File size: 2,466 Bytes
2fca478 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | 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
|