| import Mathlib |
| import OAI.Geometry.SmoothYau.DifferentialEq.SphericalOpenFormNorm |
| import OAI.Geometry.SmoothYau.Geometry.ChartTensorNeighborhoodInst1 |
| import OAI.Geometry.SmoothYau.Geometry.FixedMetricDiagonal |
| import OAI.Geometry.SmoothYau.Smoothness.CompactSmoothJetMap |
| import OAI.Geometry.SmoothYau.Spectrum.InverseFrequencyWeightedZero |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace YauCounterexamples |
| section |
| open Set Filter Function Manifold Bundle TopologicalSpace BoxIntegral |
| open scoped Topology ContDiff Distributions ENNReal NNReal |
| local instance sphericalStageFormNorm : NormedAddCommGroup (MetricForm (Euclidean 3)) := inferInstanceAs (NormedAddCommGroup (Euclidean 3 →L[ℝ] Euclidean 3 →L[ℝ] ℝ)) |
| local instance sphericalStageFormSpace : NormedSpace ℝ (MetricForm (Euclidean 3)) := inferInstanceAs (NormedSpace ℝ (Euclidean 3 →L[ℝ] Euclidean 3 →L[ℝ] ℝ)) |
|
|
| theorem exists_spherical_simple_stage |
| (g₀ : SmoothMetric (Euclidean 3) (Sphere 3)) (hg₀ : IsRound g₀) |
| (g : SmoothMetric (Euclidean 3) (Sphere 3)) (hg : g ∈ sphericalProfileMetricNeighborhood) |
| {R t s r a b c : ℝ} (hR : (1/5:ℝ) ≤ R) (hRt : R<t) (hts : t<s) (hsr : s<r) |
| (hra : r<a) (hab : a<b) (hbc : b<c) (hc : c<3/10) |
| (hext : sphericalExterior g₀ g R) |
| (hmetric : ∀ y ∈ sphericalCoverage sourcePole sourceAxisOne sourceAxisTwo (3/10), |
| ∀ v : Euclidean 3, selfMetricFlat (sphereChartMetric g sourcePole) y v v ≤ ‖v‖^2) |
| (A : ℝ) (M : ℕ) (U : Set SphericalTensor) (hU : IsOpen U) |
| (hgU : sphericalTensorOf g₀ g (sphericalExterior_mono (by linarith) hext) ∈ U) : |
| ∃ k : ℕ, M ≤ k ∧ 3 ≤ k ∧ ∃ h : SmoothMetric (Euclidean 3) (Sphere 3), |
| ∃ he : sphericalExterior g₀ h c, |
| sphericalTensorOf g₀ h (sphericalExterior_mono hc.le he) ∈ U ∧ |
| ∃ u : Sphere 3 → ℝ, ContMDiff 𝓘(ℝ,Euclidean 3) 𝓘(ℝ,ℝ) ∞ u ∧ u ≠ 0 ∧ |
| (∀ q, -laplaceBeltrami h u q=sphereFrequency k*u q) ∧ |
| SphericalRegular u ∧ SphericalSimple h u (sphereFrequency k) ∧ |
| SphericalSignAbove u (A*(k:ℝ)) := by |
| classical |
| have hRc : R < c := hRt.trans (hts.trans (hsr.trans (hra.trans (hab.trans hbc)))) |
| have hRsmall : R≤3/10 := (hRc.trans hc).le |
| let eg := sphericalExterior_support hRsmall hext |
| let N : Set (SmoothMetric (Euclidean 3) (Sphere 3)) := {h | ∀ eh, |
| chartTensorOfMetric g₀ sourcePole sphericalTensorSupport (sphere_chart_target sourcePole) h eh ∈ U} |
| have hN : IsSmoothNeighborhood g N := chartTensorRepresentation_neighborhood g₀ sourcePole |
| sphericalTensorSupport (sphere_chart_target sourcePole) g eg U (hU.mem_nhds hgU) |
| obtain ⟨partition,hpart,ell,hell,ε,hε,hpacket⟩ := exists_spherical_exact_stage g₀ hg₀ g hg |
| (hR.trans hRt.le) hts hsr hra hab (hbc.trans hc) |
| (isClosed_le (sphericalRadius_continuous _ _) continuous_const) |
| (fun q hq => (show sphericalRadius sourceAxisOne sourceAxisTwo q≤R from hq).trans_lt hRt) |
| (fun q hq v w => congrArg (fun T => T v w) (hext q (lt_of_not_ge hq))) hmetric A N hN |
| obtain ⟨k,⟨hkpacket,hkM⟩,hk3⟩ := (hpacket.and (eventually_ge_atTop M) |>.and (eventually_ge_atTop 3)).exists |
| obtain ⟨ĝ,hĝN,u,hu,hue,hreg,hout,hscore⟩ := hkpacket |
