| import Mathlib | |
| import OAI.Geometry.BallPacking.Annuli.AnnularHandleCurvature | |
| namespace OAI | |
| noncomputable section | |
| namespace PackingSufficiencySupport.Hamiltonian.AnnularHandleData | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] | |
| @[simp] theorem handleAnnularCover_eq_densityChart | |
| (D : AnnularHandleData Plane M) | |
| : D.handleAnnularCover = D.densityChart := rfl | |
| theorem handleAnnularCover_rectangle_domain | |
| (D : AnnularHandleData Plane M) | |
| {U : Set Plane} (hU : U ⊆ Ioo (-D.width) D.width ×ˢ Ioo (0:ℝ) 1) : | |
| U ⊆ D.annularCoverDomain := by | |
| intro q hq | |
| change cylinderCover q ∈ D.chart.source | |
| rw [D.source_eq] | |
| exact ⟨(hU hq).1,mem_univ _⟩ | |
| theorem annularPackingChart_smoothOn | |
| (D : AnnularHandleData Plane M) | |
| {U : Set (Plane × V)} (hU : ∀ p ∈ U, p.1 ∈ D.annularCoverDomain) : | |
| ContMDiffOn 𝓘(ℝ,Plane × V) 𝓘(ℝ,Plane × V) ∞ | |
| (flatProductMap (D.handleAnnularCover)) U := by | |
| intro p hp | |
| exact (flatProductMap_smoothAt (D.handleAnnularCover_smoothAt (hU p hp))).contMDiffWithinAt | |
| omit [NormedSpace ℝ V] in | |
| theorem annularPackingChart_isEmbedding | |
| (D : AnnularHandleData Plane M) | |
| {U : Set (Plane × V)} (hU : ∀ p ∈ U, p.1 ∈ Ioo (-D.width) D.width ×ˢ Ioo (0:ℝ) 1) : | |
| Topology.IsEmbedding (fun p : U => flatProductMap (D.handleAnnularCover) p.val) := by | |
| let c := D.densityChart.toOpenPartialHomeomorph.prod (OpenPartialHomeomorph.refl V) | |
| have hsub : U ⊆ c.source := by | |
| intro p hp | |
| change p.1 ∈ D.densityChart.source ∧ p.2 ∈ (univ : Set V) | |
| rw [D.densityChart_source] | |
| exact ⟨hU p hp,mem_univ _⟩ | |
| exact c.isEmbedding_restrict.comp (Topology.IsEmbedding.inclusion hsub) | |
| end PackingSufficiencySupport.Hamiltonian.AnnularHandleData | |
| namespace PackingSufficiencySupport.Hamiltonian | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| open MeasureTheory | |
| section | |
| variable {E M : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [TopologicalSpace M] [ChartedSpace E M] | |
| def handleDensityCore (b : ℝ) (f : Circle → ℝ) : Set HandleTorus := | |
| (circleTurn '' Icc (-b) b) ×ˢ tsupport f | |
| theorem handleDensityCore_compact (b : ℝ) (f : Circle → ℝ) : | |
| IsCompact (handleDensityCore b f) := | |
| circleBand_compact.prod isClosed_closure.isCompact | |
| theorem handleDensityCore_subset_band (b : ℝ) (f : Circle → ℝ) : | |
| handleDensityCore b f⊆compactHandleBand b := by | |
| rintro z ⟨⟨s,hs,he⟩,_⟩ | |
| exact ⟨(s,z.2),⟨hs,mem_univ _⟩,Prod.ext he rfl⟩ | |
| theorem globalHandleDensity_inside | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (b : ℝ) (f : Circle → ℝ) {x : M} (hx : x∈e.target) (v w : E) : | |
| globalHandleDensity e b f x v w= | |
| torusAnnularDensity b f (e.symm x) | |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,TorusModel) e.symm x v) | |
| (mfderiv 𝓘(ℝ,E) 𝓘(ℝ,TorusModel) e.symm x w) := by | |
| simp only [globalHandleDensity,globalHandleRadialForm,globalHandleForm, | |
| handlePushforwardOneForm,extendManifoldOneForm_inside hx,manifoldPullbackOneForm, | |
| manifoldMapDifferential,manifoldWedge_apply, | |
| torusAnnularDensity] | |
| rfl | |
| theorem globalHandleDensity_core_zero | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| {b : ℝ} (hb : 0<b) (f : Circle → ℝ) {x : M} | |
| (hx : x∉e '' handleDensityCore b f) : globalHandleDensity e b f x=0 := by | |
| by_cases ht : x∈e.target | |
| · have hn : e.symm x∉handleDensityCore b f := by | |
| intro hn | |
| exact hx ⟨e.symm x,hn,e.right_inv ht⟩ | |
| have hz : torusAnnularDensity b f (e.symm x)=0 := | |
| image_eq_zero_of_notMem_tsupport (fun hs => hn (torusAnnularDensity_tsupport hb f hs)) | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| apply ContinuousLinearMap.ext | |
| intro w | |
| change globalHandleDensity e b f x v w=0 | |
| rw [globalHandleDensity_inside e b f ht,hz] | |
| rfl | |
| · have hz : globalHandleRadialForm e b x=0 := extendManifoldOneForm_outside ht | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| apply ContinuousLinearMap.ext | |
| intro w | |
| change globalHandleDensity e b f x v w=0 | |
| simp only [globalHandleDensity,manifoldWedge_apply,hz,zero_apply,zero_mul,mul_zero,sub_self] | |
| theorem globalHandleDensity_core_support [T2Space M] | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| {b : ℝ} (hb : 0<b) (he : compactHandleBand b⊆e.source) (f : Circle → ℝ) : | |
| tsupport (globalHandleDensity e b f)⊆e '' handleDensityCore b f := by | |
| apply closure_minimal _ ((handleDensityCore_compact b f).image_of_continuousOn | |
| (e.contMDiffOn.continuousOn.mono ((handleDensityCore_subset_band b f).trans he))).isClosed | |
| intro x hx | |
| by_contra hn | |
| exact hx (globalHandleDensity_core_zero e hb f hn) | |
| end | |
| section | |
