| import Mathlib |
| import OAI.Geometry.CubicTorus.Profiles.FirstAxisShift |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| namespace Periodic |
|
|
| theorem isMinimizerAt_selected_nodal_irreducible {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) : |
| ∃ (μ : FiniteMeasure NormalBundle) (N : Cover → Cover) |
| (hN : ContDiff ℝ (⊤ : ℕ∞) N) (hp : ∀ m x,N (x+latticeVector m)=N x), |
| IsNormalMeasure E μ ∧ (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) ∧ |
| ∀ (v : Cover) (D A : Set Torus), |
| IsPositiveNodalPiece (directionalNormalScalar N hN hp v) (frontier (openRepresentative E)) D → |
| IsPositiveNodalPiece (directionalNormalScalar N hN hp v) (frontier (openRepresentative E)) A → |
| D.Nonempty → A.Nonempty → ¬Disjoint D A := by |
| obtain ⟨μ,H,N,hN,hp,hμ,hae,hb,hs,hj,hunused⟩ := isMinimizerAt_H1_translation_jacobi hV hE hmin |
| refine ⟨μ,N,hN,hp,hμ,hae,?_⟩ |
| intro v D A hD hA hDn hAn hDA |
| let f := directionalNormalScalar N hN hp v |
| have hjN := directionalNormalScalar_global_jacobi μ N hN hp (fun i ↦ (hj i).2) |
| have hjf := hjN v |
| obtain ⟨hDm,hdmem⟩ := isMinimizerAt_positive_nodal_piece_H1 hV hE hmin hμ f hD |
| obtain ⟨hAm,hamem⟩ := isMinimizerAt_positive_nodal_piece_H1 hV hE hmin hμ f hA |
| let d := nodalSurfaceJet μ f D hDm |
| let a := nodalSurfaceJet μ f A hAm |
| let u := surfaceNodalBalance μ d a |
| have hdnull := nodalSurfaceJet_null μ (surfaceCurvatureLp μ N hN) f D hDm hdmem hjf |
| have hanull := nodalSurfaceJet_null μ (surfaceCurvatureLp μ N hN) f A hAm hamem hjf |
| have horth := surfaceJacobiB_disjoint_nodal μ (surfaceCurvatureLp μ N hN) f f D A hDm hAm hDA |
| have humem : u∈surfaceH1 μ := surfaceNodalBalance_mem μ hdmem hamem |
| have hunull : surfaceJacobiB μ (surfaceCurvatureLp μ N hN) u u=0 := |
| surfaceNodalBalance_null μ _ hdnull hanull horth |
| obtain ⟨U,hU,hUn,hUD,hUA⟩ := isMinimizerAt_negative_unused_patch hV hE hmin hμ f |
| (sibling_directionalNormalScalar_mean_zero hμ hN hp hae v) hD hA hDn |
| have hju : ∀ t∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) u t=0 := |
| hunused u humem (surfaceNodalBalance_mean μ d a) hunull U hU hUn |
| (surfaceNodalBalance_zero_patch μ (nodalSurfaceJet_zero_outside μ f D U hDm hUD) |
| (nodalSurfaceJet_zero_outside μ f A U hAm hUA)) |
| let F : Torus → ℝ := fun x ↦ surfaceMean μ a*nodalTorusScalar f D x- |
| surfaceMean μ d*nodalTorusScalar f A x |
| have hFm : Measurable F := |
| (measurable_const.mul (nodalTorusScalar_measurable f hDm)).sub |
| (measurable_const.mul (nodalTorusScalar_measurable f hAm)) |
| have hFc : ContinuousOn F (frontier (openRepresentative E)) := |
| (continuousOn_const.mul (positiveNodalPiece_continuousOn f hD)).sub |
| (continuousOn_const.mul (positiveNodalPiece_continuousOn f hA)) |
| have hFa : ∀ᵐ z ∂(μ : Measure NormalBundle),u.1 z=F (normalProjection z) := |
| surfaceNodalBalance_scalar_ae μ f f D A hDm hAm |
| have hother : (frontier (openRepresentative E)\D).Nonempty := by |
| obtain ⟨x,hx⟩ := hAn |
| exact ⟨x,positiveNodalPiece_subset f hA hx,fun hd ↦ Set.disjoint_left.mp hDA hd hx⟩ |
| obtain ⟨x,hcl,hnot⟩ := positiveNodalPiece_boundary_nonempty f hc hD hDn hother |
| have hk : surfaceMean μ a≠0 := (isMinimizerAt_nodal_mean_pos hV hE hmin hμ f hAm hA hAn).ne' |
| apply isMinimizerAt_nodal_truncation_impossible hV hE hmin hμ hN hp hae hjN f hjf |
| humem hju hFm hFc hFa hD hcl hnot hk |
| · intro x hx |
| have hxA : x∉A := fun ha ↦ Set.disjoint_left.mp hDA hx ha |
| simp only [F,nodalTorusScalar,Set.indicator_of_mem hx,Set.indicator_of_notMem hxA, |
| mul_zero,sub_zero] |
| · intro x hx hle |
| have hxD : x∉D := fun hd ↦ ((positiveNodalPiece_superlevels f isClosed_frontier hD).2.1 x hd).not_ge hle |
| have hxA : x∉A := fun ha ↦ ((positiveNodalPiece_superlevels f isClosed_frontier hA).2.1 x ha).not_ge hle |
| simp only [F,nodalTorusScalar,Set.indicator_of_notMem hxD,Set.indicator_of_notMem hxA, |
| mul_zero,sub_zero] |
|
|
| end Periodic |
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma positiveNodalPiece_relative_compl (f : smoothPeriodicScalars) {S D : Set Torus} |
| (hD : IsPositiveNodalPiece f S D) : |
| IsPositiveNodalPiece f S ({x | x∈S ∧ 0<f.val (representative x)}\D) := by |
| obtain ⟨⟨U,hU,hDU⟩,⟨C,hC,hDC⟩⟩ := hD |
| constructor |
| · refine ⟨Cᶜ,hC.isOpen_compl,?_⟩ |
| rw [hDC] |
| ext x |
| simp only [Set.mem_sdiff,mem_inter_iff,mem_ofPred_eq,mem_compl_iff] |
| tauto |
| · refine ⟨Uᶜ,hU.isClosed_compl,?_⟩ |
