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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import Mathlib
import OAI.Geometry.PrescribedPotential.FrozenEllipticJets
import OAI.Geometry.PrescribedPotential.NonlinearHessianCommutator
import OAI.Geometry.PrescribedPotential.NonlinearRemainderStrong
import OAI.Geometry.PrescribedPotential.RealSobolev
import OAI.Geometry.PrescribedPotential.RegularityCoordinates
namespace OAI
/-! Commutator Support. -/
section
noncomputable section
open Set Filter Topology Matrix LineDeriv
open scoped ContDiff SchwartzMap Classical Matrix.Norms.Elementwise
namespace GlobalElliptic
open Anticanonical SourceSmooth EllipticKernel SobolevChart
variable {d : ℕ} {X : Type*} [TopologicalSpace X] [T2Space X] [CompactSpace X]
{A : ComplexAtlas d X} {ι : Type*} [Fintype ι]
lemma hessianLocalizedError_zero {κ : 𝓢(EC d, ℂ)} (v : EC d) (f : 𝓢(EC d, ℂ))
{y : EC d} (hy : y ∉ tsupport (κ : EC d → ℂ)) (i j : Fin d) :
hessianLocalizedError κ v i j f y = 0 := by
have hz := notMem_tsupport_iff_eventuallyEq.mp hy
have he : (SchwartzMap.smulLeftCLM ℂ κ (∂_{v} f) : EC d → ℂ) =ᶠ[𝓝 y]
(0 : 𝓢(EC d, ℂ)) := by
filter_upwards [hz] with z hzz
rw [SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth]
change κ z * _ = 0
simp only [hzz,Pi.zero_apply,zero_mul]
have hh := congrArg (fun u : 𝓢(EC d, ℂ) => u y) (hessianLocalizedError_identity κ f v i j)
rw [hessianEntrySchwartz_congr he,map_zero,_root_.zero_apply] at hh
simp only [_root_.add_apply] at hh
rw [SchwartzMap.smulLeftCLM_apply_apply κ.hasTemperateGrowth] at hh
change 0 = κ y * _ + hessianLocalizedError κ v i j f y at hh
simp only [hz.eq_of_nhds,Pi.zero_apply,zero_mul,zero_add] at hh
exact hh.symm
omit [T2Space X] [CompactSpace X] in
lemma source_hessian_zero_of_local {φ : SmoothRealFunction A} {x : X}
(he : φ.value =ᶠ[𝓝 x] (fun _ => 0)) (q : Fin A.count)
(hx : x ∈ (A.chart q).source) : φ.hessian q (A.chart q x) = 0 := by
have ht := (A.chart q).mapsTo hx
have hc : Tendsto (A.chart q).symm (𝓝 (A.chart q x)) (𝓝 x) := by
simpa only [(A.chart q).left_inv hx] using
((A.chart q).symm.continuousAt ht).tendsto
have hh : φ.localExpression q =ᶠ[𝓝 (A.chart q x)] (fun _ => 0) := he.comp_tendsto hc
have h2 := (hh.fderiv (𝕜 := ℝ)).fderiv_eq (𝕜 := ℝ)
change PotentialKaehler.hermitianPartMatrix (fderiv ℝ (fderiv ℝ (φ.localExpression q))
(A.chart q x)) = 0
rw [h2]
ext i j
simp [PotentialKaehler.hermitianPartMatrix_apply,PotentialKaehler.hermitianPart]
rfl
namespace GluingData
variable {g : KaehlerMetric A} (D : GluingData g ι)
def derivativeSupport (p : ι) : Set X :=
(A.euclideanChart (D.patch p).index).symm '' tsupport (D.cutoff p : EC d → ℂ)
lemma localizedDerivative_tsupport (p : ι) (v : EC d) (f : Smooth A) :
tsupport (D.localizedDerivative p v f : X → ℂ) ⊆ D.derivativeSupport p := by
rw [D.localizedDerivative_regularity]
exact (globalize_tsupport _ _ _).trans (image_mono (D.cutoff p).realPart_support)
omit [T2Space X] [CompactSpace X] in
lemma derivativeSupport_source {p : ι} {x : X} (hx : x ∈ D.derivativeSupport p) :
x ∈ (A.euclideanChart (D.patch p).index).source ∧
A.euclideanChart (D.patch p).index x ∈ tsupport (D.cutoff p : EC d → ℂ) := by
