| import OAI.Geometry.KahlerSplitting.Grid | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped Manifold ContDiff | |
| open Set | |
| namespace UniversalCoverSplitting | |
| open scoped Manifold ContDiff | |
| open Set | |
| def IsBox {n r₁ r₂ : ℕ} {X : ComplexManifold n} | |
| (S : HolomorphicSplitting X r₁ r₂) (U : Set (ℂ × ℂ)) | |
| (f : ℂ × ℂ → X.Carrier) : Prop := | |
| MDifferentiableOn 𝓘(ℂ, ℂ × ℂ) (model n) f U ∧ | |
| ∀ z ∈ U, | |
| S.projection (f z) (mfderiv 𝓘(ℂ, ℂ × ℂ) (model n) f z (1, 0)) = | |
| mfderiv 𝓘(ℂ, ℂ × ℂ) (model n) f z (1, 0) ∧ | |
| S.projection (f z) (mfderiv 𝓘(ℂ, ℂ × ℂ) (model n) f z (0, 1)) = 0 | |
| def hartogsFigure (A : (ℂ × ℂ) ≃ᵃ[ℂ] (ℂ × ℂ)) (e a c b : ℝ) : Set (ℂ × ℂ) := | |
| {z | (‖(A z).1‖ < e ∧ ‖(A z).2‖ < b) ∨ | |
| (‖(A z).1‖ < a ∧ c < ‖(A z).2‖ ∧ ‖(A z).2‖ < b)} | |
| def hartogsEnvelope (A : (ℂ × ℂ) ≃ᵃ[ℂ] (ℂ × ℂ)) (a b : ℝ) : Set (ℂ × ℂ) := | |
| {z | ‖(A z).1‖ < a ∧ ‖(A z).2‖ < b} | |
| def HartogsBoxExtension {n r₁ r₂ : ℕ} {X : ComplexManifold n} | |
| (S : HolomorphicSplitting X r₁ r₂) : Prop := | |
| ∀ (A : (ℂ × ℂ) ≃ᵃ[ℂ] (ℂ × ℂ)) (e a c b : ℝ), | |
| 0 < e → e < a → 0 < c → c < b → | |
| ∀ f : ℂ × ℂ → X.Carrier, IsBox S (hartogsFigure A e a c b) f → | |
| ∃ F : ℂ × ℂ → X.Carrier, | |
| IsBox S (hartogsEnvelope A a b) F ∧ | |
| EqOn F f (hartogsFigure A e a c b) ∧ | |
| ∀ G : ℂ × ℂ → X.Carrier, | |
| IsBox S (hartogsEnvelope A a b) G → | |
| EqOn G f (hartogsFigure A e a c b) → EqOn G F (hartogsEnvelope A a b) | |
| section ProjectionEstimate | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E] | |
| theorem mixed_direction_identity (P : E →L[ℂ] E) (D : (ℂ × ℂ) →L[ℂ] E) | |
| (h₁ : P (D (1, 0)) = D (1, 0)) (h₂ : P (D (0, 1)) = 0) | |
| (a₁ a₂ b₁ b₂ : ℂ) (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) : | |
| D (a₁, a₂) = (a₁ / b₁) • P (D (b₁, b₂)) + | |
| (a₂ / b₂) • ((ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂))) := by | |
| have split (c₁ c₂ : ℂ) : D (c₁, c₂) = c₁ • D (1, 0) + c₂ • D (0, 1) := by | |
| calc | |
| D (c₁, c₂) = D (c₁ • (1, 0) + c₂ • (0, 1)) := by congr 1; ext <;> simp | |
| _ = c₁ • D (1, 0) + c₂ • D (0, 1) := by simp only [map_add, map_smul] | |
| rw [split a₁ a₂, split b₁ b₂] | |
| simp only [map_add, map_smul, h₁, h₂, smul_zero, add_zero, | |
| sub_apply, ContinuousLinearMap.id_apply] | |
| simp only [sub_self, sub_zero, smul_zero, zero_add, smul_smul, | |
| div_mul_cancel₀ _ hb₁, div_mul_cancel₀ _ hb₂] | |
| theorem mixed_direction_norm_le (P : E →L[ℂ] E) (D : (ℂ × ℂ) →L[ℂ] E) | |
| (h₁ : P (D (1, 0)) = D (1, 0)) (h₂ : P (D (0, 1)) = 0) | |
| (a₁ a₂ b₁ b₂ : ℂ) (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) : | |
| ‖D (a₁, a₂)‖ ≤ | |
| (‖a₁ / b₁‖ * ‖P‖ + ‖a₂ / b₂‖ * ‖ContinuousLinearMap.id ℂ E - P‖) * | |
| ‖D (b₁, b₂)‖ := by | |
| calc | |
| ‖D (a₁, a₂)‖ = ‖(a₁ / b₁) • P (D (b₁, b₂)) + | |
| (a₂ / b₂) • ((ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂)))‖ := by | |
| rw [mixed_direction_identity P D h₁ h₂ a₁ a₂ b₁ b₂ hb₁ hb₂] | |
| _ ≤ ‖(a₁ / b₁) • P (D (b₁, b₂))‖ + | |
| ‖(a₂ / b₂) • ((ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂)))‖ := norm_add_le _ _ | |
