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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 | import Mathlib
import OAI.Geometry.PrescribedPotential.KaehlerClosedDerivatives
import OAI.Geometry.PrescribedPotential.PatchCutoffs
import OAI.Geometry.PrescribedPotential.RealVolume
import OAI.Geometry.PrescribedPotential.RealPoisson
import OAI.Geometry.PrescribedPotential.RealPoissonEquivalence
namespace OAI
/-! Augmented Volume. -/
section
noncomputable section
open Set Filter Topology
open scoped ContDiff Classical
namespace ExponentialAugment
variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F]
lemma derivative (V : E → F) (L : E →L[ℝ] F) (C : ℝ →L[ℝ] F) (e : E →L[ℝ] ℝ)
(hv : HasStrictFDerivAt V L 0) (h0 : V 0 = C 1) :
HasStrictFDerivAt (fun z : E × ℝ => (Real.exp (-z.2) • V z.1, e z.1))
(((L ∘L ContinuousLinearMap.fst ℝ E ℝ) - (C ∘L ContinuousLinearMap.snd ℝ E ℝ)).prod
(e ∘L ContinuousLinearMap.fst ℝ E ℝ)) 0 := by
let π₁ := ContinuousLinearMap.fst ℝ E ℝ
let π₂ := ContinuousLinearMap.snd ℝ E ℝ
have hv' : HasStrictFDerivAt (fun z : E × ℝ => V z.1) (L ∘L π₁) 0 :=
hv.comp 0 π₁.hasStrictFDerivAt
have he : HasStrictFDerivAt (fun z : E × ℝ => Real.exp (-z.2)) (-π₂) 0 := by
simpa [π₂] using ((-π₂).hasStrictFDerivAt (x := (0 : E × ℝ))).exp
have hh := he.smul hv'
have hn := (e ∘L π₁).hasStrictFDerivAt (x := (0 : E × ℝ))
have hj := hh.prodMk hn
have hlin : (Real.exp (-(0 : E × ℝ).2) • (L ∘L π₁) +
(-π₂).smulRight (V (0 : E × ℝ).1)).prod (e ∘L π₁) =
((L ∘L π₁) - (C ∘L π₂)).prod (e ∘L π₁) := by
apply ContinuousLinearMap.ext
intro z
apply Prod.ext
· change Real.exp (-0) • L z.1 + (-z.2) • V 0 = L z.1 - C z.2
rw [neg_zero, Real.exp_zero, one_smul, h0, ← map_smul]
simp [sub_eq_add_neg]
· rfl
rw [hlin] at hj
exact hj
end ExponentialAugment
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]
namespace GluingData
variable {g : KaehlerMetric A} (D : GluingData g ι)
local instance augmentedVolumeNormedAddCommGroup (s : ℝ) :
NormedAddCommGroup (D.localizers.RealSobolev s) :=
(D.localizers.realCompletion s).normedAddCommGroup
local instance augmentedVolumeNormedSpace (s : ℝ) : NormedSpace ℝ (D.localizers.RealSobolev s) :=
(D.localizers.realCompletion s).normedSpace
local instance augmentedVolumeIsTopologicalAddGroup (s : ℝ) :
IsTopologicalAddGroup (D.localizers.RealSobolev s) :=
Submodule.isTopologicalAddGroup _
local instance augmentedVolumeContinuousSMul (s : ℝ) :
ContinuousSMul ℝ (D.localizers.RealSobolev s) :=
SMulMemClass.continuousSMul _
def augmentedVolume (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X)
(z : D.localizers.RealSobolev ((k : ℝ)+2) × ℝ) :
D.localizers.RealSobolev (k : ℝ) × ℝ :=
(Real.exp (-z.2) • D.realVolume k hk z.1, D.realEvaluation ((k : ℝ)+2) x₀ z.1)
lemma augmentedVolume_zero (k : ℕ) (hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
D.augmentedVolume k hk x₀ 0 = (D.realConstants (k : ℝ) 1, 0) := by
simp [augmentedVolume, D.realVolume_zero]
lemma augmentedVolume_hasStrictFDerivAt_zero (k : ℕ)
(hk : Module.finrank ℝ (EC d) < k) (x₀ : X) :
HasStrictFDerivAt (D.augmentedVolume k hk x₀) (D.realAugmentedL k x₀) 0 := by
exact ExponentialAugment.derivative (D.realVolume k hk) (D.realLOrder k)
(D.realConstants (k : ℝ)) (D.realEvaluation ((k : ℝ)+2) x₀)
(D.realVolume_hasStrictFDerivAt_zero k hk) (D.realVolume_zero k hk)
end GluingData
end GlobalElliptic
end
end
end OAI
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