFn_holder ((graphCovector_memLp (ν := ν) hn hg).toLp (graphCovector n g)) u (r := 2) filter_upwards [ha,hb] with y ha hb change (graphCovectorLp hn hg u) y = _ at hb simpa only [ContinuousLinearMap.id_apply,ha] using hb end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma graphCovector_duality {n : Cover} (hn : ‖n‖=1) (g : Plane → ℝ) (y v : Plane) (w : Cover) : ⟪graphCovector n g y w,v⟫ = ⟪w,normalPlaneEmbedding n v+⟪gradient g y,v⟫ • n⟫ := by rw [graphCovector_apply,inner_add_left,real_inner_smul_left, inner_add_right,inner_smul_right] have he : ⟪normalPlaneCoordinates n w,v⟫=⟪w,normalPlaneEmbedding n v⟫ := by calc _ = ⟪v,normalPlaneCoordinates n w⟫ := real_inner_comm _ _ _ = ⟪normalPlaneEmbedding n v,w⟫ := (normalPlaneEmbedding_inner_coordinates hn v w).symm _ = _ := real_inner_comm _ _ rw [he] have hi : ⟪n,w⟫=⟪w,n⟫ := real_inner_comm _ _ rw [hi] ring lemma graphCovector_normal_zero {n : Cover} (hn : ‖n‖=1) (g : Plane → ℝ) (y : Plane) : graphCovector n g y (graphSlopeNormal n (gradient g y))=0 := by rw [graphSlopeNormal_weight hn,map_smul,map_sub,graphCovector_apply, graphCovector_apply,normalPlaneCoordinates_normal hn, normalPlaneCoordinates_embedding,real_inner_self_eq_norm_sq,hn,one_pow, normalPlaneEmbedding_inner hn,one_smul,zero_smul,zero_add,add_zero,sub_self,smul_zero] lemma gradient_graph_trace (a : Cover) {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {y : Plane} (hg : DifferentiableAt ℝ g y) {f : Cover → ℝ} (hf : DifferentiableAt ℝ f (a+normalGraph n g y)) : gradient (fun z => f (a+normalGraph n g z)) y = graphCovector n g y (gradient f (a+normalGraph n g y)) := by apply ext_inner_right ℝ intro v rw [inner_gradient_left,graphCovector_duality hn] have hc := hf.hasFDerivAt.comp y ((normalGraph_hasFDerivAt hg).const_add a) change (fderiv ℝ (f ∘ fun x => a+normalGraph n g x) y) v = _ rw [hc.fderiv,ContinuousLinearMap.comp_apply,← inner_gradient_left] congr 1 simp only [add_apply,ContinuousLinearMap.smulRight_apply, inner_gradient_left] rfl lemma actualNormalGraphChart_position (a : Cover) {n : Cover} (hn : ‖n‖=1) (g : Plane → ℝ) {y : Plane} (hy : ∀ i, (a+normalGraph n g y) i ∈ Icc 0 1) : ((actualNormalGraphChart a hn g y).1 : Cover)=a+normalGraph n g y := by change ((clampCell (a+normalGraph n g y)) : Cover)=_ ext i change max 0 (min 1 ((a+normalGraph n g y) i))=(a+normalGraph n g y) i rw [min_eq_right (hy i).2,max_eq_right (hy i).1] lemma actualNormalGraphChart_normal (a : Cover) {n : Cover} (hn : ‖n‖=1) (g : Plane → ℝ) (y : Plane) : ((actualNormalGraphChart a hn g y).2 : Cover)=graphSlopeNormal n (gradient g y) := graphUnitNormalExtension_graph a hn g g y lemma smoothScalarGradient_graph_covector (a : Cover) {n : Cover} (hn : ‖n‖=1) (f : smoothPeriodicScalars) {g : Plane → ℝ} {y : Plane} (hgy : DifferentiableAt ℝ g y) (hy : ∀ i, (a+normalGraph n g y) i ∈ Icc 0 1) : graphCovector n g y (smoothScalarGradient f (actualNormalGraphChart a hn g y))= gradient (fun z => f.val (a+normalGraph n g z)) y := by change graphCovector n g y (gradient f.val (actualNormalGraphChart a hn g y).1- ⟪((actualNormalGraphChart a hn g y).2 : Cover),gradient f.val (actualNormalGraphChart a hn g y).1⟫ • ((actualNormalGraphChart a hn g y).2 : Cover))=_ rw [actualNormalGraphChart_position a hn g hy,actualNormalGraphChart_normal, map_sub,map_smul,graphCovector_normal_zero hn,smul_zero,sub_zero] exact (gradient_graph_trace a hn hgy (f.property.1.differentiable (by simp) _)).symm def PlaneWeakGradientOn (S : Set Plane) (u : Plane → ℝ) (G : Plane → Plane) : Prop := ∀ (v : Plane) (ψ : Plane → ℝ), ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → tsupport ψ⊆S → (∫ y in S, ⟪G y,v⟫*ψ y)= -(∫ y in S,u y*fderiv ℝ ψ y v) lemma PlaneWeakGradientOn.congr {S : Set Plane} {u u' : Plane → ℝ} {G G' : Plane → Plane} (h : PlaneWeakGradientOn S u G) (hu : u'=ᵐ[volume.restrict S] u) (hG : G'=ᵐ[volume.restrict S] G) : PlaneWeakGradientOn S u' G' := by intro v ψ hψ hc hs have hleft : (∫ y in S, ⟪G' y,v⟫*ψ y)=(∫ y in S, ⟪G y,v⟫*ψ y) := integral_congr_ae (hG.mono (fun _ he => by dsimp only; rw [he])) have hright : (∫ y in S, u' y*fderiv ℝ ψ y v)=(∫ y in S, u y*fderiv ℝ ψ y v) := integral_congr_ae (hu.mono (fun _ he => by dsimp only; rw [he])) rw [hleft,hright] exact h v ψ hψ hc hs lemma C1_weak_gradient_on {S : Set Plane} (hS : IsOpen S) {f : Plane → ℝ} (hf : Continuous f) (hfc : ContDiffOn ℝ 1 f S) : PlaneWeakGradientOn S f (gradient f) := by intro v ψ hψ hc hs have hi₁ : (∫ y in S, ⟪gradient f y,v⟫*ψ y)=(∫ y, ⟪gradient f y,v⟫*ψ y) := by apply setIntegral_eq_integral_of_forall_compl_eq_zero intro y hy rw [image_eq_zero_of_notMem_tsupport (fun h => hy (hs h)),mul_zero] have hi₂ : (∫ y in S, f y*fderiv ℝ ψ y v)=(∫ y, f y*fderiv ℝ ψ y v) := by apply setIntegral_eq_integral_of_forall_compl_eq_zero intro y hy rw [image_eq_zero_of_notMem_tsupport (fun h => hy (hs ((tsupport_fderiv_apply_subset (f := ψ) ℝ v) h))),mul_zero] rw [hi₁,hi₂] simpa only [inner_gradient_left] using C1_local_integration_by_parts hS hf hfc (hψ.of_le (by simp)) hc hs v end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma L2_pair_scalar_measure {ν : Measure Plane} {u : Lp ℝ 2 ν} {φ : Plane → ℝ} (hφ : MemLp φ 2 ν) : ⟪u,hφ.toLp φ⟫=(∫ y,u y*φ y ∂ν) := by rw [L2.inner_def] apply integral_congr_ae filter_upwards [hφ.coeFn_toLp] with y hy simp only [hy,RCLike.inner_apply,conj_trivial] ring lemma L2_pair_vector_measure {ν : Measure Plane} {G : Lp Plane 2 ν} {f : Plane → Plane} (hf : MemLp f 2 ν) : ⟪G,hf.toLp f⟫=(∫ y,⟪G