| variable {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] | |
| def handleDensityChart | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a : ℝ) : PartialDiffeomorph 𝓘(ℝ,Plane) 𝓘(ℝ,Plane) Plane M ∞ := | |
| restrictHandleDiffeomorph (torusRectangle.trans e) (Ioo (-a) a ×ˢ Ioo (0:ℝ) 1) | |
| (isOpen_Ioo.prod isOpen_Ioo) | |
| @[simp] theorem handleDensityChart_apply | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a : ℝ) (y : Plane) : handleDensityChart e a y=e (torusTurns y) := rfl | |
| theorem handleDensityChart_source | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| {a : ℝ} (ha : a<1/2) (he : compactHandleBand a⊆e.source) : | |
| (handleDensityChart e a).source=Ioo (-a) a ×ˢ Ioo (0:ℝ) 1 := by | |
| apply inter_eq_right.mpr | |
| intro y hy | |
| refine ⟨?_,he ?_⟩ | |
| · exact ⟨⟨by linarith [hy.1.1],by linarith [hy.1.2]⟩,hy.2⟩ | |
| · exact ⟨(y.1,circleTurn y.2),⟨⟨hy.1.1.le,hy.1.2.le⟩,mem_univ _⟩,rfl⟩ | |
| theorem handleDensityChart_contains | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| {a b : ℝ} (ha : a<1/2) (hab : b<a) (he : compactHandleBand a⊆e.source) | |
| {f : Circle → ℝ} (hf : tsupport f⊆({1}ᶜ : Set Circle)) : | |
| e '' handleDensityCore b f⊆(handleDensityChart e a).target := by | |
| rintro x ⟨z,⟨⟨s,hs,heq⟩,hz⟩,rfl⟩ | |
| have ht : z.2∈positiveCircleCoordinate.target := by | |
| rw [positiveCircleCoordinate_target] | |
| exact hf hz | |
| obtain ⟨t,ht,heq'⟩ := ht | |
| have hy : (s,t)∈(handleDensityChart e a).source := by | |
| rw [handleDensityChart_source e ha he] | |
| exact ⟨⟨by linarith [hs.1],by linarith [hs.2]⟩,ht⟩ | |
| have hi := (handleDensityChart e a).map_source hy | |
| have hev : handleDensityChart e a (s,t)=e z := congrArg e (Prod.ext heq heq') | |
| exact hev ▸ hi | |
| theorem handleDensityChart_coefficient | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) {y : Plane} (hy : y∈(handleDensityChart e a).source) : | |
| partialChartCoefficient (handleDensityChart e a) (globalHandleDensity e b f) y= | |
| circleClockDensity b (circleTurn y.1)*f (circleTurn y.2) := by | |
| have hh := globalHandleDensity_turns e b f hy.1.2 (1,0) (0,1) | |
| unfold partialChartCoefficient euclideanPullbackTwoForm | |
| have heq : (handleDensityChart e a : Plane → M)=e ∘ torusTurns := rfl | |
| rw [heq] | |
| simpa only [mul_one, mul_zero, sub_zero] using! hh | |
| end | |
| section | |
| variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| {Ω : V →L[ℝ] V →L[ℝ] ℝ} | |
| def handleFirstCorrection | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) | |
| (p : ℝ × V) : ManifoldOneForm E M := fun x => | |
| surfaceSuspensionCoefficient Φ (intervalClock (-b) b) | |
| (p,((handleAnnularChart e a).symm x).1) • globalHandleRadialForm e b x + | |
| surfaceFirstCorrectionCoefficient Φ (intervalClock (-b) b) H | |
| (p,((handleAnnularChart e a).symm x).1) • globalHandleForm e f x | |
| def globalSurfaceFirstPrimitive | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (p : ℝ × V) : ManifoldOneForm E M := | |
| Γ p.2 + extendManifoldOneForm (handleAnnularChart e a).target | |
| (handleFirstCorrection e a b f Φ H p) | |
| omit [NormedAddCommGroup V] [NormedSpace ℝ V] [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem handleAnnularChart_off_band | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) {x : M} (hx : x∈(handleAnnularChart e a).target) | |
| (hn : x∉e '' compactHandleBand b) : | |
| ((handleAnnularChart e a).symm x).1∉Icc (-b) b := by | |
| intro hs | |
| apply hn | |
| exact ⟨torusAnnulusMap ((handleAnnularChart e a).symm x), | |
| ⟨(handleAnnularChart e a).symm x,⟨hs,mem_univ _⟩,rfl⟩, | |
| (handleAnnularChart e a).right_inv hx⟩ | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem handleFirstCorrection_off_band | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) {b : ℝ} (hb : 0<b) (f : Circle → ℝ) | |
| (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) (p : ℝ × V) {x : M} | |
| (hx : x∈(handleAnnularChart e a).target) (hn : x∉e '' compactHandleBand b) : | |
| handleFirstCorrection e a b f Φ H p x=0 := by | |
| have hr : globalHandleRadialForm e b x=0 := | |
| handlePushforwardOneForm_zero e (fun _ hy => torusFirstForm_clock_zero hb hy) hn | |
| have ht : intervalClock (-b) b ((handleAnnularChart e a).symm x).1=0 ∨ | |
| intervalClock (-b) b ((handleAnnularChart e a).symm x).1=1 := by | |
| have hs := handleAnnularChart_off_band e a b hx hn | |
| simp only [mem_Icc,not_and_or,not_le] at hs | |
| rcases hs with hs|hs | |
| · exact Or.inl (intervalClock_zero (by linarith) hs.le) | |
| · exact Or.inr (intervalClock_one (by linarith) hs.le) | |
| simp only [handleFirstCorrection,hr,smul_zero, | |
| surfaceFirstCorrectionCoefficient_tails Φ _ H p _ ht,zero_smul,add_zero] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem globalSurfaceFirstPrimitive_off_band | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) {b : ℝ} (hb : 0<b) (f : Circle → ℝ) | |
