| import Mathlib |
| import OAI.Geometry.PrescribedPotential.FrameCompactBounds |
| import OAI.Geometry.PrescribedPotential.FrameSecondCalculus |
| import OAI.Geometry.PrescribedPotential.MatrixFrameNormalization |
| import OAI.Geometry.PrescribedPotential.ScalarFrameCalculus |
|
|
| namespace OAI |
|
|
| /-! Chern Lu Frame Inequality. -/ |
|
|
| section |
|
|
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise |
| namespace KaehlerCalculus |
| variable {n : ℕ} |
|
|
| lemma inverseTrace_log_inequality_frame [Nonempty (Fin n)] |
| {U : Set (V n)} (hU : IsOpen U) |
| (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) |
| {M G : V n → Matrix (Fin n) (Fin n) ℂ} (hM : ContDiffOn ℝ ∞ M U) |
| (hp : ∀ y ∈ U, (M y).PosDef) |
| (hclosed : ∀ y ∈ U, ∀ u v w : V n, |
| fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) v w) y u + |
| fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) w u) y v + |
| fderiv ℝ (fun q => Anticanonical.ComplexAtlas.fundamentalForm (M q) u v) y w = 0) |
| (hG : ContDiffOn ℝ ∞ G U) (hGp : ∀ y ∈ U, (G y).PosDef) |
| {w : V n} (hw : B w ∈ U) (hM1 : pullMetric B M w = 1) : |
| frameResidual (PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) (B w)) |
| (G (B w)) (curvatureDiagonal G (B w)) (frameMatrix B) ≤ |
| inverseTrace M G (B w) * |
| (PotentialKaehler.potentialMatrix (fun y => Real.log (inverseTrace M G (B y))) w).trace.re := by |
| have hi := inverseTrace_log_inequality_at_one (hU.preimage B.continuous) |
| (pullMetric_smooth hU B hM) (pullMetric_positive B hB hp) |
| (pullMetric_closed hU B hM hclosed) (pullMetric_smooth hU B hG) (pullMetric_positive B hB hGp) hw hM1 |
| rw [potentialMatrix_logdet_pullMetric hU B hB hM hp hw] at hi |
| simp_rw [curvatureDiagonal_pullMetric hU B hB hG hw] at hi |
| have hs : inverseTrace (pullMetric B M) (pullMetric B G) = fun y => inverseTrace M G (B y) := |
| funext (fun y => inverseTrace_pullMetric B hB M G y) |
| rw [hs] at hi |
| simpa only [frameResidual,pullMetric,frameMatrix_mulVec] using hi |
|
|
| lemma inverseTrace_eq_frameGram (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) |
| (M G : V n → Matrix (Fin n) (Fin n) ℂ) (w : V n) (hM1 : pullMetric B M w = 1) : |
| inverseTrace M G (B w) = frameGram (G (B w)) (frameMatrix B) := by |
| rw [← inverseTrace_pullMetric B hB M G w] |
| unfold inverseTrace |
| rw [hM1,inv_one,one_mul] |
| rfl |
| end KaehlerCalculus |
|
|
| end |
| end |
|
|
| end OAI |
|
|
