| import Mathlib | |
| import OAI.Geometry.PrescribedPotential.ChernLuFrameInequality | |
| import OAI.Geometry.PrescribedPotential.PotentialLinearCalculus | |
| namespace OAI | |
| /-! Chern Lu Maximum. -/ | |
| section | |
| noncomputable section | |
| open Set Filter Topology Matrix | |
| open scoped ContDiff ComplexOrder Matrix.Norms.Elementwise | |
| namespace KaehlerCalculus | |
| variable {n : ℕ} | |
| lemma pullMetric_potential_relation (B : V n →L[ℂ] V n) | |
| (M G : V n → Matrix (Fin n) (Fin n) ℂ) (f : V n → ℝ) (w : V n) | |
| (hf : ContDiffAt ℝ ∞ f (B w)) | |
| (he : M (B w) = G (B w)+PotentialKaehler.potentialMatrix f (B w)) : | |
| pullMetric B M w = pullMetric B G w+PotentialKaehler.potentialMatrix (fun y => f (B y)) w := by | |
| rw [potentialMatrix_linear B hf] | |
| unfold pullMetric | |
| rw [he,mul_add,add_mul] | |
| lemma trace_potential_in_frame (B : V n →L[ℂ] V n) (hB : IsUnit (frameMatrix B)) | |
| (M G : V n → Matrix (Fin n) (Fin n) ℂ) (f : V n → ℝ) (w : V n) | |
| (hf : ContDiffAt ℝ ∞ f (B w)) | |
| (he : M (B w) = G (B w)+PotentialKaehler.potentialMatrix f (B w)) | |
| (hM1 : pullMetric B M w = 1) : | |
| (PotentialKaehler.potentialMatrix (fun y => f (B y)) w).trace.re = | |
| (n:ℝ)-inverseTrace M G (B w) := by | |
| have hh := congrArg (fun H : Matrix (Fin n) (Fin n) ℂ => H.trace.re) | |
| (pullMetric_potential_relation B M G f w hf he) | |
| rw [hM1,Matrix.trace_one,Matrix.trace_add,Complex.add_re] at hh | |
| have hs := inverseTrace_eq_frameGram B hB M G w hM1 | |
| change inverseTrace M G (B w) = (pullMetric B G w).trace.re at hs | |
| simp only [Fintype.card_fin,Complex.natCast_re] at hh | |
| linarith | |
| lemma scalar_trace_maximum {C S L D : ℝ} (hp : 0 < S) | |
| (hr : -C*S^2 ≤ S*L) (hm : L-(C+1)*(D-S) ≤ 0) : S ≤ (C+1)*D := by | |
| have he : S*(S-(C+1)*D) ≤ 0 := by nlinarith | |
| by_contra h | |
| have hmul := mul_pos hp (sub_pos.mpr (lt_of_not_ge h)) | |
| linarith | |
| lemma inverseTrace_at_localMax_bound [Nonempty (Fin n)] | |
| {U : Set (V n)} (hU : IsOpen U) | |
| {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) | |
| {f : V n → ℝ} (hf : ContDiffOn ℝ ∞ f U) | |
| {z : V n} (hz : z ∈ U) | |
| (he : M z = G z+PotentialKaehler.potentialMatrix f z) | |
| (C : ℝ) (hbound : ∀ B : Matrix (Fin n) (Fin n) ℂ, | |
| -C*(frameGram (G z) B)^2 ≤ frameResidual | |
| (PotentialKaehler.potentialMatrix (fun y => Real.log (M y).det.re) z) | |
| (G z) (curvatureDiagonal G z) B) | |
| (hmax : IsLocalMax (fun y => Real.log (inverseTrace M G y)-(C+1)*f y) z) : | |
| inverseTrace M G z ≤ (C+1)*n := by | |
| obtain ⟨B,w,hB,hw,hM1⟩ := normalized_frame M z (hp z hz) | |
| have hwU : B w ∈ U := hw ▸ hz | |
| have hfs := hf.contDiffAt (hU.mem_nhds hz) | |
| have hSpos : 0 < inverseTrace M G z := inverseTrace_pos (hp z hz) (hGp z hz) | |
| have hSs := (inverseTrace_smooth hU hM hp hG).contDiffAt (hU.mem_nhds hz) | |
| have hlog := hSs.log (ne_of_gt hSpos) | |
| have hbℝ := (B.restrictScalars ℝ).contDiff.contDiffAt (x := w) (n := ∞) | |
| have hfc : ContDiffAt ℝ ∞ (fun y => f (B y)) w := by | |
| exact (hw ▸ hfs).comp w hbℝ | |
| have hlc : ContDiffAt ℝ ∞ (fun y => Real.log (inverseTrace M G (B y))) w := by | |
| exact (hw ▸ hlog).comp w hbℝ | |
| have hmaxc : IsLocalMax (fun y => Real.log (inverseTrace M G (B y))-(C+1)*f (B y)) w := by | |
| have ht : Tendsto B (𝓝 w) (𝓝 z) := hw ▸ B.continuous.continuousAt | |
| change ∀ᶠ y in 𝓝 w, Real.log (inverseTrace M G (B y))-(C+1)*f (B y) ≤ | |
| Real.log (inverseTrace M G (B w))-(C+1)*f (B w) | |
| rw [hw] | |
| exact ht.eventually hmax | |
| have ht := EllipticKernel.trace_potentialMatrix_nonpos_at_localMax | |
| (hlc.sub (contDiffAt_const.mul hfc)) hmaxc (1 : Matrix (Fin n) (Fin n) ℂ) Matrix.PosDef.one | |
| rw [inv_one,one_mul,potentialMatrix_trace_sub_smul hlc hfc] at ht | |
| have hphi := trace_potential_in_frame B hB M G f w (hw ▸ hfs) (hw ▸ he) hM1 | |
| rw [hphi,hw] at ht | |
| have hi := inverseTrace_log_inequality_frame hU B hB hM hp hclosed hG hGp hwU hM1 | |
| rw [hw] at hi | |
| have hb := hbound (frameMatrix B) | |
| have hgram := inverseTrace_eq_frameGram B hB M G w hM1 | |
| rw [hw] at hgram | |
| rw [← hgram] at hb | |
| exact scalar_trace_maximum hSpos (hb.trans hi) ht | |
| end KaehlerCalculus | |
| end | |
| end | |
| end OAI | |