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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 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import Mathlib
import OAI.Geometry.PrescribedRicci.CofactorBilin
import OAI.Geometry.PrescribedRicci.CofactorCore
import OAI.Geometry.PrescribedPotential.GlobalHessian
namespace OAI
/-! Cofactor Localization. -/
section
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
end OAI
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