| import Mathlib |
| import OAI.Geometry.BallPacking.Necessity.DensityResidual |
| import OAI.Geometry.BallPacking.Necessity.OrbitRegularity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped ContDiff Topology |
| open Set Function Filter |
| open scoped ContDiff Topology Manifold |
| open Set Function Filter MeasureTheory |
| open Set Function MeasureTheory |
| open Set Function |
| open SymplecticBallPacking.Hamiltonian (Plane planarCurl) |
| open SymplecticBallPacking.Hamiltonian (Plane planarCurl angularOneForm radiusSq planarArea planarArea_apply) |
| open SymplecticBallPacking.Hamiltonian (Plane planarCurl angularOneForm) |
| open SymplecticBallPacking.Hamiltonian (Plane angularOneForm) |
| open SymplecticBallPacking.Hamiltonian |
| open SymplecticBallPacking.Hamiltonian (Plane) |
| open Set Filter Function |
| open Set Filter MeasureTheory |
| open scoped Topology |
| open Set Filter Finset |
| open scoped ContDiff Topology Classical |
| open Set Filter |
| open scoped BoundedContinuousFunction ContDiff Topology |
| open Set Function Filter Topology |
| open scoped NNReal |
| open scoped ContDiff Topology BoundedContinuousFunction |
| open Function |
| open scoped Topology ContDiff |
| open scoped ContDiff Topology Convolution |
| open Set Filter Function MeasureTheory _root_.ContinuousLinearMap _root_.OAI.ContinuousLinearMap |
| open Set Filter Function MeasureTheory |
|
|
| open scoped ContDiff Topology BoundedContinuousFunction |
| open Set Filter Function |
| namespace HigherDimensionalBallPacking.Rigidity |
| open HigherDimensionalBallPacking.Rigidity FramedCR |
| local instance densityRegularHGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (α : ℝ) : |
| NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance |
| local instance densityRegularHSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (α : ℝ) : |
| NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance |
| local instance densityRegularCGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (α : ℝ) : |
| NormedAddCommGroup (C1HolderSpace E α) := inferInstance |
| local instance densityRegularCSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (α : ℝ) : |
| NormedSpace ℝ (C1HolderSpace E α) := inferInstance |
| local instance densityRegularCompactGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (R : ℝ) : |
| NormedAddCommGroup (CompactHolderSpace E R) := inferInstance |
| local instance densityRegularCompactSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (R : ℝ) : |
| NormedSpace ℝ (CompactHolderSpace E R) := inferInstance |
|
|
| theorem densityCurve_deriv_one {n : ℕ} (p q : Phase n) (R : ℝ) |
| (g : CompactHolderSpace (Phase n) R) (z : ℂ) : |
| fderiv ℝ (densityCurve p q R g) z 1=densitySlope p q R g+ |
| holderValue ((1:ℝ)/3) (c1HolderDeriv ((1:ℝ)/3) (compactCRInverse R g)) z 1 := by |
| rw [densityCurve_eq_profile,c1HolderAffineCurve_fderiv] |
| change (1:ℂ) • densitySlope p q R g+_=_ |
| rw [one_smul] |
| rfl |
|
|
| theorem densityCurve_C1_curl {n : ℕ} (p q : Phase n) (R : ℝ) |
| (g : CompactHolderSpace (Phase n) R) (J : Phase n → End n) |
| (hCR : ∀ z, PseudoHolomorphicAt J (densityCurve p q R g) z) (z : ℂ) : |
| holderValue ((1:ℝ)/3) (c1StandardCurl (compactCRInverse R g)) z= |
