| import Mathlib |
| import OAI.Geometry.SmoothYau.Estimates.ComplexMul |
| import OAI.Geometry.SmoothYau.Estimates.ExistsAnnularSimplePinnedFamily |
| import OAI.Geometry.SmoothYau.Smoothness.ChartTensorCoordLocal1 |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Function Manifold |
| open scoped Topology ContDiff BoundedContinuousFunction |
| variable {P E M : Type*} [TopologicalSpace P] |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
| [CompactSpace M] [T2Space M] |
| namespace CompactMetricAtlas |
| variable {g : SmoothMetric E M} {k : ℕ} {hs : Module.finrank ℝ E < 2*(2*(k:ℝ))} |
| variable (A : CompactMetricAtlas g k hs) |
| local instance persistence_smulComm (s : ℝ) : SMulCommClass ℝ ℝ (A.realH s) := |
| ⟨fun a b c => by simp only [smul_smul,mul_comm]⟩ |
| local instance persistence_scalarTower (s : ℝ) : IsScalarTower ℝ ℝ (A.realH s) := |
| ⟨fun a b c => mul_smul a b c⟩ |
|
|
| theorem actual_simple_eigenpair_persistence |
| (I : MetricIntegralAtlas (E:=E) (M:=M)) |
| (B : ∀ i, A.PatchCoefficients i) (α : ℝ) (hα0 : 0 < α) |
| (hα : ∀ i, 1 ≤ α*(A.localInverse i).radius^2) (he : A.errorBound B α ≤ 1/2) |
| (lam : ℝ) (hn : α+lam ≠ 0) |
| (u : M → ℝ) (hu : ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ u) (hu0 : u ≠ 0) |
| (hue : ∀ x, -laplaceBeltrami g u x = lam*u x) |
| (hsimple : ∀ v : M → ℝ, ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ v → |
| (∀ x, -laplaceBeltrami g v x = lam*v x) → ∃ c : ℝ, v = fun x => c*u x) |
| (q : P → SmoothMetric E M) (p₀ : P) (hq0 : q p₀ = g) |
| (hq : ∀ i j l, ContinuousSmoothFamilyOn |
| (fun t y => metricCoefficients (q t) (A.p i) y j l) (chartAt E (A.p i)).target) : |
| ∃ (e : P → ℝ) (w : P → A.realH (2*((k+1:ℕ):ℝ))), |
| e p₀ = lam ∧ A.realValue (2*((k+1:ℕ):ℝ)) (w p₀) = u ∧ |
| ContinuousAt e p₀ ∧ ContinuousAt w p₀ ∧ |
| ∀ᶠ p in 𝓝 p₀, |
| ContMDiff 𝓘(ℝ,E) 𝓘(ℝ,ℝ) ∞ (A.realValue (2*((k+1:ℕ):ℝ)) (w p)) ∧ |
| A.realValue (2*((k+1:ℕ):ℝ)) (w p) ≠ 0 ∧ |
| ∀ x, -laplaceBeltrami (q p) (A.realValue (2*((k+1:ℕ):ℝ)) (w p)) x = |
| e p*A.realValue (2*((k+1:ℕ):ℝ)) (w p) x := by |
| classical |
| let C : ∀ p i, A.VaryingPatchCoefficients (q p) i := fun p i => |
| (A.nonempty_varyingPatchCoefficients (q p) i).some |
| let C₀ : ∀ i, A.VaryingPatchCoefficients g i := hq0 ▸ C p₀ |
| have hc := A.continuousFamily_varyingLaplacian_tendsto q hq C p₀ |
| have hbase : A.varyingLaplacianCLM (q p₀) (C p₀) = A.varyingLaplacianCLM g C₀ := by |
| subst g |
| rfl |
| rw [hbase] at hc |
