| import OAI.Geometry.Kahler.BaseRadialSubstitution |
|
|
| namespace OAI |
|
|
| open Complex |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise |
| open scoped ContDiff Matrix Matrix.Norms.Elementwise ComplexOrder |
| open scoped ContDiff ComplexOrder |
| open scoped ContDiff ENNReal |
| open Set Filter Topology |
| open scoped ContDiff |
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| noncomputable section |
|
|
| open Set Filter Topology MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| namespace PinchedHartogs.BaseConstruction |
|
|
| def adaptedHopf (p : Sphere) (t : unitInterval) (z : Circle × Circle) : Sphere := |
| sphereAction (adaptedIsometry p).symm (hopf (t,z)) |
|
|
| lemma adaptedHopf_continuous (p : Sphere) : Continuous (fun x : unitInterval × (Circle × Circle) => adaptedHopf p x.1 x.2) := |
| (sphereAction (adaptedIsometry p).symm).continuous.comp hopf_continuous |
|
|
| lemma adaptedHopf_integral {F : Sphere → ℂ} (hF : Continuous F) (p : Sphere) : |
| (∫ ξ, F ξ ∂sigma) = ∫ t : unitInterval, ∫ z, F (adaptedHopf p t z) ∂torusMeasure := by |
| rw [← sigma_integral_unitary (adaptedIsometry p).symm F] |
| exact hopf_integral_complex (hF.comp (sphereAction _).continuous) |
|
|
| lemma adaptedHopf_bracket (p : Sphere) (t : unitInterval) (z : Circle × Circle) : |
| bracket (adaptedHopf p t z : Base) (p:Base)=(z.1:ℂ)*(Real.sqrt (t:ℝ):ℂ) := by |
| rw [← adaptedIsometry_zero] |
| simp only [adaptedHopf,sphereAction_coe,LinearIsometryEquiv.apply_symm_apply,hopf,torusAction_zero,radialSphere_zero] |
|
|
| lemma adaptedHopf_bracket_norm (p : Sphere) (t : unitInterval) (z : Circle × Circle) : |
| ‖bracket (adaptedHopf p t z : Base) (p:Base)‖=Real.sqrt (t:ℝ) := by |
| rw [adaptedHopf_bracket,norm_mul,z.1.norm_coe,one_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (Real.sqrt_nonneg _)] |
|
|
| lemma adaptedHopf_phase (p : Sphere) (t : unitInterval) (ht : 0 < (t:ℝ)) (z : Circle × Circle) : |
| bracket (adaptedHopf p t z : Base) (p:Base)/(‖bracket (adaptedHopf p t z : Base) (p:Base)‖:ℂ)=(z.1:ℂ) := by |
| rw [adaptedHopf_bracket_norm,adaptedHopf_bracket,mul_div_cancel_right₀] |
| exact_mod_cast Real.sqrt_ne_zero'.mpr ht |
| |
| lemma adaptedHopf_height (k : ℕ) (p : Sphere) (t : unitInterval) (z : Circle × Circle) : |
| densityHeight k p (adaptedHopf p t z) = -(k:ℝ)/2*Real.log (t:ℝ) := by |
| rw [densityHeight,adaptedHopf_bracket_norm,Real.log_sqrt t.property.1] |
| ring |
| |
| lemma adaptedHopf_correction_expansion {W : Base → ℝ} (hW : Differentiable ℝ W) |
| (k : ℕ) (f b : ℝ → ℝ) (p : Sphere) (t : unitInterval) (ht : 0 < (t:ℝ)) |
| (s : Finset ℤ) (a : ℤ → ℂ) (he : ∀ z : Circle, (W ((z:ℂ) • (p:Base)):ℂ)=phaseSum s a z) |
| (z : Circle × Circle) : |
| densityCorrectionRaw k f b W p (adaptedHopf p t z) = |
| (∑ n ∈ s, ((t:ℝ)⁻¹:ℂ)*a n* |
| ((f (-(k:ℝ)/2*Real.log (t:ℝ))-(n:ℝ)/k*b (-(k:ℝ)/2*Real.log (t:ℝ))):ℂ)*phaseChar ((k:ℤ)+n) z.1).re := by |
| have hT := phaseDerivative_expansion hW p s a he z.1 |
| rw [densityCorrectionRaw_eq_re,adaptedHopf_phase p t ht z,adaptedHopf_bracket_norm,Real.sq_sqrt t.property.1,adaptedHopf_height] |
| have hcentral : centralPoint p (adaptedHopf p t z)=(z.1:ℂ) • (p:Base) := by |
| rw [centralPoint,adaptedHopf_phase p t ht z] |
| rw [hcentral] |
| push_cast |
| rw [he,hT] |
| unfold phaseSum |
| simp only [Finset.mul_sum] |
| rw [← Finset.sum_add_distrib,Finset.mul_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro n hn |
| rw [phaseChar_add] |
| simp only [phaseChar,Circle.coe_zpow,zpow_natCast] |
| push_cast |
| ring_nf |
| simp only [Complex.I_sq] |
| ring |
| |
| lemma phase_shifted_real_moment (s : Finset ℤ) (a : ℤ → ℂ) (k : ℤ) (N : ℕ) |
| (hs : ∀ n ∈ s, 0 < k+n) : |
| (∫ z : Circle, phaseChar N z*((∑ n ∈ s, a n*phaseChar (k+n) z).re:ℂ) ∂circleMeasure) = |
| if (N:ℤ)-k ∈ s then star (a ((N:ℤ)-k))/2 else 0 := by |
| classical |
| have he : ∀ z : Circle, phaseChar N z*((∑ n ∈ s, a n*phaseChar (k+n) z).re:ℂ) = |
| ∑ n ∈ s, phaseChar N z*((a n*phaseChar (k+n) z).re:ℂ) := by |
| intro z |
| simp only [Complex.re_sum,Complex.ofReal_sum,Finset.mul_sum] |
| simp_rw [he] |
| rw [MeasureTheory.integral_finsetSum s (fun n hn => compact_continuous_integrable (by |
| exact (phaseChar_continuous _).mul (Complex.continuous_ofReal.comp |
| (Complex.continuous_re.comp (continuous_const.mul (phaseChar_continuous _))))))] |
| have hsum : (∑ n ∈ s, ∫ z : Circle, phaseChar N z*((a n*phaseChar (k+n) z).re:ℂ) ∂circleMeasure) = |
| ∑ n ∈ s, if k+n=(N:ℤ) then star (a n)/2 else 0 := |
| Finset.sum_congr rfl (fun n hn => phase_real_moment _ _ _ (hs n hn)) |
| rw [hsum] |
| have hh : ∀ n : ℤ, k+n=(N:ℤ) ↔ n=(N:ℤ)-k := by intro n; omega |
| simp_rw [hh] |
| simp |
| |
| end PinchedHartogs.BaseConstruction |
| |
| end |
| |
| end OAI |
| |