| rw [hDU] |
| ext x |
| simp only [Set.mem_sdiff,mem_inter_iff,mem_ofPred_eq,mem_compl_iff] |
| tauto |
|
|
| lemma positiveBoundaryDomain_preconnected (f : smoothPeriodicScalars) (S : Set Torus) |
| (a : positiveBoundaryLocus f S) : IsPreconnected (positiveBoundaryDomain f S a) := |
| isPreconnected_connectedComponent.image _ |
| (continuous_subtype_val.comp continuous_subtype_val).continuousOn |
|
|
| lemma positiveNodalPiece_eq_positive_locus (f : smoothPeriodicScalars) {S D : Set Torus} |
| (hD : IsPositiveNodalPiece f S D) (_hDn : D.Nonempty) |
| (hirr : ∀ A : Set Torus,IsPositiveNodalPiece f S A → A.Nonempty → ¬Disjoint D A) : |
| D={x | x∈S ∧ 0<f.val (representative x)} := by |
| apply Set.Subset.antisymm |
| · intro x hx |
| rw [hD.1.choose_spec.2] at hx |
| exact hx.1 |
| · intro x hx |
| by_contra hxD |
| apply hirr _ (positiveNodalPiece_relative_compl f hD) ⟨x,hx,hxD⟩ |
| exact Set.disjoint_left.mpr (fun y hy hA ↦ hA.2 hy) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| namespace Periodic |
|
|
| theorem isMinimizerAt_translation_signs_preconnected {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) : |
| ∃ (μ : FiniteMeasure NormalBundle) (N : Cover → Cover) |
| (_hN : ContDiff ℝ (⊤ : ℕ∞) N) (_hp : ∀ m x,N (x+latticeVector m)=N x), |
| IsNormalMeasure E μ ∧ (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) ∧ |
| ∀ v : Cover, |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<⟪v,N (representative x)⟫} ∧ |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| ⟪v,N (representative x)⟫<0} := by |
| obtain ⟨μ,N,hN,hp,hμ,hae,hirr⟩ := |
| isMinimizerAt_selected_nodal_irreducible hV hE hmin hc |
| have hpos (v : Cover) : |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<⟪v,N (representative x)⟫} := by |
| let f := directionalNormalScalar N hN hp v |
| by_cases hn : ({x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<f.val (representative x)} : Set Torus).Nonempty |
| · obtain ⟨x,hxS,hxp⟩ := hn |
| let a : positiveBoundaryLocus f (frontier (openRepresentative E)) := ⟨⟨x,hxS⟩,hxp⟩ |
| let D := positiveBoundaryDomain f (frontier (openRepresentative E)) a |
| have hD := isMinimizerAt_actual_nodal_piece hV hE hmin f a |
| have hDn := positiveBoundaryDomain_nonempty f (frontier (openRepresentative E)) a |
| have heq := positiveNodalPiece_eq_positive_locus f hD hDn (fun A hA hAn ↦ |
| hirr v D A hD hA hDn hAn) |
| have hpD := positiveBoundaryDomain_preconnected f (frontier (openRepresentative E)) a |
| rw [heq] at hpD |
| simpa only [f,directionalNormalScalar_apply] using hpD |
| · have he := Set.not_nonempty_iff_eq_empty.mp hn |
| have hem : IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<f.val (representative x)} := he ▸ isPreconnected_empty |
| simpa only [f,directionalNormalScalar_apply] using hem |
| refine ⟨μ,N,hN,hp,hμ,hae,fun v ↦ ⟨hpos v,?_⟩⟩ |
| simpa only [inner_neg_left,neg_pos] using hpos (-v) |
|
|
| end Periodic |
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
| namespace Periodic |
|
|
| theorem isMinimizerAt_slab_or_translation_signs {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hmin : IsMinimizerAt V E) : |
| (perimeter E=2 ∧ 1/Real.pi≤ min V (1-V) ∧ StandardRegionAt V E) ∨ |
| (IsPreconnected (frontier (openRepresentative E)) ∧ |
| ∃ (μ : FiniteMeasure NormalBundle) (N : Cover → Cover) |
| (_hN : ContDiff ℝ (⊤ : ℕ∞) N) (_hp : ∀ m x,N (x+latticeVector m)=N x), |
| IsNormalMeasure E μ ∧ (∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) ∧ |
| ∀ v : Cover, |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<⟪v,N (representative x)⟫} ∧ |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| ⟪v,N (representative x)⟫<0}) := by |
| rcases isMinimizerAt_connected_or_standard_slab hV hmin with hc | hsl |
| · right |
| refine ⟨hc,?_⟩ |
| let F := toMeasurable torusVolume E |
| have hF : MeasurableSet F := measurableSet_toMeasurable _ _ |
| have hae : F =ᵐ[torusVolume] E := hmin.1.1.toMeasurable_ae_eq |
| have hminF : IsMinimizerAt V F := (isMinimizerAt_congr_ae hae).mpr hmin |
| have hcF : IsPreconnected (frontier (openRepresentative F)) := by |
| rwa [openRepresentative_congr_ae hae] |
| obtain ⟨μ,N,hN,hp,hμ,hn,hsg⟩ := |
| isMinimizerAt_translation_signs_preconnected hV hF hminF hcF |
| refine ⟨μ,N,hN,hp,hμ.congr_ae hae,hn,?_⟩ |
| simpa only [openRepresentative_congr_ae hae] using hsg |
| · exact Or.inl hsl |
|
|
| end Periodic |
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma odd_preconnected_sign_half {X : Type*} [TopologicalSpace X] |