obtain ⟨y,hy,rfl⟩ := hx
have ht := (D.cutoff p).support_sub hy
exact ⟨(A.euclideanChart (D.patch p).index).symm.mapsTo ht,
by rwa [(A.euclideanChart (D.patch p).index).right_inv ht]⟩
omit [T2Space X] [CompactSpace X] in
lemma outsideDerivativeSupport_coord {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
(hs : x ∈ (A.euclideanChart (D.patch p).index).source) :
A.euclideanChart (D.patch p).index x ∉ tsupport (D.cutoff p : EC d → ℂ) := by
intro hh
exact hx ⟨_,hh,(A.euclideanChart (D.patch p).index).left_inv hs⟩
lemma localizedDerivative_zero {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
(v : EC d) (f : Smooth A) : D.localizedDerivative p v f x = 0 :=
image_eq_zero_of_notMem_tsupport (fun hh => hx (D.localizedDerivative_tsupport p v f hh))
lemma regularityError_zero {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
(v : EC d) (f : Smooth A) (i j : Fin d) : D.regularityError p v i j f x = 0 := by
change globalize (D.patch p).index (D.regularityCutoff p) _ x = 0
rw [globalize_apply]
split_ifs with hs
· have hh := D.outsideDerivativeSupport_coord hx hs
have hz := hessianLocalizedError_zero v
(localize A (D.patch p).index (cutoffGlobal (D.patch p).index (D.regularityOuter p))
(cutoffGlobal_support _ _) f)
(fun hh' => hh ((D.cutoff p).realPart_support hh')) i j
exact mul_eq_zero_of_right _ hz
· rfl
lemma nonlinearRemainder_zero {p : ι} {x : X} (hx : x ∉ D.derivativeSupport p)
(k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (v : EC d) (f : Smooth A) :
D.localizers.strong (k : ℝ)
(D.nonlinearRemainder k hk p v (D.localizers.embed ((k : ℝ)+2) f)) x = 0 := by
rw [D.nonlinearRemainder_strong_embed]
have hm : (fun (i j : Fin d) => D.regularityError p v i j f x -
D.localizedDerivative p v (D.regularityMetric p i j) x) = 0 := by
ext i j
rw [D.regularityError_zero hx,D.localizedDerivative_zero hx,sub_self]
rfl
let M : Matrix (Fin d) (Fin d) ℂ := fun i j =>
D.regularityMetric p i j x + D.regularityHessian p i j f x
change D.regularityInverseDet p x * NonlinearHessian.detDifferential M _ -
D.localizedDerivative p v (D.regularityInverseDet p) x * M.det = 0
rw [hm]
change D.regularityInverseDet p x * NonlinearHessian.detDifferential M (0 : Matrix (Fin d) (Fin d) ℂ) -
D.localizedDerivative p v (D.regularityInverseDet p) x * M.det = 0
rw [(NonlinearHessian.detDifferential M).map_zero,D.localizedDerivative_zero hx,
mul_zero,zero_mul,sub_self]
lemma localizedDerivative_source_hessian_zero {p : ι} {x : X}
(hx : x ∉ D.derivativeSupport p) (v : EC d) (φ : SmoothRealFunction A)
(q : Fin A.count) (hq : x ∈ (A.chart q).source) :
((D.realLocalizedDerivative p v (RealSmooth.ofReal φ)).source).hessian q (A.chart q x) = 0 := by
apply source_hessian_zero_of_local _ q hq
have hn : x ∉ tsupport (D.localizedDerivative p v (Smooth.ofReal φ) : X → ℂ) :=
fun hh => hx (D.localizedDerivative_tsupport p v (Smooth.ofReal φ) hh)
have he := notMem_tsupport_iff_eventuallyEq.mp hn
filter_upwards [he] with y hy
change (D.localizedDerivative p v (Smooth.ofReal φ) y).re = 0
rw [hy]
rfl
end GluingData
end GlobalElliptic
end
end
end OAI
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