| import Mathlib | |
| import OAI.Geometry.PrescribedPotential.CalabiCoordinateCoercivity | |
| import OAI.Geometry.PrescribedPotential.CircleRadialCalculus | |
| import OAI.Geometry.PrescribedPotential.HolomorphicDirection | |
| import OAI.Geometry.PrescribedPotential.TracePositivity | |
| namespace OAI | |
| /-! Calabi Coordinate Bound. -/ | |
| section | |
| noncomputable section | |
| open Set Filter Topology Matrix | |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise | |
| namespace KaehlerCalculus | |
| variable {n : ℕ} | |
| lemma matrix_mul_elementwise_bound (M N : Matrix (Fin n) (Fin n) ℂ) : | |
| ‖M*N‖ ≤ n*‖M‖*‖N‖ := by | |
| apply (Matrix.norm_le_iff (by positivity)).mpr | |
| intro i j | |
| rw [Matrix.mul_apply] | |
| calc | |
| _ ≤ ∑ k, ‖M i k*N k j‖ := norm_sum_le _ _ | |
| _ ≤ ∑ _k : Fin n, ‖M‖*‖N‖ := by | |
| apply Finset.sum_le_sum | |
| intro k _ | |
| rw [norm_mul] | |
| exact mul_le_mul (Matrix.norm_entry_le_entrywise_sup_norm M) | |
| (Matrix.norm_entry_le_entrywise_sup_norm N) (norm_nonneg _) (norm_nonneg _) | |
| _ = _ := by simp [mul_assoc] | |
| lemma hol_matrix_stencil_bound (v : V n) (T : ConnectionTensor n) : | |
| ‖∑ i, v i • T i‖ ≤ n*‖v‖*‖T‖ := by | |
| calc | |
| _ ≤ ∑ i, ‖v i • T i‖ := norm_sum_le _ _ | |
| _ ≤ ∑ _i : Fin n, ‖v‖*‖T‖ := by | |
| apply Finset.sum_le_sum | |
| intro i _ | |
| rw [norm_smul] | |
| exact mul_le_mul (norm_le_pi_norm v i) (norm_le_pi_norm T i) (norm_nonneg _) (norm_nonneg _) | |
| _ = _ := by simp [mul_assoc] | |
| namespace LocalKaehlerField | |
| variable (K : LocalKaehlerField n) | |
| lemma holDerivative_connection {z : V n} (hz : z ∈ K.domain) (i : Fin n) : | |
| mderiv (-Complex.I) (e i) K.matrix z = K.matrix z*K.connection i z := by | |
| rw [connection,← mul_assoc,Matrix.mul_nonsing_inv _ | |
| (isUnit_iff_ne_zero.mpr (ne_of_gt (K.positive z hz).det_pos)),one_mul] | |
| lemma fderiv_eq_hol_add_star {z : V n} (hz : z ∈ K.domain) (v : V n) : | |
| fderiv ℝ K.matrix z v = mderiv (-Complex.I) v K.matrix z + | |
| (mderiv (-Complex.I) v K.matrix z)ᴴ := by | |
| rw [← mderiv_bar_eq_conjTranspose K.isOpen K.smooth | |
| (fun y hy => (K.positive y hy).isHermitian) hz v] | |
| have hd := (K.smooth.contDiffAt (K.isOpen.mem_nhds hz)).differentiableAt (by simp) | |
| ext i j | |
| have he : fderiv ℝ K.matrix z v i j = fderiv ℝ (fun y => K.matrix y i j) z v := by | |
| change fderiv ℝ (fun y a b => K.matrix y a b) z v i j = _ | |
| rw [fderiv_pi (fun a => differentiableAt_pi.mp hd a)] | |
| simp only [ContinuousLinearMap.pi_apply] | |
| rw [fderiv_pi (fun b => differentiableAt_pi.mp (differentiableAt_pi.mp hd i) b)] | |
| rfl | |
| rw [he] | |
| change _ = wderiv (-Complex.I) v (fun y => K.matrix y i j) z + | |
| wderiv Complex.I v (fun y => K.matrix y i j) z | |
| unfold wderiv | |
| ring | |
| lemma matrix_fderiv_bound {z : V n} (hz : z ∈ K.domain) : | |
| ‖fderiv ℝ (fun y i j => K.matrix y i j) z‖ ≤ 2*n^2*‖K.matrix z‖*‖fun i => K.connection i z‖ := by | |
| apply ContinuousLinearMap.opNorm_le_bound _ (by positivity) | |
| intro v | |
| have he : fderiv ℝ (fun y i j => K.matrix y i j) z v = | |
| (fun i j => (mderiv (-Complex.I) v K.matrix z + | |
| (mderiv (-Complex.I) v K.matrix z)ᴴ) i j) := K.fderiv_eq_hol_add_star hz v | |
| rw [he] | |
| change ‖mderiv (-Complex.I) v K.matrix z + (mderiv (-Complex.I) v K.matrix z)ᴴ‖ ≤ _ | |
| have ht : mderiv (-Complex.I) v K.matrix z = | |
| K.matrix z*(∑ i, v i • K.connection i z) := by | |
| rw [mderiv_hol_coordinates,Matrix.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| rw [K.holDerivative_connection hz,Matrix.mul_smul] | |
| have hb := (matrix_mul_elementwise_bound (K.matrix z) (∑ i, v i • K.connection i z)).trans | |
