| import Mathlib |
| import OAI.Geometry.PrescribedRicci.CofactorBilin |
| import OAI.Geometry.PrescribedRicci.CofactorCore |
| import OAI.Geometry.PrescribedPotential.GlobalHessian |
|
|
| namespace OAI |
|
|
| /-! Cofactor Localization. -/ |
|
|
| section |
|
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| |
|
|
| noncomputable section |
| open Set Filter Topology Matrix _root_.MeasureTheory _root_.OAI.MeasureTheory |
| 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 outerCore (p : ι) (f : Smooth A) : 𝓢(EC d,ℂ) := |
| localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.outerCutoff p)) |
| (cutoffGlobal_support _ _) f |
|
|
| def weightCore (p : ι) : 𝓢(EC d,ℂ) := D.forcing p (Smooth.const 1) |
|
|
| lemma outerCore_eventually (p : ι) (f : Smooth A) {x : X} |
| (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) : |
| (D.outerCore p f : EC d → ℂ) =ᶠ[𝓝 (A.euclideanChart (D.patch p).index x)] |
| f ∘ (A.euclideanChart (D.patch p).index).symm := |
| localize_cutoff_eventually _ (D.cutoff p) (D.outerCutoff p) (D.outerCutoff_one p) f |
| ((A.euclideanChart _).mapsTo (D.patch_source p hx)) |
| (by rw [D.cutoff_one p x hx]; exact one_ne_zero) |
| |
| lemma outerHessian_source (p : ι) (i j : Fin d) (φ : SmoothRealFunction A) {x : X} |
| (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) : |
| hessianEntrySchwartz i j (D.outerCore p (Smooth.ofReal φ)) |
| (A.euclideanChart (D.patch p).index x) = |
| φ.hessian (D.patch p).index (A.chart (D.patch p).index x) i j := by |
| have hh := D.localizedHessian_source p i j φ hx |
| change globalize (D.patch p).index (D.cutoff p) |
| (hessianEntrySchwartz i j (D.outerCore p (Smooth.ofReal φ))) x = _ at hh |
| rw [globalize_apply,ite_eq_left (D.patch_source p hx),D.cutoff_one p x hx,one_mul] at hh |
| exact hh |
|
|
| lemma cofactorDifferential_source (p : ι) (φ ψ : SmoothRealFunction A) {x : X} |
| (hx : x ∈ tsupport (D.localizers.weight p : X → ℂ)) : |
| schwartzDifferential |
| (extendedCoefficient (D.patch p).matrix (coefficientBCF (D.cofactorLocalCore p · · (Smooth.ofReal φ)))) |
| (D.outerCore p (Smooth.ofReal ψ)) (A.euclideanChart (D.patch p).index x) = |
| (((g.matrix (D.patch p).index (A.chart (D.patch p).index x)).det⁻¹ * |
| NonlinearHessian.detDifferential |
| (g.matrix (D.patch p).index (A.chart (D.patch p).index x) + |
| φ.hessian (D.patch p).index (A.chart (D.patch p).index x)) |
| (ψ.hessian (D.patch p).index (A.chart (D.patch p).index x))).re : ℂ) := by |
| let q := (D.patch p).index |
| let e := A.euclideanChart q |
| let U := ψ.localExpression q ∘ coordinateEquiv d |
| have hs := D.patch_source p hx |
| have ht := e.mapsTo hs |
| have hU : ContDiffAt ℝ ∞ U (e x) := (ψ.euclidean_smooth q).contDiffAt |
| (by simpa only [e,ComplexAtlas.euclideanChart_target] using e.open_target.mem_nhds ht) |
| have he : (D.outerCore p (Smooth.ofReal ψ) : EC d → ℂ) =ᶠ[𝓝 (e x)] Complex.ofRealCLM ∘ U := |
| D.outerCore_eventually p (Smooth.ofReal ψ) hx |
| rw [schwartzDifferential_apply _ (extendedCoefficient_temperate _ _ |
| (fun i j => (D.cofactorLocalCore p i j (Smooth.ofReal φ)).hasTemperateGrowth))] |
| simp_rw [D.cofactorLocalCore_source p _ _ _ hx,D.cofactorScalarCore_apply, |
| D.metricEntry_apply p _ _ hx,D.inverseDet_apply p hx,D.localizedHessian_source p _ _ φ hx] |
| have hh := cofactor_smooth_expression hU (D.outerCore p (Smooth.ofReal ψ)) he |
| (g.matrix q (A.chart q x)).det⁻¹ |
| (g.matrix q (A.chart q x)+φ.hessian q (A.chart q x)) |
| dsimp only [q,e] at hh |
| simp_rw [D.outerHessian_source p _ _ ψ hx] at hh |
| exact hh.symm |
|
|
| lemma weightCore_apply (p : ι) (y : EC d) : D.weightCore p y = |
| if y ∈ (A.euclideanChart (D.patch p).index).target then |
| D.localizers.weight p ((A.euclideanChart (D.patch p).index).symm y) else 0 := by |
| change localize A (D.localizers.index p) (D.localizers.weight p) (D.localizers.support_sub p) |
| (Smooth.const 1) y = _ |
| simp only [localize_apply,localizeFun,D.index_eq] |
| have hc (z : X) : (Smooth.const (A:=A) 1) z = 1 := rfl |
| simp only [hc,mul_one] |
|
|
| lemma weightCore_outerCore (p : ι) (f : Smooth A) : |
| SchwartzMap.smulLeftCLM ℂ (D.weightCore p) (D.outerCore p f) = D.forcing p f := by |
| ext y |
| rw [SchwartzMap.smulLeftCLM_apply_apply (D.weightCore p).hasTemperateGrowth,smul_eq_mul, |
| D.weightCore_apply] |
| change _ = localizeFun A (D.localizers.index p) (D.localizers.weight p) f y |
| simp only [localizeFun,D.index_eq] |
| split_ifs with hy |
| · by_cases hw : D.localizers.weight p ((A.euclideanChart (D.patch p).index).symm y) = 0 |
| · rw [hw,zero_mul,zero_mul] |
| · have hx := subset_tsupport (D.localizers.weight p : X → ℂ) hw |
| have he := (A.euclideanChart (D.patch p).index).right_inv hy |
| have hh := (D.outerCore_eventually p f hx).self_of_nhds |
| rw [he] at hh |
| rw [hh] |
| rfl |
| · exact zero_mul _ |
|
|
| end GluingData |
| end GlobalElliptic |
|
|
| end |
| end |
|
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| end OAI |
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|