| (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) (Γ : V → ManifoldOneForm E M) | |
| (p : ℝ × V) {x : M} (hn : x∉e '' compactHandleBand b) : | |
| globalSurfaceFirstPrimitive e a b f Φ H Γ p x=Γ p.2 x := by | |
| by_cases hx : x∈(handleAnnularChart e a).target | |
| · simp [globalSurfaceFirstPrimitive,extendManifoldOneForm_inside hx, | |
| handleFirstCorrection_off_band e a hb f Φ H p hx hn] | |
| · simp [globalSurfaceFirstPrimitive,extendManifoldOneForm_outside hx] | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| @[simp] theorem globalSurfaceFirstPrimitive_zero | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (v : V) : | |
| globalSurfaceFirstPrimitive e a b f Φ H Γ (0,v)=Γ v := by | |
| funext x | |
| by_cases hx : x∈(handleAnnularChart e a).target | |
| · simp [globalSurfaceFirstPrimitive,extendManifoldOneForm_inside hx,handleFirstCorrection] | |
| · simp [globalSurfaceFirstPrimitive,extendManifoldOneForm_outside hx] | |
| theorem globalSurfaceFirstPrimitive_smooth [T2Space M] | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| {a b : ℝ} (ha : a<1/2) (hb : 0<b) (hab : b<a) (he : compactHandleBand a⊆e.source) | |
| {f : Circle → ℝ} (hf : ContMDiff 𝓘(ℝ,CircleModel) 𝓘(ℝ,ℝ) ∞ f) | |
| (hfs : tsupport (torusHandleForm f)⊆e.source) | |
| (Φ : CompactHamiltonianIsotopy Ω) {H : V → ℝ} (hH : ContDiff ℝ ∞ H) | |
| {Γ : V → ManifoldOneForm E M} (hΓ : SmoothOneFormFamily Γ) : | |
| SmoothOneFormFamily (globalSurfaceFirstPrimitive e a b f Φ H Γ) := by | |
| have hbe : compactHandleBand b⊆e.source := by | |
| rintro z ⟨w,hw,rfl⟩ | |
| exact he ⟨w,⟨⟨by linarith [hw.1.1],by linarith [hw.1.2]⟩,mem_univ _⟩,rfl⟩ | |
| have hr := handlePushforwardOneForm_smooth e (compactHandleBand_isCompact b) hbe | |
| (torusFirstForm_smooth (circleClockDensity_smooth hb (lt_trans hab ha))) | |
| (fun _ hx => torusFirstForm_clock_zero hb hx) | |
| have hβ := (globalHandleForm_properties e hf hfs).1 | |
| have hα := extendManifoldOneForm_smooth (handleAnnularChart e a).open_target | |
| ((handleAnnularChart_compact e hab he).isClosed) | |
| (handleAnnularChart_compact_subset e ha hab he) | |
| (α := handleFirstCorrection e a b f Φ H) | |
| (fun c => by | |
| have h1 := annular_coefficient_smooth_on (handleAnnularChart e a).open_target | |
| (annular_inverse_fst_smooth (handleAnnularChart e a)) | |
| (surfaceSuspensionCoefficient_smooth Φ (intervalClock_smooth (-b) b)) | |
| (α := globalHandleRadialForm e b) (fun c => (hr c).comp | |
| (f := fun y => ((0:ℝ),y)) (contDiffOn_const.prodMk contDiffOn_id) | |
| (fun _ hy => ⟨mem_univ _,hy⟩)) c | |
| have h2 := annular_coefficient_smooth_on (handleAnnularChart e a).open_target | |
| (annular_inverse_fst_smooth (handleAnnularChart e a)) | |
| (surfaceFirstCorrectionCoefficient_smooth Φ (intervalClock_smooth (-b) b) hH) | |
| (α := globalHandleForm e f) (fun c => (hβ c).comp | |
| (f := fun y => ((0:ℝ),y)) (contDiffOn_const.prodMk contDiffOn_id) | |
| (fun _ hy => ⟨mem_univ _,hy⟩)) c | |
| simpa only [handleFirstCorrection,chartOneForm,ContinuousLinearMap.add_comp] using h1.add h2) | |
| (fun p x hx hn => handleFirstCorrection_off_band e a hb f Φ H p hx hn) | |
| intro c | |
| have hmap : ContDiffOn ℝ ∞ (fun q : (ℝ × V) × E => (q.1.2,q.2)) | |
| (univ ×ˢ (extChartAt 𝓘(ℝ,E) c).target) := | |
| ((contDiff_snd.comp contDiff_fst).prodMk contDiff_snd).contDiffOn | |
| have hbg := (hΓ c).comp hmap (fun _ hq => ⟨mem_univ _,hq.2⟩) | |
| simpa only [globalSurfaceFirstPrimitive,chartOneForm_add,Function.comp_apply] using hbg.add (hα c) | |
| end | |
| section | |
| variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] | |
| {Ω : V →L[ℝ] V →L[ℝ] ℝ} {ι : Type*} [Fintype ι] | |
| def handleSecondExtra | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (p : ℝ × V) : ManifoldOneForm E M := fun x => | |
| surfaceSecondExtraCoefficient Φ (intervalClock (-b) b) K H h ρ ρ₀ | |
| (p,((handleAnnularChart e a).symm x).1) • globalHandleForm e f x | |
| def globalSurfaceSecondPrimitive | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (p : ℝ × V) : ManifoldOneForm E M := | |
| globalSurfaceFirstPrimitive e a b f Φ H Γ (1,p.2) + | |
| (p.1*(K p.2-H p.2)) • globalHandleForm e f + | |
| extendManifoldOneForm (handleAnnularChart e a).target | |
| (handleSecondExtra e a b f Φ K H h ρ ρ₀ p) | |
| @[simp] theorem globalSurfaceSecondPrimitive_zero | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (v : V) : | |
| globalSurfaceSecondPrimitive e a b f Φ K H h ρ ρ₀ Γ (0,v)= | |
| globalSurfaceFirstPrimitive e a b f Φ H Γ (1,v) := by | |
| funext x | |
| by_cases hx : x∈(handleAnnularChart e a).target | |
| · simp [globalSurfaceSecondPrimitive,extendManifoldOneForm_inside hx,handleSecondExtra] | |
| · simp [globalSurfaceSecondPrimitive,extendManifoldOneForm_outside hx] | |