| {S : Set X} {R : X → X} {s u : X → ℝ} (hs : Continuous s) |
| (hR : ∀ x∈S,R x∈S) (hsR : ∀ x,s (R x)= -s x) |
| (huR : ∀ x∈S,u (R x)= -u x) |
| (hz : ∀ x∈S,s x=0 → u x=0) |
| (hc : IsPreconnected {x | x∈S ∧ 0<u x}) : |
| (∀ x∈S,s x<0 → u x≤0) ∨ (∀ x∈S,s x<0 → 0≤u x) := by |
| classical |
| by_cases hn : ∃ x∈S,s x<0 ∧ 0<u x |
| · right |
| obtain ⟨a,ha,has,hau⟩ := hn |
| have hP : {x | x∈S ∧ 0<u x}⊆{x | s x<0}∪{x | 0<s x} := by |
| rintro x ⟨hx,hu⟩ |
| have hne : s x≠0 := fun h ↦ hu.ne' (hz x hx h) |
| rcases lt_or_gt_of_ne hne with h|h |
| · exact Or.inl h |
| · exact Or.inr h |
| have hsep : Disjoint {x | s x<0} {x | 0<s x} := by |
| rw [Set.disjoint_left] |
| intro x hx hy |
| exact (show s x<0 from hx).not_gt (show 0<s x from hy) |
| have hsub := hc.subset_left_of_subset_union (isOpen_lt hs continuous_const) |
| (isOpen_lt continuous_const hs) hsep hP ⟨a,⟨ha,hau⟩,has⟩ |
| intro x hx hxs |
| by_contra hneg |
| have hu : u x<0 := lt_of_not_ge hneg |
| have hh := hsub (show R x∈{x | x∈S ∧ 0<u x} from |
| ⟨hR x hx,by rw [huR x hx]; exact neg_pos.mpr hu⟩) |
| change s (R x)<0 at hh |
| rw [hsR] at hh |
| linarith |
| · left |
| intro x hx hxs |
| exact le_of_not_gt (fun hu ↦ hn ⟨x,hx,hxs,hu⟩) |
|
|
| theorem isMinimizerAt_any_normal_signs {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) (v : Cover) : |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<⟪v,N (representative x)⟫} ∧ |
| IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| ⟪v,N (representative x)⟫<0} := by |
| obtain ⟨ν,M,hM,hpM,hν,haM,hs⟩ := |
| Periodic.isMinimizerAt_translation_signs_preconnected hV hE hmin hc |
| have he : ∀ x∈frontier (openRepresentative E),N (representative x)=M (representative x) := by |
| intro x hx |
| obtain ⟨w,hw⟩ := isMinimizerAt_normal_unique hV hE hmin (representative x) |
| (by simpa only [quotientMap_representative] using hx) |
| exact (hw μ hμ N hN hp hae).trans (hw ν hν M hM hpM haM).symm |
| have hpos : {x : Torus | x∈frontier (openRepresentative E) ∧ 0<⟪v,N (representative x)⟫}= |
| {x : Torus | x∈frontier (openRepresentative E) ∧ 0<⟪v,M (representative x)⟫} := by |
| ext x |
| simp only [Set.mem_ofPred_eq] |
| by_cases hx : x∈frontier (openRepresentative E) |
| · simp only [hx,true_and,he x hx] |
| · simp only [hx,false_and] |
| have hneg : {x : Torus | x∈frontier (openRepresentative E) ∧ ⟪v,N (representative x)⟫<0}= |
| {x : Torus | x∈frontier (openRepresentative E) ∧ ⟪v,M (representative x)⟫<0} := by |
| ext x |
| simp only [Set.mem_ofPred_eq] |
| by_cases hx : x∈frontier (openRepresentative E) |
| · simp only [hx,true_and,he x hx] |
| · simp only [hx,false_and] |
| rw [hpos,hneg] |
| exact hs v |
|
|
| theorem isMinimizerAt_reflection_weak_normal_sign {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| (hR : E=ᵐ[torusVolume](torusFirstReflection ⁻¹' E)) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) : |
| (∀ x∈frontier (openRepresentative E),firstSeamFunction (representative x)<0 → |
| N (representative x) 0≤0) ∨ |
| (∀ x∈frontier (openRepresentative E),firstSeamFunction (representative x)<0 → |
| 0≤N (representative x) 0) := by |
| have hsign := (isMinimizerAt_any_normal_signs hV hE hmin hc hμ hN hp hae |
| (EuclideanSpace.single 0 1)).1 |
| have hsign' : IsPreconnected {x : Torus | x∈frontier (openRepresentative E) ∧ |
| 0<N (representative x) 0} := by |
| simpa only [EuclideanSpace.inner_single_left,map_one,one_mul] using hsign |
| exact odd_preconnected_sign_half |
| (continuous_periodic_descent firstSeamFunction_contDiff.continuous firstSeamFunction_periodic) |
| (fun x hx ↦ (isMinimizerAt_reflection_boundary hV hE hmin hR x).mpr hx) |
| firstSeam_representative_reflection |
| (isMinimizerAt_reflection_normal_scalar hV hE hmin hR hμ hN hp hae) |
| (isMinimizerAt_firstSeam_normal_zero hV hE hmin hR hμ hN hp hae) hsign' |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_normal_strong_minimum {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (v : Cover) {x : Torus} (hx : x∈frontier (openRepresentative E)) |
| (hpos : ∀ᶠ y in 𝓝 x,y∈frontier (openRepresentative E) → 0≤⟪v,N (representative y)⟫) |
| (hz : ⟪v,N (representative x)⟫=0) : |
| ∀ y∈frontier (openRepresentative E),⟪v,N (representative y)⟫=0 := by |
| obtain ⟨ν,H,M,hM,hpM,hν,haM,hb,hs,hj,hu⟩ := Periodic.isMinimizerAt_H1_translation_jacobi hV hE hmin |
| have he : ∀ y∈frontier (openRepresentative E),N (representative y)=M (representative y) := by |
