| import Mathlib |
| import OAI.Geometry.PrescribedRicci.FiniteCofactor |
| import OAI.Geometry.PrescribedRicci.FiniteCofactorPatch |
|
|
| namespace OAI |
|
|
| /-! Cofactor Core. -/ |
|
|
| section |
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| |
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|
| noncomputable section |
| open Set Filter Topology Matrix |
| open scoped ContDiff SchwartzMap Classical BoundedContinuousFunction Matrix.Norms.Elementwise |
| namespace GlobalElliptic |
| open Anticanonical SourceSmooth EllipticKernel SobolevChart FrozenPoisson MetricLocalization |
| variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X] |
| {A : ComplexAtlas d X} {ι : Type*} [Fintype ι] |
| namespace GluingData |
| variable {g : KaehlerMetric A} (D : GluingData g ι) |
|
|
| def cofactorComplexCore (p : ι) (B : Bilin (EC d)) (f : Smooth A) : Smooth A := |
| ⟨fun x => D.inverseDet p x * NonlinearHessian.detDifferential |
| (fun i j => D.metricEntry p i j x + D.localizedHessian p i j f x) (cofactorMatrix B), |
| fun q => by |
| apply (D.inverseDet p).smooth q |>.mul |
| apply ContDiffOn.clm_apply |
| · apply ((NonlinearHessian.determinant_contDiff d).fderiv_right (m:=∞) (by simp)).comp_contDiffOn |
| apply contDiffOn_pi.mpr |
| intro i |
| apply contDiffOn_pi.mpr |
| intro j |
| exact ((D.metricEntry p i j).smooth q).add ((D.localizedHessian p i j f).smooth q) |
| · exact contDiffOn_const⟩ |
|
|
| def cofactorScalarCore (p : ι) (B : Bilin (EC d)) (f : Smooth A) : Smooth A := |
| Smooth.realify (D.cofactorComplexCore p B f) |
|
|
| lemma cofactorScalarCore_apply (p : ι) (B : Bilin (EC d)) (f : Smooth A) (x : X) : |
| D.cofactorScalarCore p B f x = |
| ((D.inverseDet p x * NonlinearHessian.detDifferential |
| (fun i j => D.metricEntry p i j x + D.localizedHessian p i j f x) (cofactorMatrix B)).re : ℂ) := by |
| exact Smooth.realify_apply _ x |
|
|
| lemma cofactorScalar_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) |
| (p : ι) (B : Bilin (EC d)) (f : Smooth A) : |
| D.cofactorScalar k hk p B (D.localizers.embed ((k:ℝ)+2) f) = |
| D.localizers.embed (k:ℝ) (D.cofactorScalarCore p B f) := by |
| have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by |
| have hh : (Module.finrank ℝ (EC d):ℝ) < (k:ℝ) := by exact_mod_cast hk |
| linarith [Nat.cast_nonneg (α:=ℝ) k] |
| apply D.localizers.strong_injective (k:ℝ) hs |
| ext x |
| rw [D.cofactorScalar_strong,D.localizers.strong_embed _ hs] |
| simp_rw [D.completedHessian_embed,D.localizers.strong_embed _ hs] |
| exact (D.cofactorScalarCore_apply p B f x).symm |
|
|
| def cofactorLocalCore (p : ι) (i j : BasisIndex d) (f : Smooth A) : 𝓢(EC d,ℂ) := |
| (D.patch p).coefficient i j + |
| localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.cutoff p)) |
| (cutoffGlobal_support _ _) |
| (D.cofactorScalarCore p (rankTwo (stdOrthonormalBasis ℝ (EC d) i) |
| (stdOrthonormalBasis ℝ (EC d) j)) f - |
| D.cofactorScalarCore p (rankTwo (stdOrthonormalBasis ℝ (EC d) i) |
| (stdOrthonormalBasis ℝ (EC d) j)) 0) |
|
|
| lemma cofactorLocal_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) |
| (p : ι) (i j : BasisIndex d) (f : Smooth A) : |
| D.cofactorLocal k hk p i j (D.localizers.embed ((k:ℝ)+2) f) = |