| theorem handleSecondExtra_off_band | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) {b : ℝ} (hb : 0<b) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (hlo : ∀ s, s ≤ -b → (∀ i,ρ i s=0) ∧ ρ₀ s=0) | |
| (hhi : ∀ s, b ≤ s → (∀ i,ρ i s=1) ∧ ρ₀ s=1) | |
| (p : ℝ × V) {x : M} (hx : x∈(handleAnnularChart e a).target) | |
| (hn : x∉e '' compactHandleBand b) : handleSecondExtra e a b f Φ K H h ρ ρ₀ p x=0 := by | |
| have hs := handleAnnularChart_off_band e a b hx hn | |
| simp only [mem_Icc,not_and_or,not_le] at hs | |
| have ht : (intervalClock (-b) b ((handleAnnularChart e a).symm x).1=0 ∧ | |
| (∀ i,ρ i ((handleAnnularChart e a).symm x).1=0) ∧ | |
| ρ₀ ((handleAnnularChart e a).symm x).1=0) ∨ | |
| (intervalClock (-b) b ((handleAnnularChart e a).symm x).1=1 ∧ | |
| (∀ i,ρ i ((handleAnnularChart e a).symm x).1=1) ∧ | |
| ρ₀ ((handleAnnularChart e a).symm x).1=1) := by | |
| rcases hs with hs|hs | |
| · exact Or.inl ⟨intervalClock_zero (by linarith) hs.le,hlo _ hs.le⟩ | |
| · exact Or.inr ⟨intervalClock_one (by linarith) hs.le,hhi _ hs.le⟩ | |
| simp only [handleSecondExtra,surfaceSecondExtraCoefficient_tails Φ _ K H h ρ ρ₀ p _ ht,zero_smul] | |
| theorem globalSurfaceSecondPrimitive_off_band | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) {b : ℝ} (hb : 0<b) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (hlo : ∀ s, s ≤ -b → (∀ i,ρ i s=0) ∧ ρ₀ s=0) | |
| (hhi : ∀ s, b ≤ s → (∀ i,ρ i s=1) ∧ ρ₀ s=1) | |
| (Γ : V → ManifoldOneForm E M) (p : ℝ × V) {x : M} | |
| (hn : x∉e '' compactHandleBand b) : | |
| globalSurfaceSecondPrimitive e a b f Φ K H h ρ ρ₀ Γ p x= | |
| Γ p.2 x+(p.1*(K p.2-H p.2)) • globalHandleForm e f x := by | |
| by_cases hx : x∈(handleAnnularChart e a).target | |
| · simp [globalSurfaceSecondPrimitive,extendManifoldOneForm_inside hx, | |
| handleSecondExtra_off_band e a hb f Φ K H h ρ ρ₀ hlo hhi p hx hn, | |
| globalSurfaceFirstPrimitive_off_band e a hb f Φ H Γ (1,p.2) hn] | |
| · simp [globalSurfaceSecondPrimitive,extendManifoldOneForm_outside hx, | |
| globalSurfaceFirstPrimitive_off_band e a hb f Φ H Γ (1,p.2) hn] | |
| variable [IsManifold 𝓘(ℝ,E) ∞ M] | |
| theorem globalSurfaceSecondPrimitive_smooth [T2Space M] | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| {a b : ℝ} (ha : a<1/2) (hb : 0<b) (hab : b<a) (he : compactHandleBand a⊆e.source) | |
| {f : Circle → ℝ} (hf : ContMDiff 𝓘(ℝ,CircleModel) 𝓘(ℝ,ℝ) ∞ f) | |
| (hfs : tsupport (torusHandleForm f)⊆e.source) | |
| (Φ : CompactHamiltonianIsotopy Ω) {K H : V → ℝ} (hK : ContDiff ℝ ∞ K) | |
| (hH : ContDiff ℝ ∞ H) {h : ι → V → ℝ} (hh : ∀ i,ContDiff ℝ ∞ (h i)) | |
| {ρ : ι → ℝ → ℝ} (hρ : ∀ i,ContDiff ℝ ∞ (ρ i)) | |
| {ρ₀ : ℝ → ℝ} (hρ₀ : ContDiff ℝ ∞ ρ₀) | |
| (hlo : ∀ s, s ≤ -b → (∀ i,ρ i s=0) ∧ ρ₀ s=0) | |
| (hhi : ∀ s, b ≤ s → (∀ i,ρ i s=1) ∧ ρ₀ s=1) | |
| {Γ : V → ManifoldOneForm E M} (hΓ : SmoothOneFormFamily Γ) : | |
| SmoothOneFormFamily (globalSurfaceSecondPrimitive e a b f Φ K H h ρ ρ₀ Γ) := by | |
| have hβ := (globalHandleForm_properties e hf hfs).1 | |
| have hα := extendManifoldOneForm_smooth (handleAnnularChart e a).open_target | |
| (handleAnnularChart_compact e hab he).isClosed | |
| (handleAnnularChart_compact_subset e ha hab he) | |
| (α := handleSecondExtra e a b f Φ K H h ρ ρ₀) | |
| (fun c => annular_coefficient_smooth_on (handleAnnularChart e a).open_target | |
| (annular_inverse_fst_smooth (handleAnnularChart e a)) | |
| (surfaceSecondExtraCoefficient_smooth Φ (intervalClock_smooth (-b) b) hK hH hh hρ hρ₀) | |
| (α := globalHandleForm e f) (fun c => (hβ c).comp | |
| (f := fun y => ((0:ℝ),y)) (contDiffOn_const.prodMk contDiffOn_id) | |
| (fun _ hy => ⟨mem_univ _,hy⟩)) c) | |
| (fun p x hx hn => handleSecondExtra_off_band e a hb f Φ K H h ρ ρ₀ hlo hhi p hx hn) | |
| have hfirst := (globalSurfaceFirstPrimitive_smooth e ha hb hab he hf hfs Φ hH hΓ).comp | |
| (f := fun p : ℝ × V => (1,p.2)) (contDiff_const.prodMk (contDiff_snd : ContDiff ℝ ∞ (Prod.snd : ℝ × V → V))) | |
| have hj := (hβ.comp (f := fun _ : ℝ × V => (0:ℝ)) contDiff_const).smul | |
| (contDiff_fst.mul ((hK.sub hH).comp contDiff_snd)) | |
| exact (hfirst.add hj).add hα | |
| end | |
| section | |
| variable {V E : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] | |
| {Ω : V →L[ℝ] V →L[ℝ] ℝ} | |
| theorem globalSurfaceFirstPrimitive_normal | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (γ : ManifoldOneForm E M) (p : ℝ × V) | |
| {x : M} (hx : x∈(handleAnnularChart e a).target) | |
| (hN : Γ p.2 x=γ x+ | |
| (intervalClock (-b) b ((handleAnnularChart e a).symm x).1*H p.2) • globalHandleForm e f x) : | |
| globalSurfaceFirstPrimitive e a b f Φ H Γ p x= | |
| γ x+surfaceSuspensionCoefficient Φ (intervalClock (-b) b) | |
| (p,((handleAnnularChart e a).symm x).1) • globalHandleRadialForm e b x+ | |
| (intervalClock (-b) b ((handleAnnularChart e a).symm x).1* | |
| H ((Φ.map p.1).symm (Φ.map (p.1*intervalClock (-b) b | |
