| import Mathlib |
| import OAI.Geometry.PrescribedPotential.CalabiFrameCalculus |
| import OAI.Geometry.PrescribedPotential.CalabiPointInequalities |
| import OAI.Geometry.PrescribedPotential.CalabiRicciFrames |
| import OAI.Geometry.PrescribedPotential.HolomorphicDirection |
| import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization |
|
|
| namespace OAI |
|
|
| /-! Calabi Frame Scalars. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Set Metric Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus.LocalKaehlerField |
| variable {n : ℕ} (K : LocalKaehlerField n) |
|
|
| lemma calabiNorm_nonneg {z : V n} (hz : z ∈ K.domain) : 0 ≤ K.calabiNorm z := by |
| obtain ⟨B,w,hB,hw,hM⟩ := normalized_frame K.matrix z (K.positive z hz) |
| have hwU : B w ∈ K.domain := hw ▸ hz |
| have h := K.pull_calabiNorm B hB hwU |
| rw [hw] at h |
| rw [← h,(K.pull B hB).calabiNorm_at_one hM] |
| exact tensorSquare_nonneg _ |
|
|
| lemma pull_weightedTrace_undo (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) |
| (D : Matrix (Fin n) (Fin n) ℂ) (hCD : frameMatrix B*D = 1) (w : V n) : |
| (K.pull B hB).weightedTrace D w = (K.matrix (B w)).trace.re := by |
| change (Dᴴ*((frameMatrix B)ᴴ*K.matrix (B w)*frameMatrix B)*D).trace.re = _ |
| have he : Dᴴ*((frameMatrix B)ᴴ*K.matrix (B w)*frameMatrix B)*D = |
| (frameMatrix B*D)ᴴ*K.matrix (B w)*(frameMatrix B*D) := by |
| simp only [Matrix.conjTranspose_mul,mul_assoc] |
| rw [he,hCD,Matrix.conjTranspose_one,one_mul,mul_one] |
|
|
| lemma pull_ricciHessian_jet (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) |
| {w : V n} (hw : B w ∈ K.domain) : |
| (fun i => mderiv (-Complex.I) (e i) (K.pull B hB).ricciHessian w) = |
| frameRicciJet (fun i => mderiv (-Complex.I) (e i) K.ricciHessian (B w)) (frameMatrix B) := by |
| funext i |
| rw [K.pull_ricciHessian_hol B hB hw,mderiv_hol_coordinates] |
| rfl |
|
|
| lemma pull_cutoff_max (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) |
| {z w : V n} (hz : z ∈ K.domain) (hw : B w = z) |
| (D : Matrix (Fin n) (Fin n) ℂ) (hCD : frameMatrix B*D = 1) |
| (χ : V n → ℝ) (a : ℝ) |
| (hmax : IsLocalMax (fun y => (χ y)^2*K.calabiNorm y+a*(K.matrix y).trace.re) z) : |
| IsLocalMax (fun y => (χ (B y))^2*(K.pull B hB).calabiNorm y+a*(K.pull B hB).weightedTrace D y) w := by |
| have ht : Tendsto B (𝓝 w) (𝓝 z) := hw ▸ B.continuous.continuousAt |
| have hnormw := K.pull_calabiNorm B hB (show B w ∈ K.domain by rw [hw]; exact hz) |
| have htracew := K.pull_weightedTrace_undo B hB D hCD w |
| change ∀ᶠ y in 𝓝 w, _ ≤ _ |
| dsimp only |
| rw [hnormw,htracew,hw] |
| filter_upwards [ht.eventually hmax,ht.eventually (K.isOpen.mem_nhds hz)] with y hy hym |
| rw [K.pull_calabiNorm B hB hym,K.pull_weightedTrace_undo B hB D hCD] |
| exact hy |
| end KaehlerCalculus.LocalKaehlerField |
|
|
| end |
| end |
|
|
| end OAI |
|
|