| intro y hy |
| obtain ⟨w,hw⟩ := isMinimizerAt_normal_unique hV hE hmin (representative y) |
| (by simpa only [quotientMap_representative] using hy) |
| exact (hw μ hμ N hN hp hae).trans (hw ν hν M hM hpM haM).symm |
| have hjM := directionalNormalScalar_global_jacobi ν M hM hpM (fun i ↦ (hj i).2) |
| have hzM : (directionalNormalScalar M hM hpM v).val (representative x)=0 := by |
| rw [directionalNormalScalar_apply,←he x hx] |
| exact hz |
| have hpM' : ∀ᶠ y in 𝓝 x,y∈frontier (openRepresentative E) → |
| 0≤(directionalNormalScalar M hM hpM v).val (representative y) := by |
| filter_upwards [hpos] with y hy hyB |
| rw [directionalNormalScalar_apply,←he y hyB] |
| exact hy hyB |
| have hzall := isMinimizerAt_Jacobi_strong_minimum hV hE hmin hc hν hM hpM haM hjM |
| (directionalNormalScalar M hM hpM v) (hjM v) hx hpM' hzM |
| intro y hy |
| rw [he y hy] |
| simpa only [directionalNormalScalar_apply] using hzall y hy |
|
|
| theorem isMinimizerAt_reflection_strict_normal_sign {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hc : IsPreconnected (frontier (openRepresentative E))) |
| (hR : E=ᵐ[torusVolume](torusFirstReflection ⁻¹' E)) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) : |
| (∀ x∈frontier (openRepresentative E),N (representative x) 0=0) ∨ |
| (∀ x∈frontier (openRepresentative E),firstSeamFunction (representative x)<0 → |
| N (representative x) 0<0) ∨ |
| (∀ x∈frontier (openRepresentative E),firstSeamFunction (representative x)<0 → |
| 0<N (representative x) 0) := by |
| have hs := continuous_periodic_descent firstSeamFunction_contDiff.continuous firstSeamFunction_periodic |
| by_cases hzero : ∀ x∈frontier (openRepresentative E),N (representative x) 0=0 |
| · exact Or.inl hzero |
| · right |
| rcases isMinimizerAt_reflection_weak_normal_sign hV hE hmin hc hR hμ hN hp hae with hneg|hpos |
| · left |
| intro x hx hxs |
| refine lt_of_le_of_ne (hneg x hx hxs) ?_ |
| intro hzx |
| have hzall := isMinimizerAt_normal_strong_minimum hV hE hmin hc hμ hN hp hae |
| (-(EuclideanSpace.single 0 1)) hx (by |
| filter_upwards [(isOpen_lt hs continuous_const).mem_nhds hxs] with y hy hyB |
| simpa only [inner_neg_left,EuclideanSpace.inner_single_left,map_one,one_mul,neg_nonneg] using hneg y hyB hy) |
| (by simp only [inner_neg_left,EuclideanSpace.inner_single_left,map_one,one_mul,hzx,neg_zero]) |
| apply hzero |
| intro y hy |
| simpa only [inner_neg_left,EuclideanSpace.inner_single_left,map_one,one_mul,neg_eq_zero] using hzall y hy |
| · right |
| intro x hx hxs |
| refine lt_of_le_of_ne (hpos x hx hxs) ?_ |
| intro hzx |
| have hzall := isMinimizerAt_normal_strong_minimum hV hE hmin hc hμ hN hp hae |
| (EuclideanSpace.single 0 1) hx (by |
| filter_upwards [(isOpen_lt hs continuous_const).mem_nhds hxs] with y hy hyB |
| simpa only [EuclideanSpace.inner_single_left,map_one,one_mul] using hpos y hyB hy) |
| (by simp only [EuclideanSpace.inner_single_left,map_one,one_mul,←hzx]) |
| apply hzero |
| intro y hy |
| simpa only [EuclideanSpace.inner_single_left,map_one,one_mul] using hzall y hy |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem normal_zero_translation_ae {E : Set Torus} (hE : MeasurableSet E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {v : Cover} (hz : ∀ᵐ z ∂(μ : Measure NormalBundle),⟪(z.2 : Cover),v⟫=0) |
| (t : ℝ) : E=ᵐ[torusVolume] (fun x ↦ x+quotientMap (t • v)) ⁻¹' E := by |
| obtain ⟨M,hM,hpolar,hunit⟩ := normalMeasure_polar_unit hμ |
| have hzM : ∀ᵐ x ∂(spatialNormalMeasure μ : Measure Torus),⟪M x,v⟫=0 := by |
| change ∀ᵐ x ∂Measure.map normalProjection (μ : Measure NormalBundle),⟪M x,v⟫=0 |
| rw [ae_map_iff normalProjection_continuous.measurable.aemeasurable |
| (measurableSet_eq_fun (hM.inner measurable_const) measurable_const)] |
| filter_upwards [hpolar,hz] with z hp hz |
| change ⟪M (quotientMap z.1),v⟫=0 |
| rw [hp,hz] |
| have hmass (B : Set Torus) : directionalMass (spatialNormalMeasure μ) M (-(t • v)) B=0 := by |
| apply integral_eq_zero_of_ae |
| filter_upwards [ae_restrict_of_ae hzM] with x hx |
| simp only [inner_neg_right,real_inner_smul_right,hx,mul_zero,neg_zero,abs_zero,Pi.zero_apply] |
| obtain ⟨r,hr⟩ := (isCompact_univ : IsCompact (univ : Set Torus)).isBounded.subset_ball (0 : Torus) |
| have hall : Metric.ball (0 : Torus) r=univ := eq_univ_of_univ_subset hr |
| have hi := characteristic_directional_translation_bound hE hμ hM hpolar hunit |
| (0 : Torus) (r := r) (s := 1) (by norm_num) (-(t • v)) |
