import Mathlib import OAI.Geometry.SmoothYau.Estimates.CanonicalCutoffWave import OAI.Geometry.SmoothYau.Geometry.WaveDerivCompLinear import OAI.Geometry.SmoothYau.Limits.FrameDualNorm import OAI.Geometry.SmoothYau.Smoothness.FlowInitialFderiv namespace OAI noncomputable section namespace YauCounterexamples section open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set open scoped Topology ContDiff NNReal variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] local instance normalFamilyVecNorm : NormedAddCommGroup (E →L[ℝ] E) := inferInstance local instance normalFamilyVecSpace : NormedSpace ℝ (E →L[ℝ] E) := inferInstance local instance normalFamilyBiVecNorm : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] E) := inferInstance local instance normalFamilyBiVecSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] E) := inferInstance local instance normalFamilyDualNorm : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance local instance normalFamilyDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance local instance normalFamilyFormNorm : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance local instance normalFamilyFormSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance lemma contDiff_frameChristoffel (g : SmoothMetric E E) : ContDiff ℝ ∞ (fun q : E × (E →L[ℝ] E) => (metricChristoffel g q.1).bilinearComp q.2 q.2) := by have h1 : ContDiff ℝ ∞ (fun q : E × (E →L[ℝ] E) => (metricChristoffel g q.1).comp q.2) := ((contDiff_metricChristoffel g).comp contDiff_fst).clm_comp contDiff_snd have h2 := (ContinuousLinearMap.flipₗᵢ ℝ E E E).toContinuousLinearMap.contDiff.comp h1 have h3 := h2.clm_comp contDiff_snd exact (ContinuousLinearMap.flipₗᵢ ℝ E E E).toContinuousLinearMap.contDiff.comp h3 lemma contDiff_normalJetFamily (g : SmoothMetric E E) : ContDiff ℝ ∞ (fun q : (E × (E →L[ℝ] E)) × E => normalJetMap q.1.1 q.1.2 ((metricChristoffel g q.1.1).bilinearComp q.1.2 q.1.2) q.2) := by have hC : ContDiff ℝ ∞ (fun q : (E × (E →L[ℝ] E)) × E => (metricChristoffel g q.1.1).bilinearComp q.1.2 q.1.2) := (contDiff_frameChristoffel g).comp contDiff_fst exact (contDiff_fst.fst.add (contDiff_fst.snd.clm_apply contDiff_snd)).sub ((hC.clm_apply contDiff_snd |>.clm_apply contDiff_snd).const_smul (1/2 : ℝ)) lemma contDiff_normalMetricFamily_apply (g : SmoothMetric E E) (v w : E) : ContDiff ℝ ∞ (fun q : (E × (E →L[ℝ] E)) × E => normalMetricTensor g q.1.1 q.1.2 q.2 v w) := by have hC : ContDiff ℝ ∞ (fun q : (E × (E →L[ℝ] E)) × E => ((metricChristoffel g q.1.1).bilinearComp q.1.2 q.1.2) q.2) := ((contDiff_frameChristoffel g).comp contDiff_fst).clm_apply contDiff_snd have hD := (show ContDiff ℝ ∞ (fun q : (E × (E →L[ℝ] E)) × E => q.1.2) from contDiff_fst.snd).sub hC exact (((contDiff_selfMetricFlat g).comp (contDiff_normalJetFamily g)).clm_apply (hD.clm_apply contDiff_const)).clm_apply (hD.clm_apply contDiff_const) theorem compact_normal_coordinate_radius [Nontrivial E] (g : SmoothMetric E E) {K : Set E} (hK : IsCompact K) : ∃ r : ℝ≥0, 0 < r ∧ ∀ q ∈ metricFrameSet g K, ∃ e : OpenPartialHomeomorph E E, (e : E → E) = normalJetMap q.1 q.2 ((metricChristoffel g q.1).bilinearComp q.2 q.2) ∧ e.source = Metric.ball 0 r ∧ ContDiffOn ℝ ∞ e.symm e.target := by obtain ⟨R,hR,hRb⟩ := metricFrameSet_inverse_bound g hK obtain ⟨B,hBb⟩ := (metricFrameSet_isCompact g hK).exists_bound_of_continuousOn (f := fun q : E × (E →L[ℝ] E) => (metricChristoffel g q.1).bilinearComp q.2 q.2) (contDiff_frameChristoffel g).continuous.continuousOn let r : ℝ≥0 := ⟨(2*(R+1)*(max B 0+1))⁻¹, by positivity⟩ have hr : 0 < r := by change 0 < (2*(R+1)*(max B 0+1))⁻¹; positivity refine ⟨r,hr,?