y,f y⟫ ∂ν) := by rw [L2.inner_def] apply integral_congr_ae filter_upwards [hf.coeFn_toLp] with y hy rw [hy] lemma PlaneWeakGradientOn.L2_limit {S : Set Plane} {u : ℕ → Lp ℝ 2 (volume.restrict S)} {G : ℕ → Lp Plane 2 (volume.restrict S)} {u₀ : Lp ℝ 2 (volume.restrict S)} {G₀ : Lp Plane 2 (volume.restrict S)} (hu : Tendsto u atTop (𝓝 u₀)) (hG : Tendsto G atTop (𝓝 G₀)) (hweak : ∀ i,PlaneWeakGradientOn S (u i) (G i)) : PlaneWeakGradientOn S u₀ G₀ := by intro v ψ hψ hc hs let a : Plane → Plane := fun y => ψ y • v have ha : MemLp a 2 (volume.restrict S) := ((hψ.continuous.smul continuous_const).memLp_of_hasCompactSupport hc.smul_right (p := 2) (μ := volume)).mono_measure Measure.restrict_le_self let A := ha.toLp a let b : Plane → ℝ := fun y => fderiv ℝ ψ y v have hb : MemLp b 2 (volume.restrict S) := (((hψ.continuous_fderiv (by simp)).clm_apply continuous_const).memLp_of_hasCompactSupport (hc.fderiv_apply ℝ v) (p := 2) (μ := volume)).mono_measure Measure.restrict_le_self let B := hb.toLp b have hi (W : Lp Plane 2 (volume.restrict S)) : ⟪W,A⟫=(∫ y in S,⟪W y,v⟫*ψ y) := by rw [L2_pair_vector_measure ha] apply integral_congr_ae filter_upwards [] with y simp only [a,inner_smul_right] ring have he (i : ℕ) : ⟪G i,A⟫= -⟪u i,B⟫ := by rw [hi,L2_pair_scalar_measure hb] exact hweak i v ψ hψ hc hs have hleft := hG.inner (𝕜 := ℝ) (tendsto_const_nhds (x := A)) have hright := (hu.inner (𝕜 := ℝ) (tendsto_const_nhds (x := B))).neg have hh := tendsto_nhds_unique (hleft.congr' (Eventually.of_forall he)) hright rw [hi,L2_pair_scalar_measure hb] at hh exact hh lemma graph_trace_contDiffOn_C1 (a n : Cover) {g : Plane → ℝ} {S : Set Plane} (hg : ContDiffOn ℝ 1 g S) {f : Cover → ℝ} (hf : ContDiff ℝ 1 f) : ContDiffOn ℝ 1 (fun y => f (a+normalGraph n g y)) S := by have hgraph : ContDiffOn ℝ 1 (fun y => a+normalGraph n g y) S := contDiffOn_const.add ((normalPlaneLinearIsometry n).contDiff.contDiffOn.add (hg.smul contDiffOn_const)) exact hf.comp_contDiffOn hgraph lemma surfaceSmoothJet_chart_weak_gradient (μ : FiniteMeasure NormalBundle) (a : Cover) {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {S : Set Plane} (hS : IsOpen S) (hgc : ContDiffOn ℝ 1 g S) (hcell : ∀ y ∈ S, ∀ i, (a+normalGraph n g y) i ∈ Icc 0 1) (hdom : Measure.map (actualNormalGraphChart a hn g) (volume.restrict S) ≤ (μ : Measure NormalBundle)) (f : smoothPeriodicScalars) : PlaneWeakGradientOn S (lpPullbackLE (actualNormalGraphChart_measurable a hn hg.continuous) hdom (surfaceSmoothJet μ f).1) (graphCovectorLp hn hg (lpPullbackLE (actualNormalGraphChart_measurable a hn hg.continuous) hdom (surfaceSmoothJet μ f).2)) := by let chart := actualNormalGraphChart a hn g have hchart : Measurable chart := actualNormalGraphChart_measurable a hn hg.continuous