| _ = ‖a₁ / b₁‖ * ‖P (D (b₁, b₂))‖ + | |
| ‖a₂ / b₂‖ * ‖(ContinuousLinearMap.id ℂ E - P) (D (b₁, b₂))‖ := by | |
| rw [norm_smul, norm_smul] | |
| _ ≤ ‖a₁ / b₁‖ * (‖P‖ * ‖D (b₁, b₂)‖) + | |
| ‖a₂ / b₂‖ * (‖ContinuousLinearMap.id ℂ E - P‖ * ‖D (b₁, b₂)‖) := by | |
| exact add_le_add (mul_le_mul_of_nonneg_left (P.le_opNorm _) (norm_nonneg _)) | |
| (mul_le_mul_of_nonneg_left ((ContinuousLinearMap.id ℂ E - P).le_opNorm _) | |
| (norm_nonneg _)) | |
| _ = _ := by ring | |
| end ProjectionEstimate | |
| end UniversalCoverSplitting | |
| noncomputable section | |
| open Set Topology Filter Metric | |
| namespace UniversalCoverSplitting | |
| section CompactQuadratic | |
| variable {B E : Type*} [TopologicalSpace B] [NormedAddCommGroup E] | |
| [NormedSpace ℝ E] [FiniteDimensional ℝ E] | |
| theorem exists_quadratic_operator_bound {K : Set B} (hK : IsCompact K) | |
| (Q : B → E → ℝ) (P : B → E →L[ℝ] E) | |
| (hQ : ContinuousOn (fun p : B × E ↦ Q p.1 p.2) (K ×ˢ univ)) | |
| (hP : ContinuousOn P K) | |
| (hpos : ∀ x ∈ K, ∀ u, u ≠ 0 → 0 < Q x u) | |
| (hhom : ∀ x ∈ K, ∀ (r : ℝ) u, Q x (r • u) = r ^ 2 * Q x u) : | |
| ∃ C : ℝ, 0 < C ∧ ∀ x ∈ K, ∀ u, Q x (P x u) ≤ C * Q x u := by | |
| let L : Set (B × E) := K ×ˢ sphere 0 1 | |
| have hL : IsCompact L := hK.prod (isCompact_sphere 0 1) | |
| have hQL : ContinuousOn (fun p : B × E ↦ Q p.1 p.2) L := | |
| hQ.mono (prod_mono_right (subset_univ _)) | |
| have hnum : ContinuousOn (fun p : B × E ↦ Q p.1 (P p.1 p.2)) L := by | |
| apply hQ.comp | |
| (continuousOn_fst.prodMk ((hP.comp continuousOn_fst (fun _ h ↦ h.1)).clm_apply | |
| continuousOn_snd)) | |
| exact fun p hp ↦ ⟨hp.1, mem_univ _⟩ | |
| have hden : ∀ p ∈ L, Q p.1 p.2 ≠ 0 := by | |
| intro p hp | |
| apply ne_of_gt (hpos p.1 hp.1 p.2 _) | |
| intro he | |
| have hn : ‖p.2‖ = (1:ℝ) := by simpa using hp.2 | |
| simp [he] at hn | |
| obtain ⟨C,hC⟩ := hL.bddAbove_image (hnum.div hQL hden) | |
| refine ⟨max C 1, by positivity, fun x hx u ↦ ?_⟩ | |
| have hzero : Q x 0 = 0 := by simpa using hhom x hx 0 (0:E) | |
| by_cases hu : u = 0 | |
| · simp [hu,hzero] | |
| have hn : 0 < ‖u‖ := norm_pos_iff.mpr hu | |
| let v := ‖u‖⁻¹ • u | |
| have hvnorm : ‖v‖ = 1 := by | |
| dsimp [v] | |
| rw [norm_smul_of_nonneg (inv_nonneg.mpr hn.le), inv_mul_cancel₀ hn.ne'] | |
| have hv : v ∈ sphere (0:E) 1 := by simpa using hvnorm | |
| have hb : Q x (P x v) / Q x v ≤ max C 1 := | |
| (hC ⟨(x,v),⟨hx,hv⟩,rfl⟩).trans (le_max_left _ _) | |
| have hvpos : 0 < Q x v := hpos x hx v (by intro hh; simp [hh] at hvnorm) | |
| have hquad : Q x (P x v) ≤ max C 1 * Q x v := (div_le_iff₀ hvpos).mp hb | |
| have huv : ‖u‖ • v = u := by | |
| dsimp [v] | |
| rw [smul_smul, mul_inv_cancel₀ hn.ne', one_smul] | |
| calc | |
| Q x (P x u) = ‖u‖ ^ 2 * Q x (P x v) := by | |
| conv_lhs => rw [← huv, map_smul] | |
| exact hhom x hx ‖u‖ (P x v) | |
| _ ≤ ‖u‖ ^ 2 * (max C 1 * Q x v) := mul_le_mul_of_nonneg_left hquad (sq_nonneg _) | |
| _ = max C 1 * Q x u := by | |
| have hs : Q x u = ‖u‖ ^ 2 * Q x v := by rw [← hhom x hx ‖u‖ v, huv] | |
| rw [hs] | |
| ring | |
| end CompactQuadratic | |
| end UniversalCoverSplitting | |
| end | |
| end | |
| end | |
| end OAI | |