| schwartzCoord (k:ℝ) (D.cofactorLocalCore p i j f) := by |
| rw [cofactorLocal,D.cofactorScalar_embed] |
| have h0 := D.cofactorScalar_embed k hk p |
| (rankTwo (stdOrthonormalBasis ℝ (EC d) i) (stdOrthonormalBasis ℝ (EC d) j)) 0 |
| rw [map_zero] at h0 |
| rw [h0,← map_sub,D.completedLocalize_embed,← schwartzCoord_add] |
| rfl |
|
|
| lemma cofactorBCF_embed (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) |
| (hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ)) (p : ι) (f : Smooth A) : |
| D.cofactorBCF k hk hs p (D.localizers.embed ((k:ℝ)+2) f) = |
| coefficientBCF (D.cofactorLocalCore p · · f) := by |
| funext i j |
| rw [cofactorBCF,D.cofactorLocal_embed,strongEmbedding_schwartz] |
| rfl |
|
|
| lemma cofactorScalarCore_zero_source (p : ι) (B : Bilin (EC d)) {x : X} |
| (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) : |
| D.cofactorScalarCore p B 0 x = |
| (traceBilin (g.matrix (D.patch p).index (A.chart (D.patch p).index x)) B : ℂ) := by |
| let k := Module.finrank ℝ (EC d)+1 |
| have hk : Module.finrank ℝ (EC d) < k := by omega |
| have hs : (Module.finrank ℝ (EC d):ℝ) < 2*(k:ℝ) := by |
| dsimp [k] |
| push_cast |
| linarith [Nat.cast_nonneg (α:=ℝ) (Module.finrank ℝ (EC d))] |
| have hh := D.cofactorScalar_zero_source k hk p B hx |
| have he := D.cofactorScalar_embed k hk p B 0 |
| rw [map_zero] at he |
| rw [he,D.localizers.strong_embed _ hs] at hh |
| exact hh |
|
|
| lemma cofactorLocalCore_source (p : ι) (i j : BasisIndex d) (f : Smooth A) {x : X} |
| (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) : |
| extendedCoefficient (D.patch p).matrix (coefficientBCF (D.cofactorLocalCore p · · f)) i j |
| (A.euclideanChart (D.patch p).index x) = |
| D.cofactorScalarCore p (rankTwo (stdOrthonormalBasis ℝ (EC d) i) |
| (stdOrthonormalBasis ℝ (EC d) j)) f x := by |
| have hs := D.patch_source p hx |
| have hy := (A.euclideanChart (D.patch p).index).mapsTo hs |
| have he := (A.euclideanChart (D.patch p).index).left_inv hs |
| change (traceBilin (D.patch p).matrix (rankTwo _ _) : ℂ) + |
| D.cofactorLocalCore p i j f (A.euclideanChart (D.patch p).index x) = _ |
| rw [cofactorLocalCore,_root_.add_apply,localize_apply,localizeFun_apply A _ _ hy,he] |
| have hc : cutoffGlobal (D.patch p).index (D.cutoff p) x = 1 := by |
| change (if x ∈ (A.euclideanChart (D.patch p).index).source then _ else _) = _ |
| rw [ite_eq_left hs,D.cutoff_one p x hx] |
| rw [hc,one_mul] |
| change _ + ((D.patch p).coefficient i j (A.euclideanChart (D.patch p).index x) + |
| (D.cofactorScalarCore p _ f x-D.cofactorScalarCore p _ 0 x)) = _ |
| rw [← add_assoc] |
| have hp := (D.patch p).equals (A.euclideanChart (D.patch p).index x) |
| (D.support_patch p hx).2 i j |
| change (traceBilin (D.patch p).matrix (rankTwo _ _) : ℂ) + |
| (D.patch p).coefficient i j (A.euclideanChart (D.patch p).index x) = _ at hp |
| rw [hp,D.cofactorScalarCore_zero_source p _ hx] |
| simp only [ComplexAtlas.euclideanChart_apply,ContinuousLinearEquiv.apply_symm_apply] |
| ring |
|
|
| end GluingData |
| end GlobalElliptic |
|
|
| end |
| end |
|
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| end OAI |
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