| have hb0 : 0 < b := by linarith |
| have hc0 : 0 < c := by linarith |
| have heĝ : sphericalExterior g₀ ĝ b := by |
| intro q hq |
| ext v w |
| exact ((hout q hq).1 v w).trans (congrArg (fun T => T v w) (hext q (by linarith))) |
| let eĝ := sphericalExterior_support (hbc.trans hc).le heĝ |
| have hĝU := hĝN eĝ |
| have hu0 : u ≠ 0 := spherical_regular_nonzero u hreg |
| obtain ⟨h,he,hhU,hhe,hsimple⟩ := spherical_pin_near g₀ hg₀ ĝ hb0 hbc hc heĝ k hk3 u hu hu0 hue |
| (fun q hq => (hout q hq).2) U hU hĝU |
| refine ⟨k,hkM,hk3,h,he,hhU,u,hu,hu0,hhe,hreg,hsimple,partition,hpart,(fun j => ε/((k:ℝ)*ell j.val)),?_,hscore⟩ |
| intro j |
| exact div_pos hε (mul_pos (by exact_mod_cast (show 0<k by omega)) (hell j.val j.property)) |
| end |
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|
|
| open Set Filter Function Manifold Bundle TopologicalSpace BoxIntegral |
| open scoped Topology ContDiff Distributions ENNReal NNReal |
| local instance sphericalFixedFormNorm : NormedAddCommGroup (MetricForm (Euclidean 3)) := inferInstanceAs (NormedAddCommGroup (Euclidean 3 →L[ℝ] Euclidean 3 →L[ℝ] ℝ)) |
| local instance sphericalFixedFormSpace : NormedSpace ℝ (MetricForm (Euclidean 3)) := inferInstanceAs (NormedSpace ℝ (Euclidean 3 →L[ℝ] Euclidean 3 →L[ℝ] ℝ)) |
| local instance sphericalFixedFormComplete : CompleteSpace (MetricForm (Euclidean 3)) := by |
| let : ContinuousSMul ℝ (Euclidean 3) := IsBoundedSMul.continuousSMul |
| let : ContinuousSMul ℝ (Euclidean 3 →L[ℝ] ℝ) := IsBoundedSMul.continuousSMul |
| let : CompleteSpace (Euclidean 3 →L[ℝ] ℝ) := ContinuousLinearMap.instCompleteSpace |
| exact ContinuousLinearMap.instCompleteSpace |
|
|
| def sphericalStageRadius (n : ℕ) : ℝ := 3/10-1/(10*((n:ℝ)+1)) |
| lemma sphericalStageRadius_bounds (n : ℕ) : |
| 1/5 ≤ sphericalStageRadius n ∧ sphericalStageRadius n < sphericalStageRadius (n+1) ∧ |
| sphericalStageRadius (n+1)<3/10 := by |
| have hn : (0:ℝ) ≤ n := Nat.cast_nonneg n |
| have h1 : 0<10*((n:ℝ)+1) := by positivity |
| have h2 : 0<10*((n:ℝ)+1+1) := by positivity |
| dsimp [sphericalStageRadius] |
| rw [Nat.cast_add,Nat.cast_one] |
| constructor |
| · have hh : 1/(10*((n:ℝ)+1)) ≤ 1/10 := (div_le_div_iff₀ h1 (by norm_num)).mpr (by nlinarith) |
| linarith |
| constructor |
| · have hh : 1/(10*((n:ℝ)+1+1)) < 1/(10*((n:ℝ)+1)) := |
| (div_lt_div_iff₀ h2 h1).mpr (by linarith) |
| linarith |
| · have hh := div_pos (show (0:ℝ)<1 by norm_num) h2 |
| linarith |
|
|
| theorem exists_spherical_fixed_witnesses |
| (g₀ : SmoothMetric (Euclidean 3) (Sphere 3)) (hg₀ : IsRound g₀) |
| (N : Set (SmoothMetric (Euclidean 3) (Sphere 3))) (hN : IsSmoothNeighborhood g₀ N) : |
| ∃ g ∈ N, ∀ n : ℕ, SphericalLargeWitness g ((n:ℝ)+1) ((n:ℝ)+1) := by |
| classical |
| let : MetricSpace SphericalTensor := compactSmoothMetricSpace sphericalTensorSupport |
| let : CompleteSpace SphericalTensor := compactSmooth_complete sphericalTensorSupport |
| obtain ⟨O,hO,h0O,hON⟩ := exists_spherical_open_model g₀ hg₀ N hN |
| let P : ℕ → Set SphericalTensor := fun n => |
| {H | ∃ g : SmoothMetric (Euclidean 3) (Sphere 3), ∃ he : sphericalExterior g₀ g (sphericalStageRadius n), |
