| import Mathlib |
| import OAI.Geometry.SmoothYau.Estimates.VaryingPatchCoefficients |
| import OAI.Geometry.SmoothYau.Estimates.CoefficientPerturbationTendsto |
| import OAI.Geometry.SmoothYau.Smoothness.ChartDerivativeNeZeroTransfer |
| import OAI.Geometry.SmoothYau.Smoothness.ChartTensorCoordLocal1 |
| import OAI.Geometry.SmoothYau.Smoothness.JointMetricDet |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Function Manifold |
| open scoped Topology ContDiff SchwartzMap |
| namespace YauCounterexamples |
| namespace ContinuousSmoothFamilyOn |
| variable {P E : Type*} [TopologicalSpace P] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] {U : Set E} |
| lemma complex_mul {f g : P → E → ℂ} (hf : ContinuousSmoothFamilyOn f U) |
| (hg : ContinuousSmoothFamilyOn g U) : ContinuousSmoothFamilyOn (fun p x => f p x*g p x) U := |
| (hf.prodMk hg).postcomp (fun z : ℂ × ℂ => z.1*z.2) (fun _ _ _ => by fun_prop) |
| lemma cutoff_complex (χ : E → ℂ) (hχ : ContDiff ℝ ∞ χ) |
| {f : P → E → ℂ} (hU : IsOpen U) (hs : tsupport χ ⊆ U) |
| (hf : ContinuousSmoothFamilyOn f U) : |
| ContinuousSmoothFamilyOn (fun p x => χ x*f p x) univ := by |
| apply of_local |
| intro x hx |
| by_cases hxs : x ∈ tsupport χ |
| · exact ⟨U,hU,hs hxs,(const χ (fun y _ => hχ.contDiffAt)).complex_mul hf⟩ |
| · refine ⟨(tsupport χ)ᶜ,(isClosed_tsupport χ).isOpen_compl,hxs,?_⟩ |
| apply (scalar_const (fun _ : P => (0:ℂ)) continuous_const).congr (isClosed_tsupport χ).isOpen_compl |
| intro p y hy |
| simp only [image_eq_zero_of_notMem_tsupport hy,zero_mul] |
| end ContinuousSmoothFamilyOn |
| variable {P E M : Type*} [TopologicalSpace P] |
| [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ,E) ∞ M] |
| [CompactSpace M] [T2Space M] |
| {g : SmoothMetric E M} {k : ℕ} {hs : Module.finrank ℝ E < 2*(2*(k:ℝ))} |
| namespace CompactMetricAtlas |
| variable (A : CompactMetricAtlas g k hs) |
|
|
| omit [T2Space M] in |
| theorem continuousFamily_varyingLaplacian_tendsto (q : P → SmoothMetric E M) |
| (hq : ∀ i j l, ContinuousSmoothFamilyOn |
| (fun t y => metricCoefficients (q t) (A.p i) y j l) (chartAt E (A.p i)).target) |
| (C : ∀ t i, A.VaryingPatchCoefficients (q t) i) (p₀ : P) : |
| Tendsto (fun t => A.varyingLaplacianCLM (q t) (C t)) (𝓝 p₀) |
| (𝓝 (A.varyingLaplacianCLM (q p₀) (C p₀))) := by |
| apply A.varyingLaplacianCLM_tendsto q (q p₀) C (C p₀) |
| · intro i j l |
| apply (schwartzToSobolev (2*(k:ℝ))).continuous.tendsto _ |>.comp |
| apply Continuous.tendsto |
| apply continuous_schwartz_common_support _ ?_ (tsupport (A.χcoord i)) (A.compact_χcoord i) |
| (fun t => A.metricCoefficientSchwartz_common_support i _ _) |
| intro n |
| have he (t : P) : (A.metricCoefficientSchwartz i ((C t i).a j l) ((C t i).smooth_a j l) : E → ℂ) = |
| fun y => A.χcoord i y * ((metricCoefficients (q t) (A.p i) y)⁻¹ j l : ℂ) := |
| funext (A.varyingCoefficientSchwartz_a_value (q t) i (C t i) j l) |
| simp_rw [he] |
| have hf := ((family_metricInverse q (A.p i) (hq i) j l).postcomp |
| (fun x : ℝ => (x:ℂ)) (fun _ _ _ => Complex.ofRealCLM.contDiff.contDiffAt)).cutoff_complex |
| (A.χcoord i) (A.smooth_χcoord i) (chartAt E (A.p i)).open_target (A.χcoord_target i) |
| exact continuousOn_univ.mp (by simpa only [univ_prod_univ] using hf.jets n) |
| · intro i j |
| apply (schwartzToSobolev (2*(k:ℝ))).continuous.tendsto _ |>.comp |
| apply Continuous.tendsto |
| apply continuous_schwartz_common_support _ ?_ (tsupport (A.χcoord i)) (A.compact_χcoord i) |
| (fun t => A.metricCoefficientSchwartz_common_support i _ _) |
| intro n |
| have he (t : P) : (A.metricCoefficientSchwartz i ((C t i).c j) ((C t i).smooth_c j) : E → ℂ) = |
| fun y => A.χcoord i y * (metricFirstCoefficient (q t) (A.p i) j y : ℂ) := |
| funext (A.varyingCoefficientSchwartz_c_value (q t) i (C t i) j) |
| simp_rw [he] |
| have hf := ((family_metricFirstCoefficient q (A.p i) (hq i) j).postcomp |
| (fun x : ℝ => (x:ℂ)) (fun _ _ _ => Complex.ofRealCLM.contDiff.contDiffAt)).cutoff_complex |
| (A.χcoord i) (A.smooth_χcoord i) (chartAt E (A.p i)).open_target (A.χcoord_target i) |
| exact continuousOn_univ.mp (by simpa only [univ_prod_univ] using hf.jets n) |
| end CompactMetricAtlas |
| end YauCounterexamples |
| end |
|
|
| end OAI |
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