| obtain ⟨hunit,hRlim⟩ := A.varyingResolvent_tendsto B α hα0 hα he C₀ q C |
| (A.varyingShift_tendsto q g C C₀ α hc) |
| let R (p : P) := A.varyingResolvent B α hα0 hα he (q p) (C p) |
| let R₀ := A.realResolvent B α hα0 hα he |
| let T (p : P) := A.realInclusion ∘L R p |
| let T₀ := A.realSpectralResolvent B α hα0 hα he |
| have hR0 : R p₀ = R₀ := by |
| dsimp only [R,R₀] |
| subst g |
| exact A.varyingResolvent_base B α hα0 hα he (C p₀) |
| have hT0 : T p₀ = T₀ := congrArg (fun R => A.realInclusion ∘L R) hR0 |
| have hcomp : Continuous (fun R : A.realH (2*(k:ℝ)) →L[ℝ] A.realH (2*((k+1:ℕ):ℝ)) => |
| A.realInclusion ∘L R) := by fun_prop |
| have hTlim : Tendsto T (𝓝 p₀) (𝓝 T₀) := hcomp.tendsto R₀ |>.comp hRlim |
| let μ := (α+lam)⁻¹ |
| have hμ : μ ≠ 0 := inv_ne_zero hn |
| have hμv : μ⁻¹-α = lam := by dsimp [μ]; rw [inv_inv]; ring |
| let uz := A.realOfSmooth (2*(k:ℝ)) hs u hu |
| have huz : uz ≠ 0 := by |
| intro hz |
| apply hu0 |
| funext x |
| have hh := congrArg (fun z => A.realValue (2*(k:ℝ)) z x) hz |
| have hzv : A.realValue (2*(k:ℝ)) 0 x = 0 := by |
| simpa only [zero_smul,zero_mul] using A.realValue_smul (2*(k:ℝ)) 0 uz x |
| simpa only [uz,A.realOfSmooth_value,hzv,Pi.zero_apply] using hh |
| have hueT : T₀ uz = μ • uz := A.shiftedInverse_of_smooth_eigenfunction g α hα0 R₀ |
| (A.realResolvent_value B α hα0 hα he) lam hn u hu hue |
| have hsimpleT : ∀ z, T₀ z = μ • z → ∃ c : ℝ, z = c • uz := by |
| intro z hz |
| obtain ⟨hzs,hze⟩ := A.realSpectralResolvent_eigenfunction B α hα0 hα he hμ z hz |
| rw [hμv] at hze |
| obtain ⟨c,hc⟩ := hsimple _ hzs hze |
| refine ⟨c,?_⟩ |
| apply A.realRepresentative_injective (2*(k:ℝ)) hs |
| ext x |
| change A.realValue _ z x = A.realValue _ (c • uz) x |
| rw [A.realValue_smul,A.realOfSmooth_value] |
| exact congrFun hc x |
| obtain ⟨ev,v,hev,hv,hcev,hcv,hpair⟩ := (A.intrinsicPairing I (2*(k:ℝ)) hs).simple_eigenpair_persistence |
| T₀ (A.realSpectralResolvent_compact B α hα0 hα he) |
| (A.realSpectralResolvent_symmetric I B α hα0 hα he) hμ huz hueT hsimpleT |
| let e (p : P) := (ev (T p))⁻¹-α |
| let w (p : P) := (ev (T p))⁻¹ • R p (v (T p)) |
| have heT : Tendsto (fun p => ev (T p)) (𝓝 p₀) (𝓝 μ) := hev ▸ hcev.tendsto.comp hTlim |
| have hvT : Tendsto (fun p => v (T p)) (𝓝 p₀) (𝓝 uz) := hv ▸ hcv.tendsto.comp hTlim |
| have hinv : Tendsto (fun p => (ev (T p))⁻¹) (𝓝 p₀) (𝓝 μ⁻¹) := heT.inv₀ hμ |
| have hev0 : ev (T p₀) = μ := hT0 ▸ hev |
| have hv0 : v (T p₀) = uz := hT0 ▸ hv |
| have hw0 (x : M) : A.realValue (2*((k+1:ℕ):ℝ)) (w p₀) x = u x := by |