_⟩ intro q hq obtain ⟨A,hA⟩ := metricFrameSet_equiv g hq let C := (metricChristoffel g q.1).bilinearComp q.2 q.2 have hAi : (A.symm : E →L[ℝ] E) = q.2.inverse := by rw [←hA, ContinuousLinearMap.inverse_equiv] have hAiPos : 0 < ‖(A.symm : E →L[ℝ] E)‖₊ := by rw [nnnorm_pos] intro hz obtain ⟨v,hv⟩ := exists_ne (0 : E) have hh := congrArg (fun T : E →L[ℝ] E => T (A v)) hz simp only [ContinuousLinearEquiv.coe_coe, A.symm_apply_apply, _root_.zero_apply] at hh exact hv hh have hb : ‖(A.symm : E →L[ℝ] E)‖ * (‖C‖ * (r : ℝ)) < 1 := by have hi : ‖(A.symm : E →L[ℝ] E)‖ ≤ R := by rw [hAi]; exact hRb q hq have hc : ‖C‖ ≤ max B 0 := (hBb q hq).trans (le_max_left _ _) have hm : ‖(A.symm : E →L[ℝ] E)‖ * (‖C‖ * (r : ℝ)) ≤ R * (max B 0 * (2*(R+1)*(max B 0+1))⁻¹) := by change ‖(A.symm : E →L[ℝ] E)‖ * (‖C‖ * (2*(R+1)*(max B 0+1))⁻¹) ≤ _ gcongr apply lt_of_le_of_lt hm rw [←mul_assoc, ←div_eq_mul_inv, div_lt_one (by positivity)] nlinarith [le_max_right B 0] have hsmall : ‖C‖₊ * r < ‖(A.symm : E →L[ℝ] E)‖₊⁻¹ := by have hnn : ‖(A.symm : E →L[ℝ] E)‖₊ * (‖C‖₊ * r) < 1 := by exact_mod_cast hb simpa only [mul_one] using (lt_inv_mul_iff₀ hAiPos).mpr hnn obtain ⟨e,he,hs,hi⟩ := normalJetMap_uniform_inverse q.1 A C (fun v w => metricChristoffel_symm g q.1 (q.2 v) (q.2 w)) r hsmall exact ⟨e, by simpa only [hA] using he, hs, hi⟩ end section open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff variable {P E F G : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup G] [NormedSpace ℝ G] lemma normal_iteratedFDeriv_comp_clm_at (L : F →L[ℝ] E) {f : E → G} {x : F} (n : ℕ) (hf : ContDiffAt ℝ n f (L x)) : iteratedFDeriv ℝ n (f ∘ L) x = (iteratedFDeriv ℝ n f (L x)).compContinuousLinearMap (fun _ => L) := by obtain ⟨U,hU,hu⟩ := hf.contDiffOn le_rfl (by simp) obtain ⟨V,hVU,hVo,hx⟩ := mem_nhds_iff.mp hU have he := L.iteratedFDerivWithin_comp_right (hu.mono hVU) hVo.uniqueDiffOn (hVo.preimage L.continuous).uniqueDiffOn hx (le_refl (n : WithTop ℕ∞)) rw [iteratedFDerivWithin_of_isOpen n hVo hx, iteratedFDerivWithin_of_isOpen n (hVo.preimage L.continuous) hx] at he exact he lemma parameter_iteratedFDeriv (f : P × E → F) (t : P) (x : E) (n : ℕ) (hf : ContDiffAt ℝ n f (t,x)) : iteratedFDeriv ℝ n (fun y => f (t,y)) x = (iteratedFDeriv ℝ n f (t,x)).compContinuousLinearMap (fun _ => ContinuousLinearMap.inr ℝ P E) := by let L := ContinuousLinearMap.inr ℝ P E let q : P × E → F := fun z => f ((t,0)+z) have hq : ContDiffAt ℝ n q (L x) := by have hg : ContDiffAt ℝ n (fun z : P × E => (t,0)+z) (L x) := by fun_prop exact (show ContDiffAt ℝ n f ((t,0)+L x) by simpa [L] using hf).comp (L x) hg have hj := normal_iteratedFDeriv_comp_clm_at L n hq simpa [q,L,Function.comp_def,iteratedFDeriv_comp_add_left] using hj lemma continuousAt_parameter_iteratedFDeriv (f : P × E → F) {z : P × E} (n : ℕ) (hf : ContDiffAt ℝ ∞ f z) : ContinuousAt (fun w : P × E => iteratedFDeriv ℝ n (fun y => f (w.1,y)) w.2) z := by have hc := (ContinuousMultilinearMap.continuous_precomp (fun _ : Fin n => ContinuousLinearMap.inr ℝ P E)).continuousAt.comp (hf.continuousAt_iteratedFDeriv (k := n) (ENat.natCast_le_of_coe_top_le_withTop le_rfl n)) apply hc.congr_of_eventuallyEq filter_upwards [(hf.of_le (show (n : WithTop ℕ∞) ≤ (∞ : WithTop ℕ∞) from ENat.natCast_le_of_coe_top_le_withTop le_rfl n)).eventually (by simp)] with w hw exact parameter_iteratedFDeriv f w.1 w.2 n hw end section open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff section ParameterDerivative variable {P E F : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] lemma parameter_fderiv_apply (f : P × E → F) (hf : Differentiable ℝ f) (p : P) (x v : E) : fderiv ℝ (fun y => f (p,y)) x v = fderiv ℝ f (p,x) (0,v) := by have h := (hf (p,x)).hasFDerivAt.comp x ((hasFDerivAt_const p x).prodMk (hasFDerivAt_id x)) exact congrArg (fun T => T v) h.fderiv lemma contDiff_parameter_fderiv_apply (f : P × E → F) (hf : ContDiff ℝ ∞ f) (v : E) : ContDiff ℝ ∞ (fun q : P × E => fderiv ℝ (fun x => f (q.1,x)) q.2 v) := by simp only [parameter_fderiv_apply f (hf.differentiable (by simp))] exact (hf.fderiv_right (by simp)).clm_apply contDiff_const lemma continuous_parameter_iteratedFDeriv (f : P × E → F) (hf : ContDiff ℝ ∞ f) (k : ℕ) : Continuous (fun q : P × E => iteratedFDeriv ℝ k (fun x => f (q.1,x)) q.2) := continuous_iff_continuousAt.mpr (fun _ => continuousAt_parameter_iteratedFDeriv f k hf.contDiffAt) end ParameterDerivative variable {P ι : Type*} [NormedAddCommGroup P] [NormedSpace ℝ P] [Fintype ι] [DecidableEq ι] lemma contDiff_matrix_det (A : P → Matrix ι ι ℝ) (hA : ∀ i j, ContDiff ℝ ∞ (fun p => A p i j)) : ContDiff ℝ ∞ (fun p => (A p).det) := by simp only [Matrix.det_apply'] apply ContDiff.sum intro σ _ apply contDiff_const.mul apply contDiff_prod intro i _ exact hA _ _ lemma contDiff_matrix_adjugate (A : P → Matrix ι ι ℝ) (hA : ∀ i j, ContDiff ℝ ∞ (fun p => A p i j)) (i j : ι) : ContDiff ℝ ∞ (fun p => (A p).adjugate i j) := by simp only [Matrix.adjugate_apply, Matrix.det_apply'] apply ContDiff.sum intro σ _ apply contDiff_const.mul apply contDiff_prod intro k _ simp only [Matrix.updateRow_apply] split_ifs · exact contDiff_const · exact hA _ _ lemma contDiffOn_matrix_inv (A : P → Matrix ι ι ℝ) (hA : ∀ i j, ContDiff ℝ ∞ (fun p => A p i j)) (i j : ι) : ContDiffOn ℝ ∞ (fun p => (A p)⁻¹ i j) {p | (A p).det ≠ 0} := by simp only [Matrix.inv_def, Ring.inverse_eq_inv, Matrix.smul_apply, smul_eq_mul] exact ((contDiff_matrix_det A hA).contDiffOn.inv (fun _ hp => hp)).mul (contDiff_matrix_adjugate A hA i j).contDiffOn end section open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Matrix open scoped Topology ContDiff Matrix Matrix.Norms.Elementwise abbrev NormalWaveSpace := EuclideanSpace ℝ (Fin 3) abbrev NormalWaveParameter := NormalWaveSpace × (NormalWaveSpace →L[ℝ] NormalWaveSpace) def normalWaveMetric (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) (x : NormalWaveSpace) : Matrix (Fin 3) (Fin 3) ℝ := fun i j => normalMetricTensor g q.1 q.2 x (EuclideanSpace.basisFun (Fin 3) ℝ i) (EuclideanSpace.basisFun (Fin 3) ℝ j) lemma contDiff_normalWaveMetric (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (i j : Fin 3) : ContDiff ℝ ∞ (fun q : NormalWaveParameter × NormalWaveSpace => normalWaveMetric g q.1 q.2 i j) := contDiff_normalMetricFamily_apply g _ _ lemma normalWaveMetric_zero (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K) : normalWaveMetric g q 0 = 1 := by ext i j change normalMetricTensor g q.1 q.2 0 _ _ = _ rw [normalMetricTensor_at_zero] change selfMetricFlat g q.1 (q.2 _) (q.2 _) = _ rw [hq.2] exact (EuclideanSpace.basisFun (Fin 3) ℝ).inner_eq_ite i j lemma normalWaveMetric_first_jet (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) : HasFDerivAt (𝕜 := ℝ) (normalWaveMetric g q) 0 0 := by have hc (i j : Fin 3) : HasFDerivAt (𝕜 := ℝ) (fun x => normalWaveMetric g q x i j) 0 0 := by have h := (contDiff_normalMetricTensor_apply g q.1 q.2 (EuclideanSpace.basisFun (Fin 3) ℝ i) (EuclideanSpace.basisFun (Fin 3) ℝ j)).differentiable (by simp) 0 simpa only [normalWaveMetric, normalMetricTensor_normal_jet] using h.hasFDerivAt convert (hasFDerivAt_pi.mpr (fun i => hasFDerivAt_pi.mpr (fun j => hc i j))) using 1 <;> rfl lemma hasFDerivAt_matrix_inv_zero {ι : Type*} [Fintype ι] [DecidableEq ι] {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] (A : E → Matrix ι ι ℝ) {x : E} (hA : HasFDerivAt (𝕜 := ℝ) A 0 x) (hdet : (A x).det ≠ 0) : HasFDerivAt (𝕜 := ℝ) (fun y => (A y)⁻¹) 0 x := by have hc (i j : ι) : ContDiffAt ℝ ∞ (fun M : Matrix ι ι ℝ => M⁻¹ i j) (A x) := by have hO : IsOpen {M : Matrix ι ι ℝ | M.det ≠ 0} := isOpen_ne_fun (contDiff_matrix_det id (fun _ _ => by fun_prop)).continuous continuous_const exact (contDiffOn_matrix_inv id (fun _ _ => by fun_prop) i j _ hdet).contDiffAt (hO.mem_nhds hdet) have hi : DifferentiableAt ℝ (fun M : Matrix ι ι ℝ => M⁻¹) (A x) := by apply differentiableAt_pi.mpr intro i apply differentiableAt_pi.mpr intro j exact (hc i j).differentiableAt (by simp) have h := hi.hasFDerivAt.comp x hA erw [ContinuousLinearMap.comp_zero] at h exact h lemma normalWaveInverse_first_jet (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K) : HasFDerivAt (𝕜 := ℝ) (fun x => (normalWaveMetric g q x)⁻¹) 0 0 := hasFDerivAt_matrix_inv_zero _ (normalWaveMetric_first_jet g q) (by rw [normalWaveMetric_zero g hq]; simp) theorem exists_normal_inverse_extension (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} (hK : IsCompact K) : ∃ A : NormalWaveParameter × NormalWaveSpace → Matrix (Fin 3) (Fin 3) ℝ, ∃ d : NormalWaveParameter × NormalWaveSpace → ℝ, ContDiff ℝ ∞ A ∧ ContDiff ℝ ∞ d ∧ ∃ O : Set (NormalWaveParameter × NormalWaveSpace), IsOpen O ∧ metricFrameSet g K ×ˢ {(0 : NormalWaveSpace)} ⊆ O ∧ ∀ q ∈ O, A q = (normalWaveMetric g q.1 q.2)⁻¹ ∧ d q = (normalWaveMetric g q.1 q.2).det⁻¹ := by let G := fun q : NormalWaveParameter × NormalWaveSpace => normalWaveMetric g q.1 q.2 let V := {q | (G q).det ≠ 0} have hG (i j : Fin 3) : ContDiff ℝ ∞ (fun q => G q i j) := contDiff_normalWaveMetric g i j have hV : IsOpen V := isOpen_ne_fun (contDiff_matrix_det G hG).continuous continuous_const have hc : IsCompact (metricFrameSet g K ×ˢ {(0 : NormalWaveSpace)}) := (metricFrameSet_isCompact g hK).prod isCompact_singleton have hsub : metricFrameSet g K ×ˢ {(0 : NormalWaveSpace)} ⊆ V := by rintro ⟨q,x⟩ ⟨hq,hx⟩ have hx0 : x = 0 := mem_singleton_iff.mp hx subst x change (normalWaveMetric g q 0).det ≠ 0 rw [normalWaveMetric_zero g hq] simp have hF : ContDiffOn ℝ ∞ (fun q => ((G q)⁻¹,(G q).det⁻¹)) V := by apply ContDiffOn.prodMk · apply contDiffOn_pi.mpr intro i apply contDiffOn_pi.mpr intro j exact contDiffOn_matrix_inv G hG i j · exact (contDiff_matrix_det G hG).contDiffOn.inv (fun _ h => h) obtain ⟨ψ,hψ,he⟩ := smooth_extension_near_compact hc hV hsub hF obtain ⟨O,hO,hKO,hEq⟩ := mem_nhdsSet_iff_exists.mp he refine ⟨fun q => (ψ q).1,fun q => (ψ q).2,hψ.fst,hψ.snd,O,hO,hKO,?