| import Mathlib |
| import OAI.Geometry.PrescribedPotential.BoundedNormalFrames |
| import OAI.Geometry.PrescribedPotential.CalabiBochner |
| import OAI.Geometry.PrescribedPotential.CalabiConnection |
| import OAI.Geometry.PrescribedPotential.CalabiFrameAlgebra |
| import OAI.Geometry.PrescribedPotential.CalabiNormAlgebra |
|
|
| namespace OAI |
|
|
| /-! Calabi Coordinate Coercivity. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus |
| variable {n : ℕ} |
|
|
| def tensorTransport (C D : Matrix (Fin n) (Fin n) ℂ) (T : ConnectionTensor n) : ConnectionTensor n := |
| fun i => D*(∑ p, C p i • T p)*C |
|
|
| lemma tensorTransport_smul (C D : Matrix (Fin n) (Fin n) ℂ) (r : ℝ) (T : ConnectionTensor n) : |
| tensorTransport C D (r • T) = r • tensorTransport C D T := by |
| funext i |
| simp only [tensorTransport,Pi.smul_apply,smul_comm (_:ℂ) r,← Finset.smul_sum, |
| Matrix.mul_smul,Matrix.smul_mul] |
|
|
| lemma tensorTransport_zero (C D : Matrix (Fin n) (Fin n) ℂ) : |
| tensorTransport C D (0 : ConnectionTensor n) = 0 := by |
| funext i |
| simp [tensorTransport] |
|
|
| lemma tensorTransport_undo {C D : Matrix (Fin n) (Fin n) ℂ} |
| (hCD : C*D=1) (_hDC : D*C=1) (T : ConnectionTensor n) : |
| tensorTransport D C (tensorTransport C D T) = T := by |
| funext i |
| unfold tensorTransport |
| have he : (∑ p, D p i • (D*(∑ q, C q p • T q)*C)) = |
| D*(∑ q, (C*D) q i • T q)*C := by |
| calc |
| _ = ∑ p, D*(∑ q, (D p i*C q p) • T q)*C := by |
| apply Finset.sum_congr rfl |
| intro p _ |
| simp only [← smul_smul,← Finset.smul_sum,Matrix.mul_smul,Matrix.smul_mul] |
| _ = D*(∑ p, ∑ q, (D p i*C q p) • T q)*C := by |
| rw [Matrix.mul_sum,Matrix.sum_mul] |
| _ = _ := by |
| congr 2 |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro q _ |
| simp only [Matrix.mul_apply,Finset.sum_smul] |
| apply Finset.sum_congr rfl |
| intro p _ |
| rw [mul_comm] |
|
|
| rw [he,hCD] |
| simp only [Matrix.one_apply,ite_smul,one_smul,zero_smul,Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
| simp only [mul_assoc,hCD,mul_one] |
| rw [← mul_assoc,hCD,one_mul] |
|
|
| lemma continuous_tensorTransport : Continuous |
| (fun q : (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) × ConnectionTensor n => |
| tensorTransport q.1.1 q.1.2 q.2) := by |
| apply continuous_pi |
| intro i |
| let P := (Matrix (Fin n) (Fin n) ℂ × Matrix (Fin n) (Fin n) ℂ) × ConnectionTensor n |
| have hC : Continuous (fun q : P => q.1.1) := continuous_fst.comp continuous_fst |
| have hD : Continuous (fun q : P => q.1.2) := continuous_snd.comp continuous_fst |
| have hT (p : Fin n) : Continuous (fun q : P => q.2 p) := (continuous_apply p).comp continuous_snd |
| have hc (p : Fin n) : Continuous (fun q : P => q.1.1 p i) := |
| (continuous_apply i).comp ((continuous_apply p).comp hC) |
| exact (hD.matrix_mul (continuous_finsetSum _ (fun p _ => (hc p).smul (hT p)))).matrix_mul hC |
|
|
| lemma boundedFrame_tensor_coercivity (B : ℝ) : |
| ∃ δ : ℝ, 0 < δ ∧ ∀ p ∈ boundedFramePairs (n := n) B, ∀ T : ConnectionTensor n, |
| δ*‖T‖^2 ≤ tensorSquare (tensorTransport p.1 p.2 T) := by |
| apply compact_homogeneous_positive (boundedFramePairs_compact B) |
| (fun p T => tensorSquare (tensorTransport p.1 p.2 T)) |
| · exact (continuous_tensorSquare.comp continuous_tensorTransport).continuousOn |
| · intro p hp T hT |
| apply tensorSquare_pos |
| intro he |
| apply hT |
| rw [← tensorTransport_undo hp.2.2.1 hp.2.2.2 T,he,tensorTransport_zero] |
| · intro p _ r T |
| rw [tensorTransport_smul,tensorSquare_smul] |
|
|
| lemma connection_norm_bound_of_frame {K : LocalKaehlerField n} {z : V n} |
| (_hz : z ∈ K.domain) {C D : Matrix (Fin n) (Fin n) ℂ} |
| (hCD : C*D=1) (hDC : D*C=1) (hN : Cᴴ*K.matrix z*C=1) |
| {δ : ℝ} (_hδ : 0 < δ) |
| (hc : ∀ T : ConnectionTensor n, δ*‖T‖^2 ≤ tensorSquare (tensorTransport C D T)) : |
| δ*‖fun i => K.connection i z‖^2 ≤ K.calabiNorm z := by |
| have hC : IsUnit C := ⟨⟨C,D,hCD,hDC⟩,rfl⟩ |
| have hi : C⁻¹=D := Matrix.inv_eq_right_inv hCD |
| have hI : C⁻¹*(K.matrix z)⁻¹*(Cᴴ)⁻¹=1 := by |
| rw [← Matrix.mul_inv_rev,← Matrix.mul_inv_rev,← mul_assoc,hN,inv_one] |
| have he := tensorPairAt_changeBasis C (K.matrix z)⁻¹ (K.matrix z) hC |
| (fun i => K.connection i z) (fun i => K.connection i z) |
| rw [hI,hN] at he |
| have ht : pullConnectionTensor C (fun i => K.connection i z) = |
| tensorTransport C D (fun i => K.connection i z) := by |
| unfold pullConnectionTensor tensorTransport |
| rw [hi] |
| rw [ht] at he |
| have hs : tensorSquare (tensorTransport C D (fun i => K.connection i z)) = K.calabiNorm z := by |
| change (tensorPairAt 1 1 _ _).re = _ |
| exact congrArg Complex.re he |
| rw [← hs] |
| exact hc _ |
| end KaehlerCalculus |
|
|
| end |
| end |
|
|
| end OAI |
|
|