| dsimp only [w] |
| rw [hev0,hv0,hR0,A.realValue_smul,← A.realValue_inclusion] |
| change μ⁻¹*A.realValue (2*(k:ℝ)) (T₀ uz) x = u x |
| rw [hueT,A.realValue_smul,A.realOfSmooth_value,← mul_assoc,inv_mul_cancel₀ hμ,one_mul] |
| refine ⟨e,w,by simpa only [e,hev0] using hμv,funext hw0,?_,?_,?_⟩ |
| · change Tendsto e (𝓝 p₀) (𝓝 (e p₀)) |
| simpa only [e,hev0] using hinv.sub_const α |
| · have happly := (isBoundedBilinearMap_apply (𝕜:=ℝ) |
| (E:=A.realH (2*(k:ℝ))) (F:=A.realH (2*((k+1:ℕ):ℝ)))).continuous.tendsto (R₀,uz) |
| have hwlim := hinv.smul (happly.comp (hRlim.prodMk_nhds hvT)) |
| change Tendsto w (𝓝 p₀) (𝓝 (w p₀)) |
| simpa only [w,hev0,hv0,hR0,Function.comp_apply,R] using hwlim |
| · have hne : ∀ᶠ p in 𝓝 p₀, ev (T p) ≠ 0 := heT.eventually (isOpen_ne.mem_nhds hμ) |
| filter_upwards [hunit,hTlim.eventually hpair,hne] with p hp hpv hpev |
| have hshift := A.varyingResolvent_equation B α hα0 hα he (q p) (C p) hp |
| obtain ⟨hsm,heig⟩ := A.shiftedInverse_eigenfunction (q p) α (R p) hshift hpev (v (T p)) hpv.2.1 |
| have hval : A.realValue (2*((k+1:ℕ):ℝ)) (w p) = A.realValue (2*(k:ℝ)) (v (T p)) := by |
| funext x |
| rw [show w p = (ev (T p))⁻¹ • R p (v (T p)) from rfl,A.realValue_smul, ← A.realValue_inclusion] |
| change (ev (T p))⁻¹*A.realValue (2*(k:ℝ)) (T p (v (T p))) x = _ |
| rw [hpv.2.1,A.realValue_smul,← mul_assoc,inv_mul_cancel₀ hpev,one_mul] |
| rw [hval] |
| refine ⟨hsm,?_,heig⟩ |
| intro hz |
| apply hpv.1 |
| apply A.realRepresentative_injective (2*(k:ℝ)) hs |
| ext x |
| change A.realValue _ (v (T p)) x = A.realValue _ 0 x |
| rw [hz] |
| simpa only [zero_smul,zero_mul,Pi.zero_apply] using |
| (A.realValue_smul (2*(k:ℝ)) 0 (v (T p)) x).symm |
| end CompactMetricAtlas |
| end |
|
|
|
|
| open Set Filter Function Manifold Bundle TopologicalSpace |
| open scoped Topology ContDiff BoundedContinuousFunction ENNReal NNReal |
| variable {P : Type*} [TopologicalSpace P] |
| lemma spherical_simple_eigenpair_open (q : P → SmoothMetric (Euclidean 3) (Sphere 3)) |
| (hq : ∀ r i j, ContinuousSmoothFamilyOn (fun a y => metricCoefficients (q a) r y i j) |
| (chartAt (Euclidean 3) r).target) (a₀ : P) |
| (lam : ℝ) (hlam : 0 < lam) (u : Sphere 3 → ℝ) |
| (hu : ContMDiff 𝓘(ℝ,Euclidean 3) 𝓘(ℝ,ℝ) ∞ u) (hu0 : u ≠ 0) |
| (hue : ∀ x, -laplaceBeltrami (q a₀) u x = lam*u x) |
| (hsimple : ∀ v, ContMDiff 𝓘(ℝ,Euclidean 3) 𝓘(ℝ,ℝ) ∞ v → |
| (∀ x, -laplaceBeltrami (q a₀) v x = lam*v x) → ∃ c : ℝ, v = fun x => c*u x) |