| ((handleAnnularChart e a).symm x).1) p.2))) • globalHandleForm e f x := by | |
| simp only [globalSurfaceFirstPrimitive,Pi.add_apply,extendManifoldOneForm_inside hx,hN, | |
| handleFirstCorrection,surfaceFirstCorrectionCoefficient] | |
| ext v | |
| simp only [add_apply,smul_apply,smul_eq_mul] | |
| ring | |
| theorem globalSurfaceSecondPrimitive_normal {ι : Type*} [Fintype ι] | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (γ : ManifoldOneForm E M) (p : ℝ × V) | |
| {x : M} (hx : x∈(handleAnnularChart e a).target) | |
| (hN : Γ p.2 x=γ x+ | |
| (intervalClock (-b) b ((handleAnnularChart e a).symm x).1*H p.2) • globalHandleForm e f x) : | |
| globalSurfaceSecondPrimitive e a b f Φ K H h ρ ρ₀ Γ p x= | |
| γ x+surfaceSuspensionCoefficient Φ (intervalClock (-b) b) | |
| ((1,p.2),((handleAnnularChart e a).symm x).1) • globalHandleRadialForm e b x+ | |
| surfaceInterpolatedLayer (intervalClock (-b) b) K H h (Φ.map 1).symm ρ ρ₀ p.1 | |
| (((handleAnnularChart e a).symm x).1, | |
| Φ.map (intervalClock (-b) b ((handleAnnularChart e a).symm x).1) p.2) • | |
| globalHandleForm e f x := by | |
| simp only [globalSurfaceSecondPrimitive,globalSurfaceFirstPrimitive,Pi.add_apply,Pi.smul_apply, | |
| extendManifoldOneForm_inside hx,hN,handleFirstCorrection,handleSecondExtra, | |
| surfaceSecondExtraCoefficient,surfaceSecondCorrectionCoefficient, | |
| surfaceFirstCorrectionCoefficient,one_mul] | |
| ext v | |
| simp only [add_apply,smul_apply,smul_eq_mul] | |
| ring | |
| def annularCoverA | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) (γ : ManifoldOneForm E M) (q : Plane) : ℝ := | |
| euclideanPullbackOneForm (fun _ => γ) (handleAnnularCover e a) (0,q) (1,0) | |
| def annularCoverB | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) (γ : ManifoldOneForm E M) (q : Plane) : ℝ := | |
| euclideanPullbackOneForm (fun _ => γ) (handleAnnularCover e a) (0,q) (0,1) | |
| theorem annularCover_form_rep | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a : ℝ) (γ : ManifoldOneForm E M) (q v : Plane) : | |
| γ (handleAnnularCover e a q) (mfderiv 𝓘(ℝ,Plane) 𝓘(ℝ,E) (handleAnnularCover e a) q v)= | |
| annularCoverA e a γ q*v.1+annularCoverB e a γ q*v.2 := by | |
| let L := euclideanPullbackOneForm (fun _ => γ) (handleAnnularCover e a) (0,q) | |
| change L v=L (1,0)*v.1+L (0,1)*v.2 | |
| have hv : v=v.1 • (1,0)+v.2 • (0,1) := by ext <;> simp | |
| rw [show L v=L (v.1 • (1,0)+v.2 • (0,1)) from congrArg L hv] | |
| simp only [map_add,map_smul,smul_eq_mul] | |
| ring | |
| theorem globalSurfaceFirstPrimitive_cover | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) (H : V → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (γ : ManifoldOneForm E M) | |
| (ℓ : ℝ) (p : Plane × V) (hp : cylinderCover p.1∈(handleAnnularChart e a).source) | |
| (hN : Γ p.2 (handleAnnularCover e a p.1)=γ (handleAnnularCover e a p.1)+ | |
| (intervalClock (-b) b p.1.1*H p.2) • globalHandleForm e f (handleAnnularCover e a p.1)) : | |
| euclideanPullbackOneForm | |
| (fun _ => productHorizontalLift (fun v => globalSurfaceFirstPrimitive e a b f Φ H Γ (ℓ,v))) | |
| (flatProductMap (handleAnnularCover e a)) (0,p)= | |
| surfaceFirstPrimitive Φ (intervalClock (-b) b) H | |
| (annularCoverA e a γ) (annularCoverB e a γ) (fun q => f (circleTurn q.2)) (ℓ,p) := by | |
| have hn := globalSurfaceFirstPrimitive_normal e a b f Φ H Γ γ (ℓ,p.2) (x := handleAnnularCover e a p.1) | |
| ((handleAnnularChart e a).map_source hp) | |
| have hi := handleAnnularCover_inverse e a hp | |
| have hN' : Γ p.2 (handleAnnularCover e a p.1)=γ (handleAnnularCover e a p.1)+ | |
| (intervalClock (-b) b ((handleAnnularChart e a).symm (handleAnnularCover e a p.1)).1*H p.2) • | |
| globalHandleForm e f (handleAnnularCover e a p.1) := by | |
| simpa only [hi,cylinderCover] using hN | |
| have he := hn hN' | |
| let A : Plane →L[ℝ] E := mfderiv 𝓘(ℝ,Plane) 𝓘(ℝ,E) (handleAnnularCover e a) p.1 | |
| have hd (u : Plane) : globalHandleForm e f (handleAnnularCover e a p.1) (A u) = | |
| f (circleTurn p.1.2) * u.2 := globalHandleForm_cover e a f hp u | |
| have hr (u : Plane) : globalHandleRadialForm e b (handleAnnularCover e a p.1) (A u) = | |
| deriv (intervalClock (-b) b) p.1.1 * u.1 := globalHandleRadialForm_cover e a b hp u | |
| have hg (u : Plane) : γ (handleAnnularCover e a p.1) (A u) = | |
| annularCoverA e a γ p.1 * u.1 + annularCoverB e a γ p.1 * u.2 := | |
| annularCover_form_rep e a γ p.1 u | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| rw [productHorizontalLift_pullback_apply _ (handleAnnularCover_smoothAt e a hp)] | |
| change (globalSurfaceFirstPrimitive e a b f Φ H Γ (ℓ,p.2) (handleAnnularCover e a p.1)) | |
| (A v.1)=_ | |
| erw [he] | |
| simp only [add_apply,smul_apply,smul_eq_mul,hi,cylinderCover, | |
| hd, hr, hg, | |
| surfaceSuspensionCoefficient,surfaceFirstPrimitive,horizontalOneForm_apply] | |
| ring | |
