| import OAI.Geometry.NodalSets.Charts.SphereIndexedDivergence |
| import OAI.Geometry.NodalSets.Spectral.FiniteEigenframeCaptureLemmas |
|
|
| namespace OAI |
|
|
| namespace Yau.Target |
| open MeasureTheory Manifold Set Filter Yau.Analysis |
| open scoped ContDiff Topology |
| noncomputable section |
| local instance sphereCoverageMeasurable : MeasurableSpace Base := borel Base |
| local instance sphereCoverageBorel : BorelSpace Base := ⟨rfl⟩ |
|
|
| theorem sphere_resolvent_eigenvalue_index (d : SphereEnergyData) |
| (hrho : ∀ p : Base, ContDiff ℝ ∞ (fun x ↦ d.density (sphereChartCoordMap p x))) |
| (f : SphereWeightedL2 d) (hf : f ≠ 0) (a : ℝ) (hfa : sphereL2Resolvent d f=a • f) : |
| ∃ N : ℕ, sphereIndexedEigenvalue d N = a⁻¹-1 := by |
| have ha := (sphereL2Resolvent_eigenvalue_bounds d a f hf hfa).1 |
| obtain ⟨N,hN⟩ := ((sphereIndexedEigenvalue_tendsto_atTop d hrho).eventually |
| (eventually_gt_atTop (a⁻¹-1))).exists |
| obtain ⟨u,mu,ho,he,hanti,hmax⟩ := sphere_resolvent_finite_smooth_frame d hrho (N+1) |
| have heq := sphereIndexedEigenvalue_eq_frame d u mu ho (fun i ↦ (he i).1) |
| (fun i ↦ (he i).2.2) hanti hmax |
| rw [heq] at hN |
| have hlt : mu (Fin.last N) < a := (inv_lt_inv₀ ha (he (Fin.last N)).1).mp (by linarith) |
| obtain ⟨i,hi⟩ := finite_eigenframe_captures_above_last (sphereL2Resolvent d) |
| (sphereL2Resolvent_symmetric d) (fun i ↦ sphereEnergyL2Linear d (u i)) mu |
| (fun i ↦ (he i).2.2) hmax f hf a hfa hlt |
| refine ⟨i.val,?_⟩ |
| rw [sphereIndexedEigenvalue_eq_frame_index d u mu ho (fun i ↦ (he i).1) |
| (fun i ↦ (he i).2.2) hanti hmax i,hi] |
|
|
| theorem sphere_smooth_intrinsic_eigenvalue_index (d : SphereEnergyData) |
| (hr : ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ d.density) |
| (u : SphereEnergySmooth d) (hu : u ≠ 0) (lam : ℝ) |
| (he : ∀ p z, -intrinsicWeightedChartOperator d.tensor d.density (SphereEnergySmooth.toSmooth d u) p z = |
| lam*(SphereEnergySmooth.toSmooth d u : Base → ℝ) ((extChartAt (𝓡 4) p).symm z)) : |
| ∃ N : ℕ, sphereIndexedEigenvalue d N=lam := by |
| obtain ⟨hlam,hun,heu⟩ := sphere_smooth_intrinsic_eigen_to_resolvent d hr u hu lam he |
| obtain ⟨N,hN⟩ := sphere_resolvent_eigenvalue_index d |
| (fun p ↦ (hr.comp (sphereChartCoordMap_smooth p)).contDiff) _ hun (lam+1)⁻¹ heu |
| exact ⟨N,by simpa using hN⟩ |
|
|
| end |
| end Yau.Target |
|
|
| end OAI |
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|