| (hreg : ∀ x, u x=0 → fderiv ℝ (u ∘ (chartAt (Euclidean 3) x).symm) |
| ((chartAt (Euclidean 3) x) x) ≠ 0) |
| {J : Type*} [Fintype J] (Q : J → Set (Fin 3 → ℝ)) (d : J → ℝ) |
| (ψ : (Fin 3 → ℝ) → Sphere 3) (hψ : Continuous ψ) |
| (L : ℝ≥0∞) (hL : L < SignTests.signCertificate Q d (u ∘ ψ)) |
| {B D : ℝ} (hB : B < lam) (hD : lam < D) : |
| ∀ᶠ a in 𝓝 a₀, ∃ e : ℝ, ∃ v : Sphere 3 → ℝ, |
| B < e ∧ e < D ∧ ContMDiff 𝓘(ℝ,Euclidean 3) 𝓘(ℝ,ℝ) ∞ v ∧ v ≠ 0 ∧ |
| (∀ x, -laplaceBeltrami (q a) v x = e*v x) ∧ |
| (∀ x, v x=0 → fderiv ℝ (v ∘ (chartAt (Euclidean 3) x).symm) |
| ((chartAt (Euclidean 3) x) x) ≠ 0) ∧ |
| L < SignTests.signCertificate Q d (v ∘ ψ) := by |
| classical |
| have hs : Module.finrank ℝ (Euclidean 3) < 2*(2*(2:ℝ)) := by norm_num [Euclidean] |
| let A := (nonempty_compactMetricAtlas (q a₀) 2 hs).some |
| let I := (nonempty_metricIntegralAtlas (E:=Euclidean 3) (M:=Sphere 3)).some |
| let C : ∀ i, A.PatchCoefficients i := fun i => (A.nonempty_patchCoefficients i).some |
| obtain ⟨α,hα1,hα⟩ := A.exists_parametrix_threshold C |
| have hα0 : 0 < α := zero_lt_one.trans_le hα1 |
| obtain ⟨e,w,he,hw,hce,hcw,hpair⟩ := A.actual_simple_eigenpair_persistence I C α hα0 |
| (hα α le_rfl).1 (hα α le_rfl).2 lam (ne_of_gt (add_pos hα0 hlam)) |
| u hu hu0 hue hsimple q a₀ rfl (fun i => hq (A.p i)) |
| have hwr : ∀ᶠ a in 𝓝 a₀, ∀ x, A.realValue A.highIndex (w a) x=0 → |
| fderiv ℝ (A.realValue A.highIndex (w a) ∘ (chartAt (Euclidean 3) x).symm) |
| ((chartAt (Euclidean 3) x) x) ≠ 0 := |
| hcw.eventually (A.eventually_regular_realValue (w a₀) (by simpa only [hw] using hreg)) |
| let f (v : A.realH A.highIndex) : (Fin 3 → ℝ) → ℝ := A.realValue A.highIndex v ∘ ψ |
| have hf : LowerSemicontinuous (fun v => SignTests.signCertificate Q d (f v)) := by |
| apply SignTests.lowerSemicontinuous_signCertificate |
| · intro v |
| exact ((A.realRepresentative A.highIndex v).continuous.comp hψ).measurable |
| · intro y |
| exact (A.realRepresentative A.highIndex).continuous.eval_const (ψ y) |
| have hwl : ∀ᶠ a in 𝓝 a₀, L < SignTests.signCertificate Q d (f (w a)) := |
| hcw.eventually (hf (w a₀) L (by simpa only [f,hw] using hL)) |
| have heb : ∀ᶠ a in 𝓝 a₀, B < e a := hce.eventually (lt_mem_nhds (he ▸ hB)) |
| have hed : ∀ᶠ a in 𝓝 a₀, e a < D := hce.eventually (gt_mem_nhds (he ▸ hD)) |
| filter_upwards [hpair,hwr,hwl,heb,hed] with a ha hr hl hb hd |
| exact ⟨e a,A.realValue A.highIndex (w a),hb,hd,ha.1,ha.2.1,ha.2.2,hr,hl⟩ |
|
|
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