| theorem globalSurfaceSecondPrimitive_cover {ι : Type*} [Fintype ι] | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) | |
| (a b : ℝ) (f : Circle → ℝ) (Φ : CompactHamiltonianIsotopy Ω) | |
| (K H : V → ℝ) (h : ι → V → ℝ) (ρ : ι → ℝ → ℝ) (ρ₀ : ℝ → ℝ) | |
| (Γ : V → ManifoldOneForm E M) (γ : ManifoldOneForm E M) | |
| (τ : ℝ) (p : Plane × V) (hp : cylinderCover p.1∈(handleAnnularChart e a).source) | |
| (hN : Γ p.2 (handleAnnularCover e a p.1)=γ (handleAnnularCover e a p.1)+ | |
| (intervalClock (-b) b p.1.1*H p.2) • globalHandleForm e f (handleAnnularCover e a p.1)) : | |
| euclideanPullbackOneForm | |
| (fun _ => productHorizontalLift (fun v => globalSurfaceSecondPrimitive e a b f Φ K H h ρ ρ₀ Γ (τ,v))) | |
| (flatProductMap (handleAnnularCover e a)) (0,p)= | |
| surfaceSecondPrimitive Φ (intervalClock (-b) b) K H h ρ ρ₀ | |
| (annularCoverA e a γ) (annularCoverB e a γ) (fun q => f (circleTurn q.2)) (τ,p) := by | |
| have hn := globalSurfaceSecondPrimitive_normal e a b f Φ K H h ρ ρ₀ Γ γ (τ,p.2) (x := handleAnnularCover e a p.1) | |
| ((handleAnnularChart e a).map_source hp) | |
| have hi := handleAnnularCover_inverse e a hp | |
| have hN' : Γ p.2 (handleAnnularCover e a p.1)=γ (handleAnnularCover e a p.1)+ | |
| (intervalClock (-b) b ((handleAnnularChart e a).symm (handleAnnularCover e a p.1)).1*H p.2) • | |
| globalHandleForm e f (handleAnnularCover e a p.1) := by | |
| simpa only [hi,cylinderCover] using hN | |
| have he := hn hN' | |
| let A : Plane →L[ℝ] E := mfderiv 𝓘(ℝ,Plane) 𝓘(ℝ,E) (handleAnnularCover e a) p.1 | |
| have hd (u : Plane) : globalHandleForm e f (handleAnnularCover e a p.1) (A u) = | |
| f (circleTurn p.1.2) * u.2 := globalHandleForm_cover e a f hp u | |
| have hr (u : Plane) : globalHandleRadialForm e b (handleAnnularCover e a p.1) (A u) = | |
| deriv (intervalClock (-b) b) p.1.1 * u.1 := globalHandleRadialForm_cover e a b hp u | |
| have hg (u : Plane) : γ (handleAnnularCover e a p.1) (A u) = | |
| annularCoverA e a γ p.1 * u.1 + annularCoverB e a γ p.1 * u.2 := | |
| annularCover_form_rep e a γ p.1 u | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| rw [productHorizontalLift_pullback_apply _ (handleAnnularCover_smoothAt e a hp)] | |
| change (globalSurfaceSecondPrimitive e a b f Φ K H h ρ ρ₀ Γ (τ,p.2) (handleAnnularCover e a p.1)) | |
| (A v.1)=_ | |
| erw [he] | |
| simp only [add_apply,smul_apply,smul_eq_mul,hi,cylinderCover, | |
| hd, hr, hg, | |
| surfaceSuspensionCoefficient,surfaceSecondPrimitive,horizontalOneForm_apply,one_mul] | |
| ring | |
| end | |
| section | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] | |
| def annularCoverDomain | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) (a : ℝ) : Set Plane := | |
| cylinderCover ⁻¹' (handleAnnularChart e a).source | |
| omit [IsManifold 𝓘(ℝ,E) ∞ M] in | |
| theorem annularCoverDomain_open | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) (a : ℝ) : | |
| IsOpen (annularCoverDomain e a) := | |
| (handleAnnularChart e a).open_source.preimage cylinderCover_smooth.continuous | |
| theorem annularCover_pullback_smoothAt | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) (a : ℝ) | |
| {γ : ManifoldOneForm E M} {q : Plane} (hq : q∈annularCoverDomain e a) | |
| (hγ : ContDiffAt ℝ ∞ (chartOneForm γ (handleAnnularCover e a q)) | |
| (extChartAt 𝓘(ℝ,E) (handleAnnularCover e a q) (handleAnnularCover e a q))) : | |
| ContDiffAt ℝ ∞ (fun y => euclideanPullbackOneForm (fun _ => γ) (handleAnnularCover e a) (0,y)) q := by | |
| have hf := hγ.comp ((0:ℝ),extChartAt 𝓘(ℝ,E) (handleAnnularCover e a q) | |
| (handleAnnularCover e a q)) contDiffAt_snd | |
| have hs := parameterEuclideanPullbackOneForm_contDiffAt (α := fun _ : ℝ => γ) | |
| (p := (0,q)) hf (handleAnnularCover_smoothAt e a hq) | |
| exact hs.comp q (contDiffAt_const.prodMk contDiffAt_id) | |
| theorem annularCover_coefficients_smooth | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) (a : ℝ) | |
| {γ : ManifoldOneForm E M} | |
| (hγ : ∀ x∈(handleAnnularChart e a).target,ContDiffAt ℝ ∞ | |
| (chartOneForm γ x) (extChartAt 𝓘(ℝ,E) x x)) : | |
| ContDiffOn ℝ ∞ (annularCoverA e a γ) (annularCoverDomain e a) ∧ | |
| ContDiffOn ℝ ∞ (annularCoverB e a γ) (annularCoverDomain e a) := by | |
| constructor <;> intro q hq | |
| · exact ((annularCover_pullback_smoothAt e a hq | |
| (hγ _ ((handleAnnularChart e a).map_source hq))).clm_apply contDiffAt_const).contDiffWithinAt | |
| · exact ((annularCover_pullback_smoothAt e a hq | |
| (hγ _ ((handleAnnularChart e a).map_source hq))).clm_apply contDiffAt_const).contDiffWithinAt | |
| theorem annularCover_curl | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,E) HandleTorus M ∞) (a : ℝ) | |
| {γ : ManifoldOneForm E M} {q : Plane} (hq : q∈annularCoverDomain e a) | |
| (hγ : ContDiffAt ℝ ∞ (chartOneForm γ (handleAnnularCover e a q)) | |
| (extChartAt 𝓘(ℝ,E) (handleAnnularCover e a q) (handleAnnularCover e a q))) : | |
