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import OAI.Geometry.SmoothYau.Estimates.NormalFamilyPhase
import OAI.Geometry.SmoothYau.Geometry.ActualMetricSuperpositionSmallBall
namespace OAI
noncomputable section
namespace YauCounterexamples
section
open Set Filter Matrix
open scoped Topology ContDiff Matrix.Norms.Elementwise
def normalPhaseCovector (q : NormalWaveParameter) (z : Fin 3 → ℂ) : NormalWaveSpace →L[ℝ] ℂ :=
∑ i : Fin 3, ((ContinuousLinearMap.proj i : (Fin 3 → ℝ) →L[ℝ] ℝ).comp
(normalWaveEquiv.symm.toContinuousLinearMap.comp q.2.inverse)).smulRight (z i)
lemma normalPhaseCovector_apply (q : NormalWaveParameter) (z : Fin 3 → ℂ) (v : NormalWaveSpace) :
normalPhaseCovector q z v = ∑ i, ((normalWaveEquiv.symm (q.2.inverse v)) i : ℂ)*z i := by
simp [normalPhaseCovector,Complex.real_smul]
lemma normal_centered_fderiv_eq_inverse (g : SmoothMetric NormalWaveSpace NormalWaveSpace)
{K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K)
(ψ : NormalWaveSpace → NormalWaveSpace)
(hder : q.2.comp (fderiv ℝ ψ 0) = ContinuousLinearMap.id ℝ NormalWaveSpace) :
fderiv ℝ ψ 0 = q.2.inverse := by
obtain ⟨A,hA⟩ := metricFrameSet_equiv g hq
ext v : 1
have hd := congrArg (fun L : NormalWaveSpace →L[ℝ] NormalWaveSpace => L v) hder
change q.2 (fderiv ℝ ψ 0 v) = v at hd
rw [←hA,ContinuousLinearMap.inverse_equiv]
apply A.injective
have hdA : A (fderiv ℝ ψ 0 v) = v := by
simpa only [←hA, ContinuousLinearEquiv.coe_coe] using hd
simpa only [ContinuousLinearEquiv.coe_coe, A.apply_symm_apply] using hdA
lemma normalFamilyPhase_centered_fderiv (g : SmoothMetric NormalWaveSpace NormalWaveSpace)
(φ : NormalWaveSpace → ℝ)
(A : NormalWaveParameter × (Fin 3 → ℝ) → Matrix (Fin 3) (Fin 3) ℂ)
(hA : ContDiff ℝ ∞ A) {K : Set NormalWaveSpace} {q : NormalWaveParameter}
(hq : q ∈ metricFrameSet g K)
(hA0 : ∀ i j, A (q,0) i j = if i = j then 1 else 0)
(hAd : ∀ i j, fderiv ℝ (fun x => A (q,x) i j) 0 = 0)
(ψ : NormalWaveSpace → NormalWaveSpace) (hψ : ContDiff ℝ ∞ ψ) (hψ0 : ψ 0 = 0)
(hder : q.2.comp (fderiv ℝ ψ 0) = ContinuousLinearMap.id ℝ NormalWaveSpace)
(z : Fin 3 → ℂ) (Q : ComplexPhaseMatrix) (hn : ∑ i, z i*z i = -1)
{κ C : ℝ} (hQ : PhaseMatrixValid (actualHessianForm (normalWaveProfile g φ q)) z κ C Q)
(m D : ℕ) :
fderiv ℝ (fun t => normalFamilyPhase g φ A q z Q m D (normalWaveEquiv.symm (ψ t))) 0 =
normalPhaseCovector q z := by
let G := fun i j x => A (q,x) i j
let S := smoothPhasePolynomial G (normalWaveProfile g φ q 0 : ℂ) z Q ((2*D+3*m+6)+(D+m+1)+1)
have hG (i j) : ContDiff ℝ ∞ (G i j) :=