| H=sphericalTensorOf g₀ g (sphericalExterior_mono |
| ((sphericalStageRadius_bounds n).2.1.trans (sphericalStageRadius_bounds n).2.2).le he) ∧ H∈O} |
| let W : ℕ → Set SphericalTensor := fun n => |
| {H | ∃ hp : H∈sphericalPositiveSet g₀, |
| SphericalLargeWitness (sphericalTensorMetric g₀ ⟨H,hp⟩) ((n:ℝ)+1) ((n:ℝ)+1)} |
| have hstep : ∀ n, ∀ H ∈ P n, ∀ U : Set SphericalTensor, IsOpen U → H∈U → |
| ∃ K∈P (n+1), K∈U ∧ ∃ V : Set SphericalTensor, IsOpen V ∧ K∈V ∧ V⊆W n := by |
| intro n H hH U hU hHU |
| obtain ⟨g,he,rfl,hgO⟩ := hH |
| let e := sphericalExterior_mono |
| ((sphericalStageRadius_bounds n).2.1.trans (sphericalStageRadius_bounds n).2.2).le he |
| obtain ⟨hp,hgN,hgood⟩ := hON _ hgO |
| have hmetric : sphericalTensorMetric g₀ ⟨sphericalTensorOf g₀ g e,hp⟩=g := sphericalTensorMetric_of g₀ g e |
| rw [hmetric] at hgood |
| let R := sphericalStageRadius n |
| let C := sphericalStageRadius (n+1) |
| have hR := (sphericalStageRadius_bounds n).1 |
| have hRC : R<C := (sphericalStageRadius_bounds n).2.1 |
| have hC : C<3/10 := (sphericalStageRadius_bounds n).2.2 |
| obtain ⟨k,hkM,hk3,h,eh,hhU,u,hu,hu0,hue,hreg,hsimple,hscore⟩ := |
| exists_spherical_simple_stage g₀ hg₀ g hgood.1 (R:=R) |
| (t:=(5*R+C)/6) (s:=(4*R+2*C)/6) (r:=(3*R+3*C)/6) |
| (a:=(2*R+4*C)/6) (b:=(R+5*C)/6) (c:=C) hR |
| (by linarith) (by linarith) (by linarith) (by linarith) (by linarith) |
| (by linarith) hC he hgood.2 ((n:ℝ)+1) (n+3) (U∩O) (hU.inter hO) ⟨hHU,hgO⟩ |
| let eh' := sphericalExterior_mono hC.le eh |
| let K := sphericalTensorOf g₀ h eh' |
| let z : SphericalPositiveTensor g₀ := ⟨K,sphericalTensorOf_positive g₀ h eh'⟩ |
| have hqz : sphericalTensorMetric g₀ z=h := sphericalTensorMetric_of g₀ h eh' |
| have hkpos : (0:ℝ)<k := by exact_mod_cast (show 0<k by omega) |
| have hkreal : (n:ℝ)+3≤k := by exact_mod_cast hkM |
| have hk3real : (3:ℝ)≤k := by exact_mod_cast hk3 |
| have hepos : 0<sphereFrequency k := by dsimp [sphereFrequency]; positivity |
| have heB : (n:ℝ)+1<sphereFrequency k := by dsimp [sphereFrequency]; nlinarith |
| have heD : sphereFrequency k<2*(k:ℝ)^2 := by dsimp [sphereFrequency]; nlinarith |
| have hev := spherical_largeWitness_eventually (sphericalTensorMetric g₀) |
| (sphericalTensorMetric_family g₀) z k hk3 (sphereFrequency k) hepos u hu hu0 |
| (by simpa only [hqz] using hue) hreg (by simpa only [hqz] using hsimple) |
| ((n:ℝ)+1) ((n:ℝ)+1) heB heD hscore |
| obtain ⟨V,hV,hVo,hzV⟩ := mem_nhds_iff.mp hev |
| refine ⟨K,⟨h,eh,rfl,hhU.2⟩,hhU.1,Subtype.val '' V, |
| (sphericalPositiveSet_isOpen g₀).isOpenMap_subtype_val V hVo,⟨z,hzV,rfl⟩,?_⟩ |
| rintro H ⟨v,hv,rfl⟩ |
| exact ⟨v.property,hV hv⟩ |
| have hzP : (0:SphericalTensor)∈P 0 := by |
| refine ⟨g₀,(fun _ _ => rfl),?_,h0O⟩ |
| exact (sphericalTensorOf_zero g₀).symm |
| obtain ⟨H,hHO,hHW⟩ := fixed_metric_diagonal P W hstep 0 hzP hO h0O |
| obtain ⟨hp,hgN,hgood⟩ := hON H hHO |
| refine ⟨sphericalTensorMetric g₀ ⟨H,hp⟩,hgN,?_⟩ |
| intro n |
| exact (hHW n).choose_spec |
|
|
| end YauCounterexamples |
| end |
|
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| end OAI |
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