| import OAI.Geometry.Kahler.BaseChartUniform |
| |
| 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 MeasureTheory |
| open scoped ContDiff ENNReal Pointwise |
| open Set Filter Topology |
| open scoped ContDiff |
| noncomputable section |
|
|
| open Set Filter Topology |
| open scoped ContDiff |
| namespace PinchedHartogs.BaseConstruction |
| |
| lemma iteratedFDeriv_add_mul {f g : Base → ℝ} {z : Base} |
| (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (L : ℝ) (n : ℕ) : |
| iteratedFDeriv ℝ n (fun x => f x+L*g x) z = |
| iteratedFDeriv ℝ n f z + L • iteratedFDeriv ℝ n g z := by |
| have hfn : ContDiffAt ℝ n f z := hf.of_le (by exact_mod_cast (le_top : (n:ℕ∞) ≤ ⊤)) |
| have hgn : ContDiffAt ℝ n g z := hg.of_le (by exact_mod_cast (le_top : (n:ℕ∞) ≤ ⊤)) |
| change iteratedFDeriv ℝ n (f + L • g) z = _ |
| rw [iteratedFDeriv_add_apply hfn (by simpa only [Pi.smul_def,smul_eq_mul] using contDiffAt_const.mul hgn), |
| iteratedFDeriv_const_smul_apply hgn] |
|
|
| lemma baseHessian_add_mul {f g : Base → ℝ} {z : Base} |
| (hf : ContDiffAt ℝ ∞ f z) (hg : ContDiffAt ℝ ∞ g z) (L : ℝ) (i j : Fin 2) : |
| baseHessian (fun x => f x+L*g x) z i j = baseHessian f z i j + (L:ℂ)*baseHessian g z i j := by |
| rw [baseHessian_eq_jet (hf.add (contDiffAt_const.mul hg)),iteratedFDeriv_add_mul hf hg, |
| map_add,map_smul,← baseHessian_eq_jet hf,← baseHessian_eq_jet hg] |
| simp only [Complex.real_smul] |
|
|
| lemma psi_baseHessian_chart_eventually {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) |
| (U : Base ≃ₗᵢ[ℂ] Base) : |
| baseHessian (psi ∘ centeredChart r U) =ᶠ[𝓝 (0:Base)] baseHessian psi := by |
| have he := (psi_chart_hessian_eventually hr hr1 U (0:ℂ)).comp_tendsto |
| (continuousAt_id.prodMk continuousAt_const : ContinuousAt (fun z : Base => (z,(0:ℂ))) 0).tendsto |
| filter_upwards [he,isOpen_ball.mem_nhds zero_mem_ball] with z hz hzb |
| have hψ := psi_contDiffAt hzb |
| have hf := (psi_contDiffAt (centeredChart_mapsTo hr hr1 U hzb)).comp z |
| ((centeredChart_analyticAt U (centeredChart_den_ne_zero hr hr1 hzb)).contDiffAt.restrict_scalars ℝ) |
| ext i j |
| have hh := congrArg (fun A => A i.castSucc j.castSucc) hz |
| change liftedHessian (psi ∘ centeredChart r U) (z,0) i.castSucc j.castSucc = |
| liftedHessian psi (z,0) i.castSucc j.castSucc at hh |
| rw [liftedHessian_horizontal (hf.of_le (WithTop.coe_le_coe.mpr le_top)), |
| liftedHessian_horizontal (hψ.of_le (WithTop.coe_le_coe.mpr le_top))] at hh |
| exact hh |
|
|
| lemma psi_baseHessian_center (i j : Fin 2) : baseHessian psi 0 i j = if i=j then 1 else 0 := by |
| rw [← liftedHessian_horizontal (q := (0,(0:ℂ))) ((psi_contDiffAt zero_mem_ball).of_le (WithTop.coe_le_coe.mpr le_top)) i j] |
| change complexHessian (fun q : Ambient => psi q.1) (0,0) i.castSucc j.castSucc = _ |
| rw [psi_hessian_center] |
| fin_cases i < |
|
|
| lemma psi_baseHessian_value (v : Base) : |
| (∑ i, ∑ j, baseHessian psi 0 i j*v i*star (v j)).re = ‖v‖^2 := by |
| simp only [psi_baseHessian_center] |
| simp only [Fin.sum_univ_two,Fin.reduceEq,ite_true,ite_false,one_mul,zero_mul,zero_add,add_zero] |
| simp only [Complex.star_def,Complex.mul_conj,Complex.add_re,Complex.ofReal_re,Complex.normSq_eq_norm_sq] |
| rw [EuclideanSpace.norm_sq_eq,Fin.sum_univ_two] |
| |
| lemma baseHessian_chart_normalize {f : Base → ℝ} (hf : ContDiffOn ℝ ∞ f ball) |
| (L : ℝ) {r : ℝ} (hr : 0 ≤ r) (hr1 : r < 1) (U : Base ≃ₗᵢ[ℂ] Base) : |
| baseHessian ((fun x => psi x+L*f x) ∘ centeredChart r U) =ᶠ[𝓝 (0:Base)] |
| baseHessian (fun z => psi z+L*f (centeredChart r U z)) := by |
| filter_upwards [psi_baseHessian_chart_eventually hr hr1 U,isOpen_ball.mem_nhds zero_mem_ball] with z hz hzb |
| have hT : ContDiffAt ℝ ∞ (centeredChart r U) z := (centeredChart_analyticAt U (centeredChart_den_ne_zero hr hr1 hzb)).contDiffAt.restrict_scalars ℝ |
| have hfc := (hf.contDiffAt (isOpen_ball.mem_nhds (centeredChart_mapsTo hr hr1 U hzb))).comp z hT |
| have hψc := (psi_contDiffAt (centeredChart_mapsTo hr hr1 U hzb)).comp z hT |
| ext i j |
| change baseHessian (fun x => psi (centeredChart r U x)+L*f (centeredChart r U x)) z i j = _ |
| erw [baseHessian_add_mul hψc hfc L i j,baseHessian_add_mul (psi_contDiffAt hzb) hfc L i j] |
| exact congrArg (fun A => A i j + (L:ℂ)*baseHessian (f ∘ centeredChart r U) z i j) hz |
|
|
| lemma baseHessian_congr_derivatives {f g : Base → ℝ} {z : Base} |
| (h : baseHessian f =ᶠ[𝓝 z] baseHessian g) (i j k l : Fin 2) : |
| dzBase (fun x => baseHessian f x i j) z k = dzBase (fun x => baseHessian g x i j) z k ∧ |
| dbarBase (fun x => baseHessian f x i j) z l = dbarBase (fun x => baseHessian g x i j) z l ∧ |
| dzBase (fun x => dbarBase (fun y => baseHessian f y i j) x l) z k = |
| dzBase (fun x => dbarBase (fun y => baseHessian g y i j) x l) z k := by |
| have he : (fun x => baseHessian f x i j) =ᶠ[𝓝 z] (fun x => baseHessian g x i j) := h.fun_comp (fun A => A i j) |
| have hd : (fun x => dbarBase (fun y => baseHessian f y i j) x l) =ᶠ[𝓝 z] |
| (fun x => dbarBase (fun y => baseHessian g y i j) x l) := by |
| filter_upwards [he.fderiv (𝕜 := ℝ)] with x hx |
| simp only [dbarBase,hx] |
| exact ⟨by simp only [dzBase,he.fderiv_eq],by simp only [dbarBase,he.fderiv_eq],by simp only [dzBase,hd.fderiv_eq]⟩ |
|
|
| end PinchedHartogs.BaseConstruction |
| |
| end |
|
|
| end OAI |
|
|