(contDiff_pi.mp (contDiff_pi.mp hA i) j).comp (contDiff_const.prodMk contDiff_id)
have hs := smoothPhasePolynomial_spec G hG hA0 hAd (normalWaveProfile g φ q 0 : ℂ)
z Q (phase_vector_ne_zero _ hn) (fun i j => congrFun (congrFun hQ.1 j) i) hn
(phase_matrix_annihilation Q z hQ.2.1) ((2*D+3*m+6)+(D+m+1)+1) (by omega)
have hs1 : ∀ i, MvPolynomial.constantCoeff (MvPolynomial.pderiv i S) = z i := hs.2.1
have hψ' := normal_centered_fderiv_eq_inverse g hq ψ hder
have hc := (contDiff_realPolyEval S).differentiable (by simp)
have hd := (hψ.differentiable (by simp) 0).hasFDerivAt
have hi := normalWaveEquiv.symm.hasFDerivAt.comp 0 hd
have hh := (hc (normalWaveEquiv.symm (ψ 0))).hasFDerivAt.comp 0 hi
change fderiv ℝ ((realPolyEval S) ∘ (normalWaveEquiv.symm ∘ ψ)) 0 = _
rw [hh.fderiv]
ext v
simp only [ContinuousLinearMap.comp_apply,normalPhaseCovector_apply,hψ0,map_zero,hψ',
ContinuousLinearEquiv.coe_coe,fderiv_realPolyEval,realPolyEval_at_zero,hs1]
end
open Set Filter
open scoped Topology ContDiff RealInnerProductSpace
lemma frame_inverse_apply (g : SmoothMetric NormalWaveSpace NormalWaveSpace)
{K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K)
(v : NormalWaveSpace) : q.2 (q.2.inverse v) = v := by
obtain ⟨A,hA⟩ := metricFrameSet_equiv g hq
rw [←hA,ContinuousLinearMap.inverse_equiv]
exact A.apply_symm_apply v
lemma normalPhaseCovector_re (q : NormalWaveParameter) (z : Fin 3 → ℂ)
(v : NormalWaveSpace) :
(normalPhaseCovector q z v).re = inner ℝ (q.2.inverse v) (phaseRealVector z) := by
rw [normalPhaseCovector_apply]
exact real_phase_linear_general z (q.2.inverse v)
lemma normalPhaseCovector_real_error (g : SmoothMetric NormalWaveSpace NormalWaveSpace)
{φ : NormalWaveSpace → ℝ} (hφ : ContDiff ℝ ∞ φ)
{K : Set NormalWaveSpace} {q : NormalWaveParameter} (hq : q ∈ metricFrameSet g K)
(z : Fin 3 → ℂ) :
‖Complex.reCLM.comp (normalPhaseCovector q z)-fderiv ℝ φ q.1‖ ≤
‖q.2.inverse‖*‖phaseRealVector z-gradient (normalWaveProfile g φ q) 0‖ := by
apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg (norm_nonneg _) (norm_nonneg _))
intro v
have hv : fderiv ℝ φ q.1 v =
inner ℝ (gradient (normalWaveProfile g φ q) 0) (q.2.inverse v) := by
rw [normalWaveProfile_gradient_pairing g hφ,frame_inverse_apply g hq]
change |(normalPhaseCovector q z v).re-fderiv ℝ φ q.1 v| ≤ _
rw [normalPhaseCovector_re,hv,real_inner_comm (q.2.inverse v) (gradient (normalWaveProfile g φ q) 0),←inner_sub_right]
calc
_ ≤ ‖q.2.inverse v‖*‖phaseRealVector z-gradient (normalWaveProfile g φ q) 0‖ :=
abs_real_inner_le_norm _ _
_ ≤ (‖q.2.inverse‖*‖v‖)*‖phaseRealVector z-gradient (normalWaveProfile g φ q) 0‖ :=