| rw [hmass,hall,Measure.restrict_univ,quotientMap_neg] at hi |
| simp only [sub_neg_eq_add] at hi |
| have hint : Integrable (fun x ↦ |characteristic E (x+quotientMap (t • v))-characteristic E x|) torusVolume := by |
| apply Integrable.of_bound (((measurable_characteristic hE).comp |
| (measurable_id.add_const _)).sub (measurable_characteristic hE)).abs.aestronglyMeasurable 1 |
| exact Eventually.of_forall fun x ↦ by |
| rcases characteristic_dichotomy E (x+quotientMap (t • v)) with h|h <;> |
| rcases characteristic_dichotomy E x with h'|h' <;> simp [h,h'] |
| have he := (integral_eq_zero_iff_of_nonneg (fun x ↦ abs_nonneg _) hint).mp |
| (le_antisymm hi (integral_nonneg fun x ↦ abs_nonneg _)) |
| filter_upwards [he] with x hx |
| have heq := sub_eq_zero.mp (abs_eq_zero.mp hx) |
| change (x∈E)=(x+quotientMap (t • v)∈E) |
| apply propext |
| by_cases h : x∈E <;> by_cases h' : x+quotientMap (t • v)∈E <;> |
| simp_all [characteristic] |
|
|
| theorem isMinimizerAt_zero_normal_invariant {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (_hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (v : Cover) (hz : ∀ x∈frontier (openRepresentative E),⟪v,N (representative x)⟫=0) |
| (t : ℝ) : |
| E=ᵐ[torusVolume] (fun x ↦ x+quotientMap (t • v)) ⁻¹' E ∧ |
| openRepresentative E=(fun x ↦ x+quotientMap (t • v)) ⁻¹' openRepresentative E ∧ |
| frontier (openRepresentative E)=(fun x ↦ x+quotientMap (t • v)) ⁻¹' frontier (openRepresentative E) := by |
| have hμO : IsNormalMeasure (openRepresentative E) μ := |
| hμ.congr_ae (isMinimizerAt_openRepresentative_ae hV hE hmin).symm |
| have hflux : ∀ᵐ z ∂(μ : Measure NormalBundle),⟪(z.2 : Cover),v⟫=0 := by |
| filter_upwards [normalMeasure_frontier_ae (isOpen_openRepresentative E).measurableSet hμO,hae] with z hb hn |
| have hh := hz (quotientMap z.1) hb |
| rw [periodic_representative hp,hn,real_inner_comm] at hh |
| exact hh |
| have he := normal_zero_translation_ae hE hμ hflux t |
| have hO := openRepresentative_congr_ae he |
| rw [openRepresentative_translation hE] at hO |
| refine ⟨he,hO,?_⟩ |
| exact (congrArg frontier hO).trans |
| ((Homeomorph.addRight (quotientMap (t • v))).preimage_frontier (openRepresentative E)).symm |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma normalizedCMCCharts_zero_component_differential {B : Set Torus} |
| {N : Cover → Cover} {H : ℝ} (hcharts : HasNormalizedCMCCharts B N H) |
| (hN : ContDiff ℝ (⊤ : ℕ∞) N) {v : Cover} |
| (hz : ∀ x : Cover,quotientMap x∈B → ⟪v,N x⟫=0) |
| {a : Cover} (ha : quotientMap a∈B) {w : Cover} (hw : ⟪N a,w⟫=0) : |
| ⟪v,fderiv ℝ N a w⟫=0 := by |
| obtain ⟨hn,f,hf,hf0,hfgrad,hg,hcov⟩ := hcharts a ha |
| let Φ : Plane → Cover := fun y ↦ a+normalGraph (N a) f y |
| have hΦ0 : Φ 0=a := by |
| simp only [Φ,normalGraph,hf0,zero_smul,add_zero] |
| have he : normalPlaneEmbedding (N a) 0=0 := norm_eq_zero.mp (by rw [normalPlaneEmbedding_norm,norm_zero]) |
| rw [he,add_zero] |
| have hΦ : DifferentiableAt ℝ Φ 0 := |
| (normalGraph_hasFDerivAt (n := N a) (hf.differentiable (by simp) 0)).differentiableAt.const_add a |
| have hzero : (fun y ↦ ⟪v,N (Φ y)⟫)=ᶠ[𝓝 (0 : Plane)](fun _ ↦ (0 : ℝ)) := by |
| filter_upwards [hg] with y hy |
| exact hz _ hy.2.1 |
| have hd := hzero.fderiv_eq (𝕜 := ℝ) |
| have hcomp := ((innerSL ℝ v).hasFDerivAt.comp (Φ 0) |
| (hN.differentiable (by simp) (Φ 0)).hasFDerivAt).comp 0 hΦ.hasFDerivAt |
| change HasFDerivAt (fun y ↦ ⟪v,N (Φ y)⟫) _ 0 at hcomp |
| rw [hcomp.fderiv,hΦ0] at hd |
| have he := congrArg (fun L : Plane →L[ℝ] ℝ ↦ L (normalPlaneCoordinates (N a) w)) hd |
| have hΦd : fderiv ℝ Φ 0 (normalPlaneCoordinates (N a) w)=w := by |
| rw [fderiv_const_add] |
| rw [normalGraph_fderiv_apply hf,hfgrad,inner_zero_left,zero_smul,add_zero] |
| exact normalPlaneEmbedding_coordinates_of_inner_zero hn hw |
| simpa only [ContinuousLinearMap.comp_apply,innerSL_apply_apply,hΦd,fderiv_const_apply,zero_apply] using he |
|
|
| theorem isMinimizerAt_zero_component_shape_kernel {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| {H : ℝ} (hb : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) |
| {v : Cover} (hz : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ⟪v,N x⟫=0) |
| {a : Cover} (ha : quotientMap a∈frontier (openRepresentative E)) : |
| fderiv ℝ N a v=0 := by |
| have hcharts := isMinimizerAt_hasNormalizedCMCCharts hV hE hmin hμ hN hp hae hb |
| have hd := smooth_unit_normal_derivative (hN.differentiable (by simp) a) (hb a ha).1 |
