| import Mathlib | |
| import OAI.Geometry.PrescribedPotential.CalabiTensorAlgebra | |
| namespace OAI | |
| /-! Calabi Norm Algebra. -/ | |
| section | |
| noncomputable section | |
| open Set Filter Topology Matrix | |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise | |
| namespace KaehlerCalculus | |
| variable {n : ℕ} | |
| def tensorVector (T : ConnectionTensor n) : EuclideanSpace ℂ (Fin n × Fin n × Fin n) := | |
| WithLp.toLp 2 (fun p => T p.1 p.2.1 p.2.2) | |
| def tensorSquare (T : ConnectionTensor n) : ℝ := (tensorPairAt 1 1 T T).re | |
| lemma tensorPairAt_one_inner (T U : ConnectionTensor n) : | |
| tensorPairAt 1 1 T U = inner ℂ (tensorVector U) (tensorVector T) := by | |
| rw [tensorPairAt_one] | |
| simp only [tensorVector,PiLp.inner_apply,Fintype.sum_prod_type, | |
| Matrix.trace,Matrix.diag_apply,Matrix.mul_apply,Matrix.conjTranspose_apply, | |
| RCLike.inner_apply] | |
| rfl | |
| lemma tensorSquare_norm (T : ConnectionTensor n) : tensorSquare T = ‖tensorVector T‖^2 := by | |
| rw [tensorSquare,tensorPairAt_one_inner] | |
| exact (norm_sq_eq_re_inner (𝕜 := ℂ) (tensorVector T)).symm | |
| lemma tensorSquare_nonneg (T : ConnectionTensor n) : 0 ≤ tensorSquare T := by | |
| rw [tensorSquare_norm] | |
| positivity | |
| lemma tensorSquare_pos {T : ConnectionTensor n} (hT : T ≠ 0) : 0 < tensorSquare T := by | |
| rw [tensorSquare_norm] | |
| apply sq_pos_of_pos | |
| apply norm_pos_iff.mpr | |
| intro he | |
| apply hT | |
| funext i | |
| ext a b | |
| exact congrArg (fun v : EuclideanSpace ℂ (Fin n × Fin n × Fin n) => v (i,a,b)) he | |
| lemma tensorPairAt_cauchy (T U : ConnectionTensor n) : | |
| ‖tensorPairAt 1 1 T U‖^2 ≤ tensorSquare T*tensorSquare U := by | |
| rw [tensorPairAt_one_inner,tensorSquare_norm,tensorSquare_norm] | |
| have hh := norm_inner_le_norm (𝕜 := ℂ) (tensorVector U) (tensorVector T) | |
| have hs := sq_le_sq₀ (norm_nonneg _) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) |>.mpr hh | |
| simpa only [mul_pow,mul_comm] using hs | |
| lemma tensorSquare_smul (r : ℝ) (T : ConnectionTensor n) : | |
| tensorSquare (r • T) = r^2*tensorSquare T := by | |
| have he : tensorVector (r • T) = r • tensorVector T := rfl | |
| rw [tensorSquare_norm,he,norm_smul,mul_pow,Real.norm_eq_abs,sq_abs,tensorSquare_norm] | |
| lemma continuous_tensorSquare : Continuous (tensorSquare (n := n)) := by | |
| rw [show tensorSquare (n := n) = fun T => ‖tensorVector T‖^2 from funext tensorSquare_norm] | |
| apply Continuous.pow | |
| apply Continuous.norm | |
| exact (PiLp.continuous_toLp 2 _).comp (continuous_pi (fun p => | |
| (continuous_apply p.2.2).comp ((continuous_apply p.2.1).comp (continuous_apply p.1)))) | |
| lemma continuous_tensorVector : Continuous (tensorVector (n := n)) := by | |
| exact (PiLp.continuous_toLp 2 _).comp (continuous_pi (fun p => | |
| (continuous_apply p.2.2).comp ((continuous_apply p.2.1).comp (continuous_apply p.1)))) | |
| lemma tensorPairAt_one_continuous : Continuous | |
| (fun q : ConnectionTensor n × ConnectionTensor n => tensorPairAt 1 1 q.1 q.2) := by | |
| simp_rw [tensorPairAt_one_inner] | |
| exact Continuous.inner (𝕜 := ℂ) (continuous_tensorVector.comp continuous_snd) | |
| (continuous_tensorVector.comp continuous_fst) | |
| lemma tensorPairAt_one_smul (r s : ℝ) (T U : ConnectionTensor n) : | |
| tensorPairAt 1 1 (r • T) (s • U) = ((r*s : ℝ) : ℂ)*tensorPairAt 1 1 T U := by | |
| simp only [tensorPairAt_one,Pi.smul_apply] | |
| change (∑ i, (((r:ℂ) • T i)*((s:ℂ) • U i)ᴴ).trace) = _ | |
| simp only [Matrix.conjTranspose_smul,Complex.star_def,Complex.conj_ofReal, | |
| smul_mul_assoc,mul_smul_comm,smul_smul,Matrix.trace_smul,smul_eq_mul, | |
| ← Complex.ofReal_mul,← Finset.mul_sum] | |
| rw [mul_comm s r] | |
| end KaehlerCalculus | |
| end | |
| end | |
| end OAI | |