| import OAI.Geometry.RCD.Heat.CanonicalCompactLogLimit |
|
|
| namespace OAI |
|
|
| /-! # The logarithmic heat-kernel limit on compact endpoint types -/ |
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|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped Topology |
| noncomputable section |
| namespace RegularPointCoordinates.NoncollapsedRCD |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X) |
| (hn : 2 ≤ n) (p : X) [SeparableSpace X] |
| [IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular] |
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|
| theorem canonicalHeatKernel_endpoint_log_limit (hK : K ≤ 0) |
| {S T : Set X} (ε : ℕ → ℝ) (hε : ∀ k, 0 < ε k) (hε1 : ∀ k, ε k ≤ 1) |
| (hε0 : Tendsto ε atTop (𝓝 0)) {R : ℝ} (hR : 0 ≤ R) |
| (hdS : ∀ a : S, dist p a ≤ R) (hdT : ∀ b : T, dist p b ≤ R) : |
| TendstoUniformly (fun k (q : S × T) => |
| ε k*Real.log (hX.canonicalHeatKernel hn p (ε k/4) q.1 q.2)) |
| (fun q => -(dist (q.1 : X) (q.2 : X))^2) atTop := by |
| let inc : S × T → closedBall p R × closedBall p R := fun q => |
| (⟨q.1,by rw [mem_closedBall,dist_comm]; exact hdS q.1⟩, |
| ⟨q.2,by rw [mem_closedBall,dist_comm]; exact hdT q.2⟩) |
| have hh := (hX.canonicalHeatKernel_compact_log_limit hn p hK ε hε hε1 hε0 hR |
| (T := (1:ℝ)/2) (by norm_num) (by norm_num)).comp inc |
| convert hh using 1 |
| · funext k q |
| dsimp only [Function.comp_def,inc] |
| congr 2 |
| ring_nf |
| · funext q |
| dsimp [inc] |
| ring |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
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|
| end |
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|
| end OAI |
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|