| have hDv : ⟪N a,fderiv ℝ N a v⟫=0 := by |
| rw [←ContinuousLinearMap.adjoint_inner_left,hd,inner_zero_left] |
| have hv : ⟪N a,v⟫=0 := by rw [real_inner_comm,hz a ha] |
| have hh := isMinimizerAt_shape_tangent_symmetric hV hE hmin hμ hN hp hae a ha v |
| (fderiv ℝ N a v) hv hDv |
| rw [normalizedCMCCharts_zero_component_differential hcharts hN hz ha hDv] at hh |
| exact inner_self_eq_zero.mp hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_zero_component_shape {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| {H : ℝ} (hb : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) |
| {v : Cover} (hv : ‖v‖=1) |
| (hz : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ⟪v,N x⟫=0) |
| {a : Cover} (ha : quotientMap a∈frontier (openRepresentative E)) |
| {w : Cover} (hw : ⟪N a,w⟫=0) : |
| fderiv ℝ N a w=H • (w-⟪v,w⟫ • v) := by |
| have hd := smooth_unit_normal_derivative (hN.differentiable (by simp) a) (hb a ha).1 |
| have hD (u : Cover) : ⟪N a,fderiv ℝ N a u⟫=0 := by |
| rw [←ContinuousLinearMap.adjoint_inner_left,hd,inner_zero_left] |
| have hnv : ⟪N a,v⟫=0 := by simpa only [real_inner_comm] using hz a ha |
| have hk := isMinimizerAt_zero_component_shape_kernel hV hE hmin hμ hN hp hae hb hz ha |
| have hh := normal_unit_kernel_shape (hb a ha).1.self_of_nhds hv hnv hD |
| (isMinimizerAt_shape_tangent_symmetric hV hE hmin hμ hN hp hae a ha) hk hw |
| rw [←divergence_eq_coverTrace,(hb a ha).2] at hh |
| exact hh |
|
|
| theorem isMinimizerAt_zero_component_curvature {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| {H : ℝ} (hb : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) |
| {v : Cover} (hv : ‖v‖=1) |
| (hz : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ⟪v,N x⟫=0) |
| {a : Cover} (ha : quotientMap a∈frontier (openRepresentative E)) : |
| coverTrace ((fderiv ℝ N a).comp (fderiv ℝ N a))=H^2 := by |
| have hd := smooth_unit_normal_derivative (hN.differentiable (by simp) a) (hb a ha).1 |
| have hD (u : Cover) : ⟪N a,fderiv ℝ N a u⟫=0 := by |
| rw [←ContinuousLinearMap.adjoint_inner_left,hd,inner_zero_left] |
| have hnv : ⟪N a,v⟫=0 := by simpa only [real_inner_comm] using hz a ha |
| have hk := isMinimizerAt_zero_component_shape_kernel hV hE hmin hμ hN hp hae hb hz ha |
| have hh := normal_unit_kernel_trace_square (hb a ha).1.self_of_nhds hv hnv hD |
| (isMinimizerAt_shape_tangent_symmetric hV hE hmin hμ hN hp hae a ha) hk |
| rw [←divergence_eq_coverTrace,(hb a ha).2] at hh |
| exact hh |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def sibling_axialProjection (v : Cover) : Cover →L[ℝ] Cover := |
| ContinuousLinearMap.id ℝ Cover-(innerSL ℝ v).smulRight v |
|
|
| lemma sibling_axialProjection_apply (v w : Cover) : sibling_axialProjection v w=w-⟪v,w⟫ • v := rfl |
|
|
| lemma axialProjection_fixed {v w : Cover} (hw : ⟪v,w⟫=0) : sibling_axialProjection v w=w := by |
| rw [sibling_axialProjection_apply,hw,zero_smul,sub_zero] |
|
|
| lemma graphSlopeNormal_inner_fderiv {n : Cover} (hn : ‖n‖=1) |
| {f : Plane → ℝ} (hf : ContDiff ℝ (⊤ : ℕ∞) f) (y u : Plane) : |
| ⟪graphSlopeNormal n (gradient f y),fderiv ℝ (normalGraph n f) y u⟫=0 := by |
| have he : ⟪normalPlaneEmbedding n (gradient f y),normalPlaneEmbedding n u⟫=⟪gradient f y,u⟫ := |
| (normalPlaneLinearIsometry n).inner_map_map _ _ |
| have hen : ⟪normalPlaneEmbedding n (gradient f y),n⟫=0 := by |
| simpa only [real_inner_comm] using normalPlaneEmbedding_inner hn (gradient f y) |
| rw [normalGraph_fderiv_apply hf,graphSlopeNormal_weight hn,real_inner_smul_left, |
| inner_sub_left,inner_add_right,inner_add_right,real_inner_smul_right, |
| real_inner_smul_right,normalPlaneEmbedding_inner hn,hen,he, |
| real_inner_self_eq_norm_sq,hn,one_pow,mul_one,mul_zero,zero_add,add_zero, |
| sub_self,mul_zero] |
|
|
| theorem normalizedCMCCharts_axial_local_center {B : Set Torus} |
| {N : Cover → Cover} {H : ℝ} (hc : HasNormalizedCMCCharts B N H) |
| (hN : ContDiff ℝ (⊤ : ℕ∞) N) (hH : H≠0) {v : Cover} |
| (hs : ∀ a : Cover,quotientMap a∈B → ∀ w : Cover,⟪N a,w⟫=0 → |
| fderiv ℝ N a w=H • sibling_axialProjection v w) |
| {a : Cover} (ha : quotientMap a∈B) : |
| ∀ᶠ w in 𝓝 (0 : Cover),quotientMap (a+w)∈B → |
| sibling_axialProjection v (a+w)-H⁻¹ • N (a+w)=sibling_axialProjection v a-H⁻¹ • N a := by |
| obtain ⟨hn,f,hf,hf0,hfgrad,hg,hcov⟩ := hc a ha |
| let Φ : Plane → Cover := fun y ↦ a+normalGraph (N a) f y |
| have hΦ0 : Φ 0=a := by |
| simp only [Φ,normalGraph,hf0,zero_smul,add_zero] |
