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b37e936 | 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 | import OAI.Geometry.RCD.Heat.CanonicalCompactLogLimit
namespace OAI
/-! # The logarithmic heat-kernel limit on compact endpoint types -/
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]
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
end
end OAI
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