have hv := (lpPullbackLE_ae hchart hdom (surfaceSmoothJet μ f).1).trans (lpPullbackLE_congr_ae hchart hdom ((smoothScalarValue f).coeFn_toLp (p := 2) (𝕜 := ℝ) (μ : Measure NormalBundle))) have hw := (lpPullbackLE_ae hchart hdom (surfaceSmoothJet μ f).2).trans (lpPullbackLE_congr_ae hchart hdom ((smoothScalarGradient f).coeFn_toLp (p := 2) (𝕜 := ℝ) (μ : Measure NormalBundle))) have hv' : lpPullbackLE hchart hdom (surfaceSmoothJet μ f).1 =ᵐ[volume.restrict S] (fun y => f.val (a+normalGraph n g y)) := by filter_upwards [hv,ae_restrict_mem hS.measurableSet] with y hy hSy change (lpPullbackLE hchart hdom (surfaceSmoothJet μ f).1) y = f.val (chart y).1 at hy rw [hy] change f.val (actualNormalGraphChart a hn g y).1 = _ rw [actualNormalGraphChart_position a hn g (hcell y hSy)] have hw' : graphCovectorLp hn hg (lpPullbackLE hchart hdom (surfaceSmoothJet μ f).2) =ᵐ[volume.restrict S] gradient (fun y => f.val (a+normalGraph n g y)) := by filter_upwards [graphCovectorLp_ae hn hg (lpPullbackLE hchart hdom (surfaceSmoothJet μ f).2),hw, ae_restrict_mem hS.measurableSet] with y hmap hy hSy change (lpPullbackLE hchart hdom (surfaceSmoothJet μ f).2) y = smoothScalarGradient f (chart y) at hy rw [hmap,hy] exact smoothScalarGradient_graph_covector a hn f (((hgc y hSy).contDiffAt (hS.mem_nhds hSy)).differentiableAt (by simp)) (hcell y hSy) exact (C1_weak_gradient_on hS (graph_trace_continuous a n hg.continuous f.property.1.continuous) (graph_trace_contDiffOn_C1 a n hgc (f.property.1.of_le (by simp)))).congr hv' hw' theorem surfaceH1_chart_weak_gradient (μ : FiniteMeasure NormalBundle) (a : Cover) {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} (hg : LipschitzWith K g) {S : Set Plane} (hS : IsOpen S) (hgc : ContDiffOn ℝ 1 g S) (hcell : ∀ y ∈ S, ∀ i, (a+normalGraph n g y) i ∈ Icc 0 1) (hdom : Measure.map (actualNormalGraphChart a hn g) (volume.restrict S) ≤ (μ : Measure NormalBundle)) {u : SurfaceJet μ} (hu : u∈surfaceH1 μ) : PlaneWeakGradientOn S (lpPullbackLE (actualNormalGraphChart_measurable a hn hg.continuous) hdom u.1) (graphCovectorLp hn hg (lpPullbackLE (actualNormalGraphChart_measurable a hn hg.continuous) hdom u.2)) := by obtain ⟨w,hw,hl⟩ := mem_closure_iff_seq_limit.mp hu let hchart := actualNormalGraphChart_measurable a hn hg.continuous apply PlaneWeakGradientOn.L2_limit (((lpPullbackLE hchart hdom).continuous.tendsto _).comp ((continuous_fst.tendsto u).comp hl)) (((graphCovectorLp hn hg).continuous.tendsto _).comp (((lpPullbackLE hchart hdom).continuous.tendsto _).comp ((continuous_snd.tendsto u).comp hl))) intro i obtain ⟨f,hf⟩ := hw i simp only [Function.comp_apply] rw [← hf] exact surfaceSmoothJet_chart_weak_gradient μ a hn hg hS hgc hcell hdom f end CubicTorus end end end OAI