| fderiv ℝ (annularCoverB e a γ) q (1,0)-fderiv ℝ (annularCoverA e a γ) q (0,1)= | |
| manifoldExteriorOneForm γ (handleAnnularCover e a q) | |
| (mfderiv 𝓘(ℝ,Plane) 𝓘(ℝ,E) (handleAnnularCover e a) q (1,0)) | |
| (mfderiv 𝓘(ℝ,Plane) 𝓘(ℝ,E) (handleAnnularCover e a) q (0,1)) := by | |
| have hs := annularCover_pullback_smoothAt e a hq hγ | |
| have he : (fun y => euclideanPullbackOneForm (fun _ => γ) (handleAnnularCover e a) (0,y))= | |
| (fun y => planarCovector (annularCoverA e a γ y) (annularCoverB e a γ y)) := by | |
| funext y | |
| apply ContinuousLinearMap.ext | |
| intro v | |
| exact annularCover_form_rep e a γ y v | |
| have hd := euclidean_manifold_pullback_exterior_at (handleAnnularCover_smoothAt e a hq) hγ | |
| have hA : ContDiffAt ℝ ∞ (annularCoverA e a γ) q := hs.clm_apply contDiffAt_const | |
| have hB : ContDiffAt ℝ ∞ (annularCoverB e a γ) q := hs.clm_apply contDiffAt_const | |
| rw [he,planarCovector_exterior_at hA hB] at hd | |
| have hv := congrArg (fun Ω : Plane →L[ℝ] Plane →L[ℝ] ℝ => Ω (1,0) (0,1)) hd | |
| simp only [smul_apply,smul_eq_mul,planarArea_apply,mul_one,mul_zero,sub_zero, | |
| manifoldMapDifferential] at hv | |
| exact hv | |
| end | |
| variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] | |
| @[simp] theorem handleAnnularCover_eq_densityChart | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a : ℝ) : handleAnnularCover e a = handleDensityChart e a := rfl | |
| theorem handleAnnularCover_rectangle_domain | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| {a : ℝ} (ha : a < 1/2) (he : compactHandleBand a ⊆ e.source) | |
| {U : Set Plane} (hU : U ⊆ Ioo (-a) a ×ˢ Ioo (0:ℝ) 1) : | |
| U ⊆ annularCoverDomain e a := by | |
| intro q hq | |
| change cylinderCover q ∈ (handleAnnularChart e a).source | |
| rw [handleAnnularChart_source e ha he] | |
| exact ⟨(hU hq).1,mem_univ _⟩ | |
| theorem annularPackingChart_smoothOn | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a : ℝ) {U : Set (Plane × V)} (hU : ∀ p ∈ U, p.1 ∈ annularCoverDomain e a) : | |
| ContMDiffOn 𝓘(ℝ,Plane × V) 𝓘(ℝ,Plane × V) ∞ | |
| (flatProductMap (handleAnnularCover e a)) U := by | |
| intro p hp | |
| exact (flatProductMap_smoothAt (handleAnnularCover_smoothAt e a (hU p hp))).contMDiffWithinAt | |
| omit [NormedSpace ℝ V] in | |
| theorem annularPackingChart_isEmbedding | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| {a : ℝ} (ha : a < 1/2) (he : compactHandleBand a ⊆ e.source) | |
| {U : Set (Plane × V)} (hU : ∀ p ∈ U, p.1 ∈ Ioo (-a) a ×ˢ Ioo (0:ℝ) 1) : | |
| Topology.IsEmbedding (fun p : U => flatProductMap (handleAnnularCover e a) p.val) := by | |
| let c := (handleDensityChart e a).toOpenPartialHomeomorph.prod (OpenPartialHomeomorph.refl V) | |
| have hsub : U ⊆ c.source := by | |
| intro p hp | |
| change p.1 ∈ (handleDensityChart e a).source ∧ p.2 ∈ (univ : Set V) | |
| rw [handleDensityChart_source e ha he] | |
| exact ⟨hU p hp,mem_univ _⟩ | |
| exact c.isEmbedding_restrict.comp (Topology.IsEmbedding.inclusion hsub) | |
| variable {Ω : V →L[ℝ] V →L[ℝ] ℝ} | |
| def annularUnshearMap | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) : Plane × V → M × V := | |
| flatProductMap (handleAnnularCover e a) ∘ | |
| (hamiltonianBaseShear Φ (intervalClock (-b) b) (intervalClock_smooth (-b) b)).symm | |
| @[simp] theorem annularUnshearMap_apply | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) (p : Plane × V) : | |
| annularUnshearMap e a b Φ p = | |
| (handleAnnularCover e a p.1,(Φ.map (intervalClock (-b) b p.1.1)).symm p.2) := rfl | |
| theorem annularUnshearMap_smoothOn | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) {U : Set Plane} | |
| (hU : U ⊆ annularCoverDomain e a) : | |
| ContMDiffOn 𝓘(ℝ,Plane × V) 𝓘(ℝ,Plane × V) ∞ | |
| (annularUnshearMap e a b Φ) (U ×ˢ (univ : Set V)) := by | |
| intro p hp | |
| exact ((flatProductMap_smoothAt (x := (hamiltonianBaseShear Φ (intervalClock (-b) b) (intervalClock_smooth (-b) b)).symm p) | |
| (handleAnnularCover_smoothAt e a (hU hp.1))).comp p | |
| (hamiltonianBaseShear_inverse_smooth Φ (intervalClock_smooth (-b) b)).contMDiff.contMDiffAt).contMDiffWithinAt | |
| theorem annularUnshearMap_isEmbedding | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| {a : ℝ} (ha : a < 1/2) (he : compactHandleBand a ⊆ e.source) | |
| (b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) {U : Set Plane} | |
| (hU : U ⊆ Ioo (-a) a ×ˢ Ioo (0:ℝ) 1) : | |
| Topology.IsEmbedding (fun p : U ×ˢ (univ : Set V) => annularUnshearMap e a b Φ p.val) := by | |
| let D := hamiltonianBaseShear Φ (intervalClock (-b) b) (intervalClock_smooth (-b) b) | |
| have hd : Topology.IsEmbedding (fun p : U ×ˢ (univ : Set V) => D.symm p.val) := | |
| D.symm.isEmbedding.comp Topology.IsEmbedding.subtypeVal | |
| have hm : ∀ p : U ×ˢ (univ : Set V),D.symm p.val ∈ U ×ˢ (univ : Set V) := | |
