| import OAI.Geometry.RCD.Metric.BoundedRegionVolume |
|
|
| namespace OAI |
|
|
| /-! # Uniform polynomial normalization volumes on a bounded region -/ |
|
|
| open Set MeasureTheory Metric |
| open scoped NNReal ENNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
|
|
| theorem NoncollapsedRCD.bounded_heat_volume_power {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) (p : X) (R : ℝ) : |
| ∃ c : ℝ, 0 < c ∧ ∀ y, dist y p ≤ R → ∀ t, 0 < t → t ≤ 1 → |
| c*(Real.sqrt t)^n ≤ (hausdorff n X).real (ball y (Real.sqrt t)) := by |
| obtain ⟨m,hm,hvol⟩ := hX.ball_volume_lower_on_bounded_region hn p R 1 zero_lt_one |
| let E := 2*Real.exp (4*Real.sqrt |K|) |
| have hE : 0 < E := by dsimp only [E]; positivity |
| refine ⟨m/E^n,div_pos hm (pow_pos hE n),?_⟩ |
| intro y hy t ht ht1 |
| have hroot : Real.sqrt t ≤ 1 := by nlinarith [Real.sq_sqrt ht.le,Real.sqrt_nonneg t] |
| have hk : -|K| ≤ K*(1:ℝ)^2 := by simpa using neg_abs_le K |
| have hh := hX.rescaled_ball_volume_lower hn (abs_nonneg K) (Real.sqrt_pos.mpr ht) |
| hroot y 1 zero_lt_one hk y (by simp) |
| simp only [mul_one] at hh |
| calc |
| (m/E^n)*(Real.sqrt t)^n = (Real.sqrt t/E)^n*m := by rw [div_pow]; ring |
| _ ≤ (Real.sqrt t/E)^n*(hausdorff n X).real (ball y 1) := |
| mul_le_mul_of_nonneg_left (hvol y hy) (by positivity) |
| _ ≤ _ := hh |
|
|
| end RegularPointCoordinates |
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|
| end |
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| end OAI |
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|