| (J (densityCurve p q R g z)-standardJ n) |
| (densitySlope p q R g+holderValue ((1:ℝ)/3) (c1HolderDeriv ((1:ℝ)/3) (compactCRInverse R g)) z 1) := by |
| have hv : holderValue ((1:ℝ)/3) (c1StandardCurl (compactCRInverse R g)) z=compactHolderValue R g z := by |
| rw [c1StandardCurl_value] |
| exact compactCRInverse_rightInverse R g z |
| rw [hv,←densityCurve_deriv_one p q R g z,←densityCurve_curl p q R g z] |
| have he := (hCR z).2 (1:ℂ) |
| simp only [mul_one] at he |
| unfold standardCurl |
| rw [he] |
| rfl |
|
|
| theorem densityCurve_regular {n : ℕ} (p q : Phase n) (R : ℝ) |
| (g : CompactHolderSpace (Phase n) R) (ha : densitySlope p q R g≠0) |
| (J : Phase n → End n) (hJs : ContDiff ℝ ∞ J) |
| (hJ : ∀ x, Compatible (J x)) (hc : HasCompactSupport (fun x => J x-standardJ n)) |
| (hCR : ∀ z, PseudoHolomorphicAt J (densityCurve p q R g) z) : |
| ContDiff ℝ ∞ (densityCurve p q R g) := by |
| let v := compactCRInverse R g |
| let c := p-compactCRInverseValue R g 0 |
| let a := densitySlope p q R g |
| have he : densityCurve p q R g=c1HolderAffineCurve ((1:ℝ)/3) c (affineSlope a) v := |
| densityCurve_eq_profile p q R g |
| have hcurl (z : ℂ) : holderValue ((1:ℝ)/3) (c1StandardCurl v) z= |
| (J (c1HolderAffineCurve ((1:ℝ)/3) c (affineSlope a) v z)-standardJ n) |
| (a+holderValue ((1:ℝ)/3) (c1HolderDeriv ((1:ℝ)/3) v) z 1) := by |
| rw [←he] |
| exact densityCurve_C1_curl p q R g J hCR z |
| have hv := affine_c1_regular (fun x => J x-standardJ n) (hJs.sub contDiff_const) hc |
| (fun x => by rw [add_comm,sub_add_cancel]; exact hJ x) c a ha v hcurl |
| (compactCRInverse_tendsto_zero R g) |
| rw [he] |
| exact (contDiff_const.add (affineSlope a).contDiff).add hv |
|
|
| theorem densityCurve_affineLine {n : ℕ} (p q : Phase n) (R : ℝ) |
| (g : CompactHolderSpace (Phase n) R) (ha : densitySlope p q R g≠0) |
| (J : Phase n → End n) (hJs : ContDiff ℝ ∞ J) |
| (hJ : ∀ x, Compatible (J x)) (hc : HasCompactSupport (fun x => J x-standardJ n)) |
| (hCR : ∀ z, PseudoHolomorphicAt J (densityCurve p q R g) z) : |
| AffineLineCurve J p q (densityCurve p q R g) := |
| ⟨densityCurve_regular p q R g ha J hJs hJ hc hCR,hCR,densityCurve_zero p q R g, |
| densityCurve_one p q R g,densitySlope p q R g,ha,densityCurve_leading p q R g⟩ |
|
|
| theorem densityResidual_zero_affineLine {n : ℕ} (p q : Phase n) (R : ℝ) |
| (A : ℝ × Phase n → End n) (hA : ContDiff ℝ ∞ A) (hc : HasCompactSupport A) |
| (J : Phase n → End n) (hJs : ContDiff ℝ ∞ J) |
| (hJ : ∀ x, Compatible (J x)) (hJc : HasCompactSupport (fun x => J x-standardJ n)) |
| (he : ∀ t ∈ Icc (0:ℝ) 1, ∀ x, A (t,x)=lineHomotopy J t x-standardJ n) |
| (y : ℝ × CompactHolderSpace (Phase n) R) (hy : y.1∈Icc (0:ℝ) 1) |
| (ha : densitySlope p q R y.2≠0) (hz : densityResidual p q R A hA hc y=0) : |
| AffineLineCurve (lineHomotopy J y.1) p q (densityCurve p q R y.2) := by |
| apply densityCurve_affineLine p q R y.2 ha |
| · rw [contDiff_iff_contDiffAt] |
| intro x |
| exact (lineHomotopy_contDiffAt hJs hJ (y := (y.1,x)) hy).comp x |
| (contDiffAt_const.prodMk contDiffAt_id) |
| · exact lineHomotopy_compatible hJ hy |
| · exact lineHomotopy_compact hJc hy |
| · exact (densityResidual_zero_iff p q R A hA hc J hJ he y hy).mp hz |
|
|