| have he : normalPlaneEmbedding (N a) 0=0 := norm_eq_zero.mp (by rw [normalPlaneEmbedding_norm,norm_zero]) |
| rw [he,add_zero] |
| have hΦ : Differentiable ℝ Φ := fun y ↦ |
| (normalGraph_hasFDerivAt (n := N a) (hf.differentiable (by simp) y)).differentiableAt.const_add a |
| let F : Plane → Cover := fun y ↦ sibling_axialProjection v (Φ y)-H⁻¹ • N (Φ y) |
| have hF : Differentiable ℝ F := |
| ((sibling_axialProjection v).differentiable.comp hΦ).sub |
| (((hN.differentiable (by simp)).comp hΦ).const_smul H⁻¹) |
| obtain ⟨r,hr,hgr⟩ := Metric.mem_nhds_iff.mp hg |
| have hd : ∀ y∈Metric.ball (0 : Plane) r,fderiv ℝ F y=0 := by |
| intro y hy |
| have hmem : quotientMap (Φ y)∈B := (hgr hy).2.1 |
| have hnorm : N (Φ y)=graphSlopeNormal (N a) (gradient f y) := (hgr hy).1 |
| have ht : ∀ u : Plane,⟪N (Φ y),fderiv ℝ Φ y u⟫=0 := by |
| intro u |
| rw [hnorm] |
| have he : fderiv ℝ Φ y=fderiv ℝ (normalGraph (N a) f) y := |
| fderiv_const_add a |
| rw [he] |
| exact graphSlopeNormal_inner_fderiv hn hf y u |
| have he := (((sibling_axialProjection v).hasFDerivAt.comp y (hΦ y).hasFDerivAt).sub |
| (((hN.differentiable (by simp) (Φ y)).hasFDerivAt.comp y |
| (hΦ y).hasFDerivAt).const_smul H⁻¹)).fderiv |
| change fderiv ℝ F y=_ at he |
| rw [he] |
| ext u : 1 |
| simp only [sub_apply,ContinuousLinearMap.comp_apply,smul_apply,zero_apply] |
| rw [hs (Φ y) hmem _ (ht u),smul_smul,inv_mul_cancel₀ hH,one_smul,sub_self] |
| have hconst (y : Plane) (hy : y∈Metric.ball (0 : Plane) r) : F y=F 0 := |
| (convex_ball (0 : Plane) r).is_const_of_fderivWithin_eq_zero hF.differentiableOn |
| (fun z hz ↦ by rw [fderivWithin_of_isOpen Metric.isOpen_ball hz]; exact hd z hz) |
| hy (Metric.mem_ball_self hr) |
| have hct : Tendsto (normalPlaneCoordinates (N a)) (𝓝 (0 : Cover)) (𝓝 (0 : Plane)) := by |
| simpa only [map_zero] using (normalPlaneCoordinates (N a)).continuous.tendsto (0 : Cover) |
| filter_upwards [hcov,hct.eventually (Metric.ball_mem_nhds (0 : Plane) hr)] with w hw hy hmem |
| have he : Φ (normalPlaneCoordinates (N a) w)=a+w := by |
| dsimp only [Φ] |
| rw [hw.mp hmem] |
| have hh := hconst _ hy |
| simpa only [F,he,hΦ0] using hh |
|
|
| theorem isMinimizerAt_zero_component_local_center {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| {H : ℝ} (hH : H≠0) |
| (hb : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) |
| {v : Cover} (hv : ‖v‖=1) |
| (hz : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ⟪v,N x⟫=0) |
| {a : Cover} (ha : quotientMap a∈frontier (openRepresentative E)) : |
| ∀ᶠ w in 𝓝 (0 : Cover),quotientMap (a+w)∈frontier (openRepresentative E) → |
| sibling_axialProjection v (a+w)-H⁻¹ • N (a+w)=sibling_axialProjection v a-H⁻¹ • N a := by |
| exact normalizedCMCCharts_axial_local_center |
| (isMinimizerAt_hasNormalizedCMCCharts hV hE hmin hμ hN hp hae hb) hN hH |
| (fun a ha w hw ↦ isMinimizerAt_zero_component_shape hV hE hmin hμ hN hp hae hb hv hz ha hw) ha |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| def axialCylinder (v c : Cover) (r : ℝ) : Set Cover := |
| {x | ‖sibling_axialProjection v x-c‖=r} |
|
|
| lemma sibling_axialProjection_inner {v n : Cover} (hnv : ⟪n,v⟫=0) (w : Cover) : |
| ⟪n,sibling_axialProjection v w⟫=⟪n,w⟫ := by |
| rw [sibling_axialProjection_apply,inner_sub_right,real_inner_smul_right,hnv,mul_zero,sub_zero] |
|
|
| lemma axial_normal_squared_distance_difference {n v : Cover} (hn : ‖n‖=1) |
| (hnv : ⟪n,v⟫=0) (g : Plane → ℝ) (c w : Cover) : |
| ‖sibling_axialProjection v w-c‖^2- |
| ‖sibling_axialProjection v (normalGraph n g (normalPlaneCoordinates n w))-c‖^2= |
| (⟪n,w⟫-g (normalPlaneCoordinates n w))* |
| (⟪n,w⟫+g (normalPlaneCoordinates n w)-2*⟪n,c⟫) := by |
| let q := normalPlaneEmbedding n (normalPlaneCoordinates n w) |
| have hq : ⟪n,sibling_axialProjection v q⟫=0 := by |
| rw [sibling_axialProjection_inner hnv] |
| exact normalPlaneEmbedding_inner hn _ |
| have hvn : ⟪v,n⟫=0 := by simpa only [real_inner_comm] using hnv |
| have hw : w=q+⟪n,w⟫ • n := (normalPlaneCoordinates_reconstruct hn w).symm |
| have hg : normalGraph n g (normalPlaneCoordinates n w)=q+g (normalPlaneCoordinates n w) • n := rfl |
| rw [hg,map_add,map_smul,axialProjection_fixed hvn] |
| conv_lhs => lhs; rw [hw,map_add,map_smul,axialProjection_fixed hvn] |
| simp only [norm_sub_sq_real,norm_add_sq_real,real_inner_smul_left, |
| real_inner_smul_right,inner_add_left,norm_smul,Real.norm_eq_abs, |
| sq_abs,hn,mul_one] |
| rw [←real_inner_comm (sibling_axialProjection v q) n,hq] |
