| import Mathlib.Analysis.SpecificLimits.Basic |
| import OAI.Geometry.NodalSets.Coefficients.SphereCoefficientNormConvergence |
|
|
| namespace OAI |
|
|
| namespace Yau.Target |
| open Manifold Yau.Geometry Filter |
| open scoped ContDiff Topology |
| noncomputable section |
|
|
| theorem sphere_coefficient_diagonal_summable (P : Finset Base) (d : ℕ → SphereEnergyData) |
| (hd : ∀ k, ContMDiff (𝓡 4) 𝓘(ℝ,ℝ) ∞ (d k).density) |
| (hstep : ∀ k, sphereCoefficientDistance P (k+1) (d k).tensor (d k).density |
| (d (k+1)).tensor (d (k+1)).density < (1/2:ℝ)^(k+1)) : |
| ∀ J : ℕ, Summable (fun k ↦ sphereCoefficientDistance P J (d k).tensor (d k).density |
| (d (k+1)).tensor (d (k+1)).density) := by |
| intro J |
| have hgeom : Summable (fun k : ℕ ↦ (1/2:ℝ)^(k+1)) := |
| summable_geometric_two.comp_injective (fun _ _ h ↦ Nat.succ.inj h) |
| apply Summable.of_norm_bounded_eventually hgeom |
| rw [Nat.cofinite_eq_atTop] |
| filter_upwards [eventually_ge_atTop J] with k hk |
| have hnonneg := sphereCoefficientDistance_nonneg P J (d k).tensor (d (k+1)).tensor |
| (d k).density (d (k+1)).density |
| (fun p _ ↦ intrinsic_coefficient_chart_smooth (d k).tensor (d k).smooth (d k).symm (d k).pos (d k).density (hd k) p) |
| (fun p _ ↦ intrinsic_coefficient_chart_smooth (d (k+1)).tensor (d (k+1)).smooth (d (k+1)).symm (d (k+1)).pos |
| (d (k+1)).density (hd (k+1)) p) |
| rw [Real.norm_eq_abs,abs_of_nonneg hnonneg] |
| exact (sphereCoefficientDistance_mono_order P (d k) (d (k+1)) (hd k) (hd (k+1)) J (k+1) (by omega)).trans (hstep k).le |
|
|
| end |
| end Yau.Target |
|
|
| end OAI |
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|
