| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.LocalMetricCharts |
| import OAI.Analysis.BiholderTransport.Coordinates.PoleCoordinates |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Metric Manifold Bundle |
| open scoped Topology ContDiff NNReal |
|
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| namespace WeakMTWTransport |
| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace M] [Nonempty M] |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ,Model n) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ,Model n) ∞ (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x)] |
| [IsRiemannianManifold 𝓘(ℝ,Model n) M] |
|
|
| lemma exists_chart_hopf_gradient_bound (c:M) : |
| ∃K≥0,∀ᶠ z in 𝓝 (extChartAt 𝓘(ℝ,Model n) c c),∀t:ℝ,1/2 ≤ t → |
| ∀u:M → ℝ,Continuous u → |
| ‖fderiv ℝ (fun w=>hopfLax t u ((extChartAt 𝓘(ℝ,Model n) c).symm w)) z‖ ≤ K := by |
| let : IsContinuousRiemannianBundle (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ,Model n) x) := |
| continuousRiemannianBundle_of_smooth (IB := 𝓘(ℝ,Model n)) |
| obtain ⟨L,r,hr,hT,hL⟩ := exists_lipschitzOn_inverse_chart (E := Model n) c |
| let D:ℝ≥0:=Real.toNNReal (Metric.diam (univ:Set M)) |
| have hD:∀x y:M,dist x y ≤ D := by |
| intro x y |
| rw [Real.coe_toNNReal _ Metric.diam_nonneg] |
| exact Metric.dist_le_diam_of_mem isCompact_univ.isBounded (mem_univ x) (mem_univ y) |
| refine ⟨2*(D:ℝ)*L,by positivity,?_⟩ |
| filter_upwards [ball_mem_nhds (extChartAt 𝓘(ℝ,Model n) c c) hr] with z hz |
| intro t ht u hu |
| have ht0:0 < t := by linarith |
| have hn:=norm_fderiv_le_of_lipschitzOn ℝ (isOpen_ball.mem_nhds hz) |
| ((hopfLax_lipschitz hu hD ht0).comp_lipschitzOnWith hL) |
| have hdiv:(D:ℝ)/t ≤ 2*D := (div_le_iff₀ ht0).mpr (by nlinarith [D.coe_nonneg]) |
| calc |
| _ ≤ ↑((D/Real.toNNReal t)*L:ℝ≥0) := hn |
| _ = (D:ℝ)/t*L := by rw [NNReal.coe_mul,NNReal.coe_div,Real.coe_toNNReal _ ht0.le] |
| _ ≤ _ := mul_le_mul_of_nonneg_right hdiv L.coe_nonneg |
|
|
| end WeakMTWTransport |
|
|
| end |
|
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| end OAI |
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|