mul_le_mul_of_nonneg_right (q.2.inverse.le_opNorm v) (norm_nonneg _)
_ = _ := by ring
lemma compact_fderiv_displacement_bound {E F : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
(f : E → F) (hf : ContDiff ℝ ∞ f) {K : Set E} (hK : IsCompact K) :
∃ C : ℝ, 1 ≤ C ∧ ∀ q ∈ K, ∀ x : E, ‖x-q‖ ≤ 1 →
‖fderiv ℝ f x-fderiv ℝ f q‖ ≤ C*‖x-q‖ := by
let H := fun w : E × E => fderiv ℝ f (w.1+w.2)
have hH : ContDiff ℝ ∞ H := (hf.fderiv_right (m := ∞) (by simp)).comp
(contDiff_fst.add contDiff_snd)
obtain ⟨C,hC,hb⟩ := compact_fiber_jet_bound H hH (hK.prod (isCompact_closedBall 0 1)) 1
refine ⟨C,hC,?_⟩
intro q hq x hx
have hd (v) (hv : v ∈ Metric.closedBall (0 : E) 1) :
‖fderiv ℝ (fun t => H (q,t)) v‖ ≤ C := by
rw [←norm_iteratedFDeriv_one]
exact hb (q,v) ⟨hq,hv⟩ 1 le_rfl
have hu : ContDiff ℝ ∞ (fun t => H (q,t)) := hH.comp (contDiff_const.prodMk contDiff_id)
have hh := (convex_closedBall (0 : E) 1).norm_image_sub_le_of_norm_fderiv_le
(fun v _ => hu.differentiable (by simp) v) hd (Metric.mem_closedBall_self zero_le_one)
(show x-q ∈ Metric.closedBall (0 : E) 1 by simpa only [Metric.mem_closedBall,dist_zero_right] using hx)
simpa only [H,add_zero,add_sub_cancel,sub_zero] using hh
theorem compact_normal_real_slope_error (g : SmoothMetric NormalWaveSpace NormalWaveSpace)
{φ : NormalWaveSpace → ℝ} (hφ : ContDiff ℝ ∞ φ)
{K : Set NormalWaveSpace} (hK : IsCompact K) :
∃ C > 0, ∀ q ∈ metricFrameSet g K, ∀ z : Fin 3 → ℂ,
∀ x : NormalWaveSpace, ‖x-q.1‖ ≤ 1 →
‖Complex.reCLM.comp (normalPhaseCovector q z)-fderiv ℝ φ x‖ ≤
C*(‖phaseRealVector z-gradient (normalWaveProfile g φ q) 0‖+‖x-q.1‖) := by
obtain ⟨R,hR,hb⟩ := metricFrameSet_inverse_bound g hK
obtain ⟨D,hD,hd⟩ := compact_fderiv_displacement_bound φ hφ hK
refine ⟨R+D,by linarith,?_⟩
intro q hq z x hx
have h1 := (normalPhaseCovector_real_error g hφ hq z).trans
(mul_le_mul_of_nonneg_right (hb q hq) (norm_nonneg _))
have h2 := hd q.1 hq.1 x hx
rw [norm_sub_rev] at h2
have htriangle : ‖Complex.reCLM.comp (normalPhaseCovector q z)-fderiv ℝ φ x‖ ≤
‖Complex.reCLM.comp (normalPhaseCovector q z)-fderiv ℝ φ q.1‖ +
‖fderiv ℝ φ q.1-fderiv ℝ φ x‖ := by
simpa only [sub_add_sub_cancel] using norm_add_le
(Complex.reCLM.comp (normalPhaseCovector q z)-fderiv ℝ φ q.1)
(fderiv ℝ φ q.1-fderiv ℝ φ x)
have h := htriangle.trans (add_le_add h1 h2)
refine h.trans ?_
nlinarith [norm_nonneg (phaseRealVector z-gradient (normalWaveProfile g φ q) 0),
norm_nonneg (x-q.1)]
end YauCounterexamples
end
end OAI
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