| ring |
|
|
| lemma sibling_local_graph_cylinder_iff {n v : Cover} (hn : ‖n‖=1) (hnv : ⟪n,v⟫=0) |
| {g : Plane → ℝ} (hg : Continuous g) (hg0 : g 0=0) {c : Cover} {r : ℝ} |
| (hc : ⟪n,c⟫≠0) |
| (hs : ∀ᶠ y in 𝓝 (0 : Plane),normalGraph n g y∈axialCylinder v c r) : |
| ∀ᶠ w in 𝓝 (0 : Cover), |
| (normalGraph n g (normalPlaneCoordinates n w)=w ↔ w∈axialCylinder v c r) := by |
| have hct : Tendsto (normalPlaneCoordinates n) (𝓝 (0 : Cover)) (𝓝 (0 : Plane)) := by |
| simpa only [map_zero] using (normalPlaneCoordinates n).continuous.tendsto (0 : Cover) |
| have hnon : ∀ᶠ w in 𝓝 (0 : Cover), |
| ⟪n,w⟫+g (normalPlaneCoordinates n w)-2*⟪n,c⟫≠0 := by |
| have ht : ContinuousAt (fun w : Cover ↦ ⟪n,w⟫+g (normalPlaneCoordinates n w)-2*⟪n,c⟫) 0 := by fun_prop |
| apply ht.eventually_ne |
| simp only [inner_zero_right,map_zero,hg0,zero_add,zero_sub,neg_ne_zero,mul_ne_zero_iff] |
| exact ⟨by norm_num,hc⟩ |
| filter_upwards [hct.eventually hs,hnon] with w hw hnon |
| constructor |
| · intro he; rwa [←he] |
| · intro hws |
| have he : ‖sibling_axialProjection v w-c‖^2- |
| ‖sibling_axialProjection v (normalGraph n g (normalPlaneCoordinates n w))-c‖^2=0 := by |
| rw [show ‖sibling_axialProjection v w-c‖=r from hws, |
| show ‖sibling_axialProjection v (normalGraph n g (normalPlaneCoordinates n w))-c‖=r from hw,sub_self] |
| rw [axial_normal_squared_distance_difference hn hnv,mul_eq_zero] at he |
| exact (normalGraph_coordinates_eq_iff hn g w).mpr |
| ((sub_eq_zero.mp (he.resolve_right hnon)).symm) |
|
|
| theorem isMinimizerAt_zero_component_local_cylinder {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| {N : Cover → Cover} (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hp : ∀ m x,N (x+latticeVector m)=N x) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| {H : ℝ} (hH : H≠0) |
| (hb : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| (∀ᶠ y in 𝓝 x,‖N y‖=1) ∧ divergence N x=H) |
| {v : Cover} (hv : ‖v‖=1) |
| (hz : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → ⟪v,N x⟫=0) |
| {a : Cover} (ha : quotientMap a∈frontier (openRepresentative E)) : |
| ∀ᶠ w in 𝓝 (0 : Cover), |
| (quotientMap (a+w)∈frontier (openRepresentative E) ↔ |
| a+w∈axialCylinder v (sibling_axialProjection v a-H⁻¹ • N a) |H⁻¹|) ∧ |
| (quotientMap (a+w)∈frontier (openRepresentative E) → |
| sibling_axialProjection v (a+w)-H⁻¹ • N (a+w)=sibling_axialProjection v a-H⁻¹ • N a) := by |
| have hcharts := isMinimizerAt_hasNormalizedCMCCharts hV hE hmin hμ hN hp hae hb |
| obtain ⟨hn,f,hf,hf0,hfgrad,hg,hcov⟩ := hcharts a ha |
| have hloc := isMinimizerAt_zero_component_local_center hV hE hmin hμ hN hp hae hH hb hv hz ha |
| let c : Cover := -H⁻¹ • N a |
| have hcn : ⟪N a,c⟫≠0 := by |
| dsimp only [c] |
| rw [real_inner_smul_right,real_inner_self_eq_norm_sq,hn,one_pow,mul_one] |
| exact neg_ne_zero.mpr (inv_ne_zero hH) |
| have hnv : ⟪N a,v⟫=0 := by simpa only [real_inner_comm] using hz a ha |
| have hgzero : normalGraph (N a) f 0=0 := by |
| simp only [normalGraph,hf0,zero_smul,add_zero] |
| exact norm_eq_zero.mp (by rw [normalPlaneEmbedding_norm,norm_zero]) |
| have hgt : Tendsto (normalGraph (N a) f) (𝓝 (0 : Plane)) (𝓝 (0 : Cover)) := by |
| rw [←hgzero] |
| exact (((normalPlaneLinearIsometry (N a)).continuous).add (hf.continuous.smul continuous_const)).tendsto 0 |
| have hcs : ∀ᶠ y in 𝓝 (0 : Plane),normalGraph (N a) f y∈axialCylinder v c |H⁻¹| := by |
| filter_upwards [hg,hgt.eventually hloc] with y hy hyloc |
| have he := hyloc hy.2.1 |
| have he' : sibling_axialProjection v (normalGraph (N a) f y)-c=H⁻¹ • N (a+normalGraph (N a) f y) := by |
| rw [map_add] at he |
| dsimp only [c] |
| linear_combination (norm := module) he |
| change ‖sibling_axialProjection v (normalGraph (N a) f y)-c‖=|H⁻¹| |
| rw [he',norm_smul,Real.norm_eq_abs,(hb _ hy.2.1).1.self_of_nhds,mul_one] |
| have hiff := sibling_local_graph_cylinder_iff hn hnv hf.continuous hf0 hcn hcs |
| filter_upwards [hcov,hiff,hloc] with w hw hi hl |
| refine ⟨?_,hl⟩ |
| rw [hw,hi] |
| change (‖sibling_axialProjection v w-c‖=|H⁻¹| ↔ |
| ‖sibling_axialProjection v (a+w)-(sibling_axialProjection v a-H⁻¹ • N a)‖=|H⁻¹|) |
| rw [map_add] |
| have he : sibling_axialProjection v a+sibling_axialProjection v w-(sibling_axialProjection v a-H⁻¹ • N a)= |
| sibling_axialProjection v w-c := by dsimp only [c]; module |
| rw [he] |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| end CubicTorus |
| end |
| end |
| end |
|
|
| end OAI |
|
|