| end HigherDimensionalBallPacking.Rigidity |
|
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| |
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| |
|
|
| open scoped ContDiff Topology BoundedContinuousFunction |
| open Set Filter Function |
| namespace HigherDimensionalBallPacking.Rigidity |
| open HigherDimensionalBallPacking.Rigidity |
| local instance densityExteriorGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (R : ℝ) : NormedAddCommGroup (CompactHolderSpace E R) := inferInstance |
| local instance densityExteriorSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (R : ℝ) : NormedSpace ℝ (CompactHolderSpace E R) := inferInstance |
|
|
| theorem densityCurve_lower_bound {n : ℕ} (p q : Phase n) (R : ℝ) |
| (g : CompactHolderSpace (Phase n) R) (z : ℂ) : |
| ‖z‖*‖densitySlope p q R g‖ ≤ ‖densityCurve p q R g z‖+densityAffineSize p R g := by |
| have he : z • densitySlope p q R g=densityCurve p q R g z-(p-compactCRInverseValue R g 0)-compactCRInverseValue R g z := by |
| rw [densityCurve_eq_profile] |
| change z • densitySlope p q R g=((p-compactCRInverseValue R g 0)+z • densitySlope p q R g+compactCRInverseValue R g z)-(p-compactCRInverseValue R g 0)-compactCRInverseValue R g z |
| abel |
| have hn : ‖z • densitySlope p q R g‖≤‖densityCurve p q R g z‖+‖p-compactCRInverseValue R g 0‖+‖compactCRInverseValue R g z‖ := by |
| rw [he] |
| have h1 := norm_sub_le (densityCurve p q R g z-(p-compactCRInverseValue R g 0)) (compactCRInverseValue R g z) |
| have h2 := norm_sub_le (densityCurve p q R g z) (p-compactCRInverseValue R g 0) |
| linarith |
| rw [norm_smul] at hn |
| have hb := (compactCRInverseValue R g).norm_coe_le_norm z |
| unfold densityAffineSize |
| linarith |
|
|
| def goodDensities {n : ℕ} (p q : Phase n) (R m M : ℝ) : Set (CompactHolderSpace (Phase n) R) := |
| {g | m < ‖densitySlope p q R g‖ ∧ densityAffineSize p R g < M} |
|
|
| theorem goodDensities_isOpen {n : ℕ} (p q : Phase n) (R m M : ℝ) : IsOpen (goodDensities p q R m M) := |
| (isOpen_lt continuous_const (densitySlope_continuous p q R).norm).inter |
| (isOpen_lt (densityAffineSize_continuous p R) continuous_const) |
|
|
| theorem goodDensities_slope_ne {n : ℕ} (p q : Phase n) (R m M : ℝ) (hm : 0 < m) |
| (g : CompactHolderSpace (Phase n) R) (hg : g∈goodDensities p q R m M) : densitySlope p q R g≠0 := |
| norm_pos_iff.mp (hm.trans hg.1) |
|
|
| theorem goodDensities_exterior {n : ℕ} (p q : Phase n) (R m M C S : ℝ) (hm : 0 < m) |
| (hS : 0 < S) (hSC : M+C≤S*m) |
| (A : ℝ × Phase n → End n) (hAc : ∀ y∈tsupport A, ‖y‖≤C) |
| (t : ℝ) (g : CompactHolderSpace (Phase n) R) (hg : g∈goodDensities p q R m M) |
| (z : ℂ) (hz : S<‖z‖) : A (t,densityCurve p q R g z)=0 := by |
| apply image_eq_zero_of_notMem_tsupport |
| intro hy |
| have hb : ‖densityCurve p q R g z‖≤C := (_root_.norm_snd_le _).trans (hAc _ hy) |
| have hl := densityCurve_lower_bound p q R g z |
| have hmul : S*m<‖z‖*‖densitySlope p q R g‖ := |
| mul_lt_mul hz hg.1.le hm (le_of_lt (hS.trans hz)) |
| linarith [hg.2] |
|
|
| end HigherDimensionalBallPacking.Rigidity |
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| |
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| |
|
|
| open scoped ContDiff Topology BoundedContinuousFunction |
| open Set Filter Function |
| namespace HigherDimensionalBallPacking.Rigidity |
| open HigherDimensionalBallPacking.Rigidity |
| local instance compactMapHGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (α : ℝ) : NormedAddCommGroup (HolderSpace ℂ E α) := inferInstance |
| local instance compactMapHSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (α : ℝ) : NormedSpace ℝ (HolderSpace ℂ E α) := inferInstance |
| local instance compactMapCGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (α : ℝ) : NormedAddCommGroup (C1HolderSpace E α) := inferInstance |
| local instance compactMapCSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] (α : ℝ) : NormedSpace ℝ (C1HolderSpace E α) := inferInstance |
| local instance compactMapCompactGroup (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (R : ℝ) : NormedAddCommGroup (CompactHolderSpace E R) := inferInstance |
| local instance compactMapCompactSpace (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] (R : ℝ) : NormedSpace ℝ (CompactHolderSpace E R) := inferInstance |
|
|
| def cutoffValues {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| (b : ContDiffBump (0:ℂ)) : C1HolderSpace E ((1:ℝ)/3) →L[ℝ] HolderSpace ℂ E ((1:ℝ)/3) := |
| c1LowerOrderCLM (holderLinearPost ((1:ℝ)/3) (ContinuousLinearMap.lsmul ℝ ℝ) |
| (holderSourceCutoff ((1:ℝ)/3) (by norm_num) (by norm_num) b)) |
|
|
| theorem cutoffValues_value {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| (b : ContDiffBump (0:ℂ)) (v : C1HolderSpace E ((1:ℝ)/3)) (z : ℂ) : |
| holderValue ((1:ℝ)/3) (cutoffValues b v) z=b z • holderValue ((1:ℝ)/3) (c1HolderValue ((1:ℝ)/3) v) z := rfl |
|
|
| theorem cutoffValues_compact {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| [FiniteDimensional ℝ E] (b : ContDiffBump (0:ℂ)) : IsCompactOperator (cutoffValues (E := E) b) := by |
| apply c1LowerOrderCLM_compact_holder b.rOut |
| intro z hz |
| change b z • (ContinuousLinearMap.id ℝ E)=0 |
| rw [b.zero_of_le_dist (by simpa using hz.le),zero_smul] |
|
|
| abbrev ResponseCoords (n : ℕ) := (ℝ × Phase n × Phase n) × HolderSpace ℂ (Phase n) ((1:ℝ)/3) |
|
|
| def responseCoords {n : ℕ} (B : ContDiffBump (0:ℂ)) (p q : Phase n) (R : ℝ) |
| (y : ℝ × CompactHolderSpace (Phase n) R) : ResponseCoords n := |
| ((y.1,p-compactCRInverseValue R y.2 0,densitySlope p q R y.2),cutoffValues B (compactCRInverse R y.2)) |
|
|
| theorem responseCoords_contDiff {n : ℕ} (B : ContDiffBump (0:ℂ)) (p q : Phase n) (R : ℝ) : |
| ContDiff ℝ ∞ (responseCoords B p q R) := by |
| have h0 : ContDiff ℝ ∞ (fun y : ℝ × CompactHolderSpace (Phase n) R => compactCRInverseValue R y.2 0) := |
| (((c1HolderEval ((1:ℝ)/3) 0).comp (compactCRInverse (E := Phase n) R)).contDiff).comp contDiff_snd |
| have ha : ContDiff ℝ ∞ (fun y : ℝ × CompactHolderSpace (Phase n) R => densitySlope p q R y.2) := |
| (densitySlope_contDiff p q R).comp contDiff_snd |
| exact (contDiff_fst.prodMk ((contDiff_const.sub h0).prodMk ha)).prodMk |
| (((cutoffValues B).comp (compactCRInverse (E := Phase n) R)).contDiff.comp contDiff_snd) |
|
|
| def responseProfile {n : ℕ} (B : ContDiffBump (0:ℂ)) (x : ResponseCoords n) : |
| HolderSpace ℂ (ℝ × Phase n) ((1:ℝ)/3) := |