_⟩ intro q hq have hh := hEq hq exact ⟨(congrArg Prod.fst hh).symm,(congrArg Prod.snd hh).symm⟩ def normalWaveFirstCoefficient (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) (x : NormalWaveSpace) (j : Fin 3) : ℝ := ∑ i, (fderiv ℝ (fun y => (normalWaveMetric g q y)⁻¹ i j) x (EuclideanSpace.basisFun (Fin 3) ℝ i) + (1/2 : ℝ) * (normalWaveMetric g q x)⁻¹ i j * (normalWaveMetric g q x).det⁻¹ * fderiv ℝ (fun y => (normalWaveMetric g q y).det) x (EuclideanSpace.basisFun (Fin 3) ℝ i)) lemma normalWaveMetric_det_first_jet (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) : HasFDerivAt (𝕜 := ℝ) (fun x => (normalWaveMetric g q x).det) 0 0 := by have hd := (contDiff_matrix_det (id : Matrix (Fin 3) (Fin 3) ℝ → _) (fun _ _ => by fun_prop)).differentiable (by simp) have h := (hd (normalWaveMetric g q 0)).hasFDerivAt.comp 0 (normalWaveMetric_first_jet g q) erw [ContinuousLinearMap.comp_zero] at h exact h lemma normalWaveFirstCoefficient_zero (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K) : normalWaveFirstCoefficient g q 0 = 0 := by let : ContinuousSMul ℝ NormalWaveSpace := IsBoundedSMul.continuousSMul funext j have hi := normalWaveInverse_first_jet g hq have hc (i j : Fin 3) : fderiv ℝ (fun y => (normalWaveMetric g q y)⁻¹ i j) 0 = 0 := by have h1 := (ContinuousLinearMap.proj i : Matrix (Fin 3) (Fin 3) ℝ →L[ℝ] (Fin 3 → ℝ)).hasFDerivAt.comp 0 hi have h2 := (ContinuousLinearMap.proj j : (Fin 3 → ℝ) →L[ℝ] ℝ).hasFDerivAt.comp 0 h1 have h := h2.fderiv erw [ContinuousLinearMap.comp_zero] at h exact h simp [normalWaveFirstCoefficient,hc,(normalWaveMetric_det_first_jet g q).fderiv] theorem exists_normal_wave_coefficients (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} (hK : IsCompact K) : ∃ A : NormalWaveParameter × NormalWaveSpace → Matrix (Fin 3) (Fin 3) ℝ, ∃ b : NormalWaveParameter × NormalWaveSpace → Fin 3 → ℝ, ContDiff ℝ ∞ A ∧ ContDiff ℝ ∞ b ∧ ∃ r > 0, ∀ q ∈ metricFrameSet g K, A (q,0) = 1 ∧ HasFDerivAt (𝕜 := ℝ) (fun x => A (q,x)) 0 0 ∧ b (q,0) = 0 ∧ ∀ x ∈ Metric.ball (0 : NormalWaveSpace) r, A (q,x) = (normalWaveMetric g q x)⁻¹ ∧ b (q,x) = normalWaveFirstCoefficient g q x := by obtain ⟨A,d,hA,hd,O,hO,hsub,he⟩ := exists_normal_inverse_extension g hK let b := fun q : NormalWaveParameter × NormalWaveSpace => fun j : Fin 3 => ∑ i, (fderiv ℝ (fun x => A (q.1,x) i j) q.2 (EuclideanSpace.basisFun (Fin 3) ℝ i) + (1/2 : ℝ) * A q i j * d q * fderiv ℝ (fun x => (normalWaveMetric g q.1 x).det) q.2 (EuclideanSpace.basisFun (Fin 3) ℝ i)) have hb : ContDiff ℝ ∞ b := by apply contDiff_pi.mpr intro j dsimp only [b] apply ContDiff.sum intro i _ apply ContDiff.add · exact contDiff_parameter_fderiv_apply (fun q => A q i j) (contDiff_pi.mp (contDiff_pi.mp hA i) j) _ · exact ((contDiff_const.mul (contDiff_pi.mp (contDiff_pi.mp hA i) j)).mul hd).mul (contDiff_parameter_fderiv_apply _ (contDiff_matrix_det _ (contDiff_normalWaveMetric g)) _) have hgerm (q : NormalWaveParameter) (x : NormalWaveSpace) (hx : (q,x) ∈ O) : (fun y => A (q,y)) =ᶠ[𝓝 x] (fun y => (normalWaveMetric g q y)⁻¹) := by have ht : Tendsto (fun y : NormalWaveSpace => (q,y)) (𝓝 x) (𝓝 (q,x)) := (continuous_const.prodMk continuous_id).continuousAt filter_upwards [ht (hO.mem_nhds hx)] with y hy exact (he (q,y) hy).1 have hbe (q : NormalWaveParameter) (x : NormalWaveSpace) (hx : (q,x) ∈ O) : b (q,x) = normalWaveFirstCoefficient g q x := by funext j dsimp only [b,normalWaveFirstCoefficient] apply Finset.sum_congr rfl intro i _ have hg : (fun y => A (q,y) i j) =ᶠ[𝓝 x] (fun y => (normalWaveMetric g q y)⁻¹ i j) := (hgerm q x hx).fun_comp (fun M => M i j) rw [hg.fderiv_eq,(he (q,x) hx).1,(he (q,x) hx).2] obtain ⟨U,V,hU,hV,hKU,h0V,hUV⟩ := generalized_tube_lemma (metricFrameSet_isCompact g hK) isCompact_singleton hO hsub obtain ⟨r,hr,hrV⟩ := Metric.isOpen_iff.mp hV 0 (h0V (mem_singleton 0)) refine ⟨A,b,hA,hb,r,hr,?_⟩ intro q hq have hq0 : (q,0) ∈ O := hsub ⟨hq,mem_singleton 0⟩ refine ⟨?_,?_,?_,?_⟩ · rw [(he (q,0) hq0).1,normalWaveMetric_zero g hq]; simp · exact (normalWaveInverse_first_jet g hq).congr_of_eventuallyEq (hgerm q 0 hq0) · rw [hbe q 0 hq0,normalWaveFirstCoefficient_zero g hq] · intro x hx have hqx : (q,x) ∈ O := hUV ⟨hKU hq,hrV hx⟩ exact ⟨(he (q,x) hqx).1,hbe q x hqx⟩ end section open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Matrix open scoped Topology ContDiff Matrix.Norms.Elementwise section Density variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] lemma density_first_derivative {d a : E → ℝ} {x : E} (hd : DifferentiableAt ℝ d x) (ha : DifferentiableAt ℝ a x) (hx : 0 < d x) (v : E) : (Real.sqrt (d x))⁻¹ * fderiv ℝ (fun y => Real.sqrt (d y) * a y) x v = fderiv ℝ a x v + (1/2 : ℝ) * a x * (d x)⁻¹ * fderiv ℝ d x v := by erw [fderiv_fun_mul (hd.sqrt hx.ne') ha, _root_.add_apply] simp only [_root_.smul_apply, smul_eq_mul, fderiv_sqrt hd hx.ne'] have hs : Real.sqrt (d x) ≠ 0 := (Real.sqrt_pos.mpr hx).ne' have hs2 : Real.sqrt (d x) ^ 2 = d x := Real.sq_sqrt hx.le field_simp rw [hs2] ring end Density lemma normalWaveDensity_firstCoefficient (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) {x : NormalWaveSpace} (hx : 0 < (normalWaveMetric g q x).det) (j : Fin 3) : (Real.sqrt (normalWaveMetric g q x).det)⁻¹ * ∑ i, fderiv ℝ (fun y => Real.sqrt (normalWaveMetric g q y).det * (normalWaveMetric g q y)⁻¹ i j) x (EuclideanSpace.basisFun (Fin 3) ℝ i) = normalWaveFirstCoefficient g q x j := by rw [Finset.mul_sum] apply Finset.sum_congr rfl intro i hi have hG (i j : Fin 3) : ContDiff ℝ ∞ (fun y => normalWaveMetric g q y i j) := (contDiff_normalMetricTensor_apply g q.1 q.2 (EuclideanSpace.basisFun (Fin 3) ℝ i) (EuclideanSpace.basisFun (Fin 3) ℝ j)) have hd := (contDiff_matrix_det _ hG).differentiable (by simp) x have hO : IsOpen {y : NormalWaveSpace | (normalWaveMetric g q y).det ≠ 0} := isOpen_ne_fun (contDiff_matrix_det _ hG).continuous continuous_const have ha := (contDiffOn_matrix_inv (normalWaveMetric g q) hG i j x hx.ne').contDiffAt (hO.mem_nhds hx.ne') exact density_first_derivative hd (ha.differentiableAt (by simp)) hx _ lemma compact_family_positive_radius {P E : Type*} [TopologicalSpace P] [NormedAddCommGroup E] {K : Set P} (hK : IsCompact K) (f : P × E → ℝ) (hf : Continuous f) (hpos : ∀ p ∈ K, 0 < f (p,0)) : ∃ r > 0, ∀ p ∈ K, ∀ x ∈ Metric.ball (0 : E) r, 0 < f (p,x) := by let O : Set (P × E) := {q | 0 < f q} have hO : IsOpen O := isOpen_lt continuous_const hf have hsub : K ×ˢ {(0 : E)} ⊆ O := by rintro ⟨p,x⟩ ⟨hp,hx⟩ change 0 < f (p,x) have hx0 : x = 0 := hx rw [hx0] exact hpos p hp obtain ⟨U,V,hU,hV,hKU,h0V,hUV⟩ := generalized_tube_lemma hK isCompact_singleton hO hsub obtain ⟨r,hr,hrV⟩ := Metric.isOpen_iff.mp hV 0 (h0V (mem_singleton 0)) exact ⟨r,hr,fun p hp x hx => hUV ⟨hKU hp,hrV hx⟩⟩ lemma compact_normal_density_radius (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} (hK : IsCompact K) : ∃ r > 0, ∀ q ∈ metricFrameSet g K, ∀ x ∈ Metric.ball (0 : NormalWaveSpace) r, 0 < (normalWaveMetric g q x).det := by apply compact_family_positive_radius (metricFrameSet_isCompact g hK) (fun q : NormalWaveParameter × NormalWaveSpace => (normalWaveMetric g q.1 q.2).det) (contDiff_matrix_det _ (contDiff_normalWaveMetric g)).continuous intro q hq rw [normalWaveMetric_zero g hq] simp end section open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter Matrix open scoped Topology ContDiff Matrix.Norms.Elementwise abbrev normalWaveEquiv : (Fin 3 → ℝ) ≃L[ℝ] NormalWaveSpace := (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin 3 => ℝ)).symm lemma normalWaveEquiv_single (i : Fin 3) : normalWaveEquiv (Pi.single i 1) = EuclideanSpace.basisFun (Fin 3) ℝ i := by rw [EuclideanSpace.basisFun_apply] rfl theorem exists_normal_wave_pi_coefficients (g : SmoothMetric NormalWaveSpace NormalWaveSpace) {K : Set NormalWaveSpace} (hK : IsCompact K) : ∃ A : NormalWaveParameter × (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℂ, ∃ b : NormalWaveParameter × (Fin 3 → ℝ) → Fin 3 → ℂ, ContDiff ℝ ∞ A ∧ ContDiff ℝ ∞ b ∧ ∃ r > 0, ∀ q ∈ metricFrameSet g K, (∀ i j, A (q,0) i j = if i = j then 1 else 0) ∧ (∀ i j, fderiv ℝ (fun x => A (q,x) i j) 0 = 0) ∧ b (q,0) = 0 ∧ ∀ x, ‖normalWaveEquiv x‖ < r → (∀ i j, A (q,x) i j = ((normalWaveMetric g q (normalWaveEquiv x))⁻¹ i j : ℂ)) ∧ (∀ j, b (q,x) j = (normalWaveFirstCoefficient g q (normalWaveEquiv x) j : ℂ)) := by obtain ⟨A,b,hA,hb,r,hr,he⟩ := exists_normal_wave_coefficients g hK let Ap := fun q : NormalWaveParameter × (Fin 3 → ℝ) => fun i j : Fin 3 => (A (q.1,normalWaveEquiv q.2) i j : ℂ) let bp := fun q : NormalWaveParameter × (Fin 3 → ℝ) => fun j : Fin 3 => (b (q.1,normalWaveEquiv q.2) j : ℂ) have hT : ContDiff ℝ ∞ (fun q : NormalWaveParameter × (Fin 3 → ℝ) => (q.1,normalWaveEquiv q.2)) := contDiff_fst.prodMk (normalWaveEquiv.contDiff.comp contDiff_snd) have hAp : ContDiff ℝ ∞ Ap := by apply contDiff_pi.mpr intro i apply contDiff_pi.mpr intro j exact Complex.ofRealCLM.contDiff.comp ((contDiff_pi.mp (contDiff_pi.mp hA i) j).comp hT) have hbp : ContDiff ℝ ∞ bp := by apply contDiff_pi.mpr intro j exact Complex.ofRealCLM.contDiff.comp ((contDiff_pi.mp hb j).comp hT) refine ⟨Ap,bp,hAp,hbp,r,hr,?_⟩ intro q hq obtain ⟨h0,hd,hb0,heq⟩ := he q hq refine ⟨?_,?_,?_,?_⟩ · intro i j simp only [Ap,map_zero,h0,Matrix.one_apply] split_ifs <;> simp · intro i j let L : Matrix (Fin 3) (Fin 3) ℝ →L[ℝ] ℂ := Complex.ofRealCLM.comp ((ContinuousLinearMap.proj j : (Fin 3 → ℝ) →L[ℝ] ℝ).comp (ContinuousLinearMap.proj i : Matrix (Fin 3) (Fin 3) ℝ →L[ℝ] (Fin 3 → ℝ))) have hf := (L.hasFDerivAt.comp 0 hd).comp 0 normalWaveEquiv.hasFDerivAt have hf' : HasFDerivAt (fun x => Ap (q,x) i j) ((L.comp 0).comp normalWaveEquiv.toContinuousLinearMap) 0 := hf simpa only [ContinuousLinearMap.comp_zero,ContinuousLinearMap.zero_comp] using hf'.fderiv · ext j simp [bp,hb0] · intro x hx have h := heq (normalWaveEquiv x) (by simpa only [Metric.mem_ball,dist_zero_right] using hx) exact ⟨fun i j => by dsimp [Ap]; rw [h.1],fun j => by dsimp [bp]; rw [h.2]⟩ lemma normal_profile_family_smooth (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (φ : NormalWaveSpace → ℝ) (hφ : ContDiff ℝ ∞ φ) : ContDiff ℝ ∞ (fun q : NormalWaveParameter × NormalWaveSpace => φ (normalJetMap q.1.1 q.1.2 ((metricChristoffel g q.1.1).bilinearComp q.1.2 q.1.2) q.2)) := hφ.comp (contDiff_normalJetFamily g) end open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open MvPolynomial open Set Filter open scoped ContDiff open Set Filter open scoped Topology ContDiff open Set Filter MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter Function MvPolynomial open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open Set Filter Function open scoped Topology ContDiff open scoped Topology open Set Filter Manifold Bundle MeasureTheory open scoped Topology ContDiff ENNReal open Matrix open scoped Topology Matrix.Norms.Elementwise open Set Filter Manifold Bundle open scoped Topology ContDiff open Set Filter open scoped Topology ContDiff lemma contDiff_smoothFiniteWave {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {S : E → ℂ} {V : ℕ → E → ℂ} (hS : ContDiff ℝ ∞ S) (hV : ∀ j, ContDiff ℝ ∞ (V j)) (J : ℕ) (n : ℝ) : ContDiff ℝ ∞ (smoothFiniteWave S V J n) := by unfold smoothFiniteWave exact (contDiff_const.mul hS).cexp.mul (ContDiff.sum (fun j _ => contDiff_const.mul (hV j))) lemma contDiff_canonicalCutoffWave (A : Fin 3 → Fin 3 → (Fin 3 → ℝ) → ℂ) (b : Fin 3 → (Fin 3 → ℝ) → ℂ) (s : ℂ) (z : Fin 3 → ℂ) (Q : ComplexPhaseMatrix) (ζ : (Fin 3 → ℝ) → ℂ) (hζ : ContDiff ℝ ∞ ζ) (m D : ℕ) (n : ℝ) : ContDiff ℝ ∞ (canonicalCutoffWave A b s z Q ζ m D n) := contDiff_smoothFiniteWave (contDiff_realPolyEval _) (fun _ => hζ.mul (contDiff_realPolyEval _)) _ _ def normalWavePiOperator (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) (f : (Fin 3 → ℝ) → ℂ) (x : Fin 3 → ℝ) : ℂ := waveCoordinateOperator (EuclideanSpace.basisFun (Fin 3) ℝ) (fun i j y => ((normalWaveMetric g q y)⁻¹ i j : ℂ)) (fun j y => (normalWaveFirstCoefficient g q y j : ℂ)) (f ∘ normalWaveEquiv.symm) (normalWaveEquiv x) lemma normalWavePiOperator_eq (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) {f : (Fin 3 → ℝ) → ℂ} (hf : ContDiff ℝ ∞ f) (x : Fin 3 → ℝ) : normalWavePiOperator g q f x = waveCoordinateOperator (fun i => Pi.single i 1) (fun i j y => ((normalWaveMetric g q (normalWaveEquiv y))⁻¹ i j : ℂ)) (fun j y => (normalWaveFirstCoefficient g q (normalWaveEquiv y) j : ℂ)) f x := by have h := waveCoordinateOperator_comp_linear normalWaveEquiv.toContinuousLinearMap (fun i => Pi.single i 1) (fun i j y => ((normalWaveMetric g q y)⁻¹ i j : ℂ)) (fun j y => (normalWaveFirstCoefficient g q y j : ℂ)) (hf.comp normalWaveEquiv.symm.contDiff) x simpa only [normalWavePiOperator,Function.comp_def,ContinuousLinearEquiv.coe_coe, normalWaveEquiv_single,ContinuousLinearEquiv.symm_apply_apply] using h.symm lemma normalWavePiOperator_eq_extension (g : SmoothMetric NormalWaveSpace NormalWaveSpace) (q : NormalWaveParameter) {f : (Fin 3 → ℝ) → ℂ} (hf : ContDiff ℝ ∞ f) (A : Fin 3 → Fin 3 → (Fin 3 → ℝ) → ℂ) (b : Fin 3 → (Fin 3 → ℝ) → ℂ) (x : Fin 3 → ℝ) (hA : ∀ i j, A i j x = ((normalWaveMetric g q (normalWaveEquiv x))⁻¹ i j : ℂ)) (hb : ∀ j, b j x = (normalWaveFirstCoefficient g q (normalWaveEquiv x) j : ℂ)) : normalWavePiOperator g q f x = waveCoordinateOperator (fun i => Pi.single i 1) A b f x := by rw [normalWavePiOperator_eq g q hf] simp only [waveCoordinateOperator,hA,hb] end YauCounterexamples end end OAI