| fun p => ⟨p.property.1,mem_univ _⟩ | |
| have hc := annularPackingChart_isEmbedding e ha he (V := V) | |
| (U := U ×ˢ univ) (fun _ h => hU h.1) | |
| exact hc.comp (hd.codRestrict (U ×ˢ univ) hm) | |
| theorem smoothHomeomorph_inverse_derivative {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (e : E ≃ₜ E) (he : ContDiff ℝ ∞ e) (hi : ContDiff ℝ ∞ e.symm) (p v : E) : | |
| fderiv ℝ e (e.symm p) (fderiv ℝ e.symm p v) = v := by | |
| have hd := fderiv_comp p (he.differentiable (by simp) _) (hi.differentiable (by simp) p) | |
| have hh : (e : E → E) ∘ e.symm = id := by funext x; exact e.apply_symm_apply x | |
| rw [hh,fderiv_id] at hd | |
| exact congrArg (fun L : E →L[ℝ] E => L v) hd.symm | |
| end PackingSufficiencySupport.Hamiltonian | |
| namespace PackingSufficiencySupport.Hamiltonian.AnnularHandleData | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] | |
| {Ω : V →L[ℝ] V →L[ℝ] ℝ} | |
| def annularUnshearMap | |
| (D : AnnularHandleData Plane M) | |
| (b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) : Plane × V → M × V := | |
| flatProductMap (D.handleAnnularCover) ∘ | |
| (hamiltonianBaseShear Φ (intervalClock (-b) b) (intervalClock_smooth (-b) b)).symm | |
| @[simp] theorem annularUnshearMap_apply | |
| (D : AnnularHandleData Plane M) | |
| (b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) (p : Plane × V) : | |
| annularUnshearMap D b Φ p = | |
| (D.handleAnnularCover p.1,(Φ.map (intervalClock (-b) b p.1.1)).symm p.2) := rfl | |
| theorem annularUnshearMap_smoothOn | |
| (D : AnnularHandleData Plane M) | |
| (b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) {U : Set Plane} | |
| (hU : U ⊆ D.annularCoverDomain) : | |
| ContMDiffOn 𝓘(ℝ,Plane × V) 𝓘(ℝ,Plane × V) ∞ | |
| (annularUnshearMap D b Φ) (U ×ˢ (univ : Set V)) := by | |
| intro p hp | |
| exact ((flatProductMap_smoothAt (x := (hamiltonianBaseShear Φ (intervalClock (-b) b) (intervalClock_smooth (-b) b)).symm p) | |
| (D.handleAnnularCover_smoothAt (hU hp.1))).comp p | |
| (hamiltonianBaseShear_inverse_smooth Φ (intervalClock_smooth (-b) b)).contMDiff.contMDiffAt).contMDiffWithinAt | |
| theorem annularUnshearMap_isEmbedding | |
| (D : AnnularHandleData Plane M) | |
| (b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) {U : Set Plane} | |
| (hU : U ⊆ Ioo (-D.width) D.width ×ˢ Ioo (0:ℝ) 1) : | |
| Topology.IsEmbedding (fun p : U ×ˢ (univ : Set V) => annularUnshearMap D b Φ p.val) := by | |
| let L := hamiltonianBaseShear Φ (intervalClock (-b) b) (intervalClock_smooth (-b) b) | |
| have hd : Topology.IsEmbedding (fun p : U ×ˢ (univ : Set V) => L.symm p.val) := | |
| L.symm.isEmbedding.comp Topology.IsEmbedding.subtypeVal | |
| have hm : ∀ p : U ×ˢ (univ : Set V),L.symm p.val ∈ U ×ˢ (univ : Set V) := | |
| fun p => ⟨p.property.1,mem_univ _⟩ | |
| have hc := D.annularPackingChart_isEmbedding (V := V) | |
| (U := U ×ˢ univ) (fun _ h => hU h.1) | |
| exact hc.comp (hd.codRestrict (U ×ˢ univ) hm) | |
| end PackingSufficiencySupport.Hamiltonian.AnnularHandleData | |
| namespace PackingSufficiencySupport.Hamiltonian | |
| open scoped ContDiff Manifold Topology | |
| open Set Function Manifold | |
| theorem homeomorph_mapsTo_interior {X : Type*} [TopologicalSpace X] | |
| (e : X ≃ₜ X) {Y : Set X} (hf : MapsTo e Y Y) (hi : MapsTo e.symm Y Y) : | |
| MapsTo e (interior Y) (interior Y) := by | |
| have he : e '' Y = Y := by | |
| apply Subset.antisymm (image_subset_iff.mpr hf) | |
| intro x hx | |
| exact ⟨e.symm x,hi hx,e.apply_symm_apply x⟩ | |
| have hI : e '' interior Y = interior Y := by rw [e.image_interior,he] | |
| intro x hx | |
| rw [←hI] | |
| exact ⟨x,hx,rfl⟩ | |
| variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V] | |
| {M : Type*} [TopologicalSpace M] [ChartedSpace Plane M] | |
| {Ω : V →L[ℝ] V →L[ℝ] ℝ} | |
| theorem handleAnnularCover_mem_band | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a b : ℝ) {p : Plane} (hp : p.1 ∈ Icc (-b) b) : | |
| handleAnnularCover e a p ∈ e '' compactHandleBand b := by | |
| exact ⟨torusAnnulusMap (cylinderCover p),⟨cylinderCover p,⟨hp,mem_univ _⟩,rfl⟩,rfl⟩ | |
| theorem annularUnshearMap_mem_interior | |
| (e : PartialDiffeomorph 𝓘(ℝ,TorusModel) 𝓘(ℝ,Plane) HandleTorus M ∞) | |
| (a b : ℝ) (Φ : CompactHamiltonianIsotopy Ω) {Y : Set V} {D : Set M} | |
| (hbase : e '' compactHandleBand b ⊆ interior D) | |
| (hΦ : ∀ t∈Icc (0:ℝ) 1,MapsTo (Φ.map t) Y Y ∧ MapsTo (Φ.map t).symm Y Y) | |
| {p : Plane × V} (hp : p.1.1 ∈ Icc (-b) b) (hv : p.2 ∈ interior Y) : | |
| annularUnshearMap e a b Φ p ∈ interior (D ×ˢ Y) := by | |
| rw [interior_prod_eq] | |
| refine ⟨hbase (handleAnnularCover_mem_band e a b hp),?_⟩ | |
| have ht : intervalClock (-b) b p.1.1 ∈ Icc (0:ℝ) 1 := | |
| intervalClock_bounds _ _ _ | |
| exact homeomorph_mapsTo_interior (Φ.map _).symm (hΦ _ ht).2 (hΦ _ ht).1 hv | |
| end PackingSufficiencySupport.Hamiltonian | |
| end | |
| end OAI | |