| (mul_le_mul_of_nonneg_left (hol_matrix_stencil_bound v (fun i => K.connection i z)) (by positivity)) | |
| rw [← ht] at hb | |
| calc | |
| _ ≤ ‖mderiv (-Complex.I) v K.matrix z‖ + ‖(mderiv (-Complex.I) v K.matrix z)ᴴ‖ := norm_add_le _ _ | |
| _ ≤ 2*(n*‖K.matrix z‖*(n*‖v‖*‖fun i => K.connection i z‖)) := by | |
| rw [Matrix.norm_conjTranspose] | |
| linarith | |
| _ = _ := by ring | |
| end LocalKaehlerField | |
| theorem calabi_coordinate_connection_uniform {X : Type*} [TopologicalSpace X] | |
| {L : Set X} (hL : IsCompact L) (G : X → Matrix (Fin n) (Fin n) ℂ) | |
| (hG : ContinuousOn G L) (hp : ∀ x ∈ L, (G x).PosDef) | |
| {R H C : ℝ} (hR : 0 ≤ R) (hH : 0 ≤ H) (hC : 0 ≤ C) : | |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ x ∈ L, ∀ (K : LocalKaehlerField n) z, z ∈ K.domain → | |
| ((R:ℂ) • K.matrix z-G x).PosSemidef → ((H:ℂ) • G x-K.matrix z).PosSemidef → | |
| K.calabiNorm z ≤ C → ‖K.matrix z‖ ≤ B ∧ ‖fun i => K.connection i z‖ ≤ B := by | |
| obtain ⟨b,hb,hframe⟩ := compact_normalizing_frames hL G hG hp hR hH | |
| obtain ⟨δ,hδ,hcoercive⟩ := boundedFrame_tensor_coercivity (n := n) b | |
| let M := (n:ℝ)*b^2 | |
| let T := C/δ+1 | |
| let B := max M T | |
| have hM : 0 ≤ M := by dsimp [M]; positivity | |
| have hT : 0 ≤ T := by dsimp [T]; positivity | |
| refine ⟨B,hM.trans (le_max_left _ _),?_⟩ | |
| intro x hx K z hz hlow hupp hcal | |
| obtain ⟨F,hF,hN⟩ := whitening (K.matrix z) (K.positive z hz) | |
| have hFI : F*F⁻¹=1 := Matrix.mul_nonsing_inv _ ((Matrix.isUnit_iff_isUnit_det _).mp hF) | |
| have hIF : F⁻¹*F=1 := Matrix.nonsing_inv_mul _ ((Matrix.isUnit_iff_isUnit_det _).mp hF) | |
| obtain ⟨hf,hfi⟩ := hframe x hx (K.matrix z) F F⁻¹ hlow hupp hFI hN | |
| have hpair : (F,F⁻¹) ∈ boundedFramePairs b := ⟨hf,hfi,hFI,hIF⟩ | |
| have hcoer : ∀ U : ConnectionTensor n, δ*‖U‖^2 ≤ tensorSquare (tensorTransport F F⁻¹ U) := | |
| hcoercive (F,F⁻¹) hpair | |
| have hc : δ*‖fun i => K.connection i z‖^2 ≤ C := | |
| (connection_norm_bound_of_frame (K := K) (z := z) (C := F) (D := F⁻¹) | |
| (δ := δ) hz hFI hIF hN hδ hcoer).trans hcal | |
| have hconn : ‖fun i => K.connection i z‖ ≤ T := by | |
| have hsq : ‖fun i => K.connection i z‖^2 ≤ C/δ := | |
| (le_div_iff₀ hδ).mpr (by simpa only [mul_comm] using hc) | |
| dsimp [T] | |
| nlinarith only [hsq, sq_nonneg (‖fun i => K.connection i z‖-1)] | |
| have hm : ‖K.matrix z‖ ≤ M := by | |
| rw [unnormalize_matrix hFI hN] | |
| have hh := matrix_mul_elementwise_bound (F⁻¹)ᴴ F⁻¹ | |
| rw [Matrix.norm_conjTranspose] at hh | |
| have hs := (sq_le_sq₀ (norm_nonneg _) hb).mpr hfi | |
| dsimp [M] | |
| nlinarith only [hh,hs,Nat.cast_nonneg (α := ℝ) n] | |
| exact ⟨hm.trans (le_max_left _ _),hconn.trans (le_max_right _ _)⟩ | |
| theorem calabi_coordinate_C1_uniform {X : Type*} [TopologicalSpace X] | |
| {L : Set X} (hL : IsCompact L) (G : X → Matrix (Fin n) (Fin n) ℂ) | |
| (hG : ContinuousOn G L) (hp : ∀ x ∈ L, (G x).PosDef) | |
| {R H C : ℝ} (hR : 0 ≤ R) (hH : 0 ≤ H) (hC : 0 ≤ C) : | |
| ∃ B : ℝ, 0 ≤ B ∧ ∀ x ∈ L, ∀ (K : LocalKaehlerField n) z, z ∈ K.domain → | |
| ((R:ℂ) • K.matrix z-G x).PosSemidef → ((H:ℂ) • G x-K.matrix z).PosSemidef → | |
| K.calabiNorm z ≤ C → ‖K.matrix z‖ ≤ B ∧ | |
| ‖fderiv ℝ (fun y i j => K.matrix y i j) z‖ ≤ B := by | |
| obtain ⟨B,hB,hbound⟩ := calabi_coordinate_connection_uniform hL G hG hp hR hH hC | |
| refine ⟨max B (2*n^2*B*B),hB.trans (le_max_left _ _),?_⟩ | |
| intro x hx K z hz hl hu hs | |
| obtain ⟨hm,hc⟩ := hbound x hx K z hz hl hu hs | |
| refine ⟨hm.trans (le_max_left _ _),?_⟩ | |
| exact (K.matrix_fderiv_bound hz).trans ((mul_le_mul | |
| (mul_le_mul_of_nonneg_left hm (by positivity)) hc (norm_nonneg _) (by positivity)).trans | |
| (le_max_right _ _)) | |
| end KaehlerCalculus | |
| end | |
| end | |
| end OAI | |