| holderConstantCLM ((1:ℝ)/3) (x.1.1,0)+holderLinearPost ((1:ℝ)/3) (ContinuousLinearMap.inr ℝ ℝ (Phase n)) |
| (holderCutoffAffine ((1:ℝ)/3) (by norm_num) (by norm_num) B (x.1.2.1,affineSlope x.1.2.2)+x.2) |
|
|
| theorem responseProfile_contDiff {n : ℕ} (B : ContDiffBump (0:ℂ)) : |
| ContDiff ℝ ∞ (responseProfile (n := n) B) := by |
| have hc := (holderConstantCLM (E := ℝ × Phase n) ((1:ℝ)/3)).contDiff.comp |
| ((contDiff_fst.fst : ContDiff ℝ ∞ (fun x : ResponseCoords n => x.1.1)).prodMk (contDiff_const (c := (0:Phase n)))) |
| have ha : ContDiff ℝ ∞ (fun x : ResponseCoords n => affineSlope x.1.2.2) := |
| ((ContinuousLinearMap.lsmul ℝ ℂ : ℂ →L[ℝ] Phase n →L[ℝ] Phase n).flip).contDiff.comp contDiff_fst.snd.snd |
| have ht := (holderCutoffAffine_contDiff ((1:ℝ)/3) (by norm_num) (by norm_num) B).comp |
| ((contDiff_fst.snd.fst : ContDiff ℝ ∞ (fun x : ResponseCoords n => x.1.2.1)).prodMk ha) |
| let L : HolderSpace ℂ (Phase n) ((1:ℝ)/3) →L[ℝ] HolderSpace ℂ (ℝ × Phase n) ((1:ℝ)/3) := |
| holderLinearPost ((1:ℝ)/3) (ContinuousLinearMap.inr ℝ ℝ (Phase n)) |
| exact hc.add (L.contDiff.comp (ht.add contDiff_snd)) |
|
|
| def responseCoefficient {n : ℕ} (b B : ContDiffBump (0:ℂ)) |
| (A : ℝ × Phase n → End n) (hA : ContDiff ℝ ∞ A) (hc : HasCompactSupport A) |
| (x : ResponseCoords n) : HolderSpace ℂ (End n) ((1:ℝ)/3) := |
| (cutoffCompact b (holderPost ((1:ℝ)/3) (by norm_num) A hA hc (responseProfile B x))).1 |
|
|
| theorem responseCoefficient_contDiff {n : ℕ} (b B : ContDiffBump (0:ℂ)) |
| (A : ℝ × Phase n → End n) (hA : ContDiff ℝ ∞ A) (hc : HasCompactSupport A) : |
| ContDiff ℝ ∞ (responseCoefficient b B A hA hc) := by |
| exact ((compactHolderSubmodule (E := End n) b.rOut).subtypeL.comp (cutoffCompact b)).contDiff.comp |
| ((holderPost_contDiff ((1:ℝ)/3) (by norm_num) A hA hc).comp (responseProfile_contDiff B)) |
|
|
| theorem responseCoefficient_value {n : ℕ} (b B : ContDiffBump (0:ℂ)) (hbB : b.rOut≤B.rIn) |
| (p q : Phase n) (A : ℝ × Phase n → End n) (hA : ContDiff ℝ ∞ A) (hc : HasCompactSupport A) |
| (y : ℝ × CompactHolderSpace (Phase n) b.rOut) (z : ℂ) : |
| holderValue ((1:ℝ)/3) (responseCoefficient b B A hA hc (responseCoords B p q b.rOut y)) z= |
| b z • A (y.1,densityCurve p q b.rOut y.2 z) := by |
| change b z • holderValue ((1:ℝ)/3) (holderPost ((1:ℝ)/3) (by norm_num) A hA hc _) z=_ |
| rw [holderPost_apply] |
| by_cases hz : b z=0 |
| · rw [hz] |
| ext v |
| simp |
| have hzz : ‖z‖≤b.rOut := by |
| by_contra hn |
| exact hz (b.zero_of_le_dist (by simpa using (le_of_not_ge hn))) |
| have hB : B z=1 := B.one_of_mem_closedBall (by simpa using hzz.trans hbB) |
| congr 2 |
| rw [responseProfile,map_add,BoundedContinuousFunction.add_apply,holderConstantCLM_value, |
| holderLinearPost_apply, map_add, BoundedContinuousFunction.add_apply, holderCutoffAffine_apply] |
| change (y.1,0)+(ContinuousLinearMap.inr ℝ ℝ (Phase n)) |
| (B z • ((p-compactCRInverseValue b.rOut y.2 0)+affineSlope (densitySlope p q b.rOut y.2) z)+ |
| B z • compactCRInverseValue b.rOut y.2 z)=_ |
| rw [hB,one_smul,one_smul] |
| apply Prod.ext |
| · simp |
| · simp only [Prod.snd_add, ContinuousLinearMap.inr_apply, zero_add] |
| rw [densityCurve_eq_profile] |
| rfl |
|
|
| end HigherDimensionalBallPacking.Rigidity |
|
|
| end |
|
|
| end OAI |
|
|
