| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.SphericalBaseProfileRadius |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter Manifold Bundle MeasureTheory |
| open scoped Topology ContDiff ENNReal |
| open Matrix |
| open scoped Topology Matrix.Norms.Elementwise |
| namespace YauCounterexamples |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] |
| {M : Type*} [TopologicalSpace M] [ChartedSpace E M] [IsManifold 𝓘(ℝ, E) ∞ M] |
|
|
| local instance chartExtDualNorm : NormedAddCommGroup (E →L[ℝ] ℝ) := inferInstance |
| local instance chartExtDualSpace : NormedSpace ℝ (E →L[ℝ] ℝ) := inferInstance |
| local instance chartExtFormNorm : NormedAddCommGroup (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance |
| local instance chartExtFormSpace : NormedSpace ℝ (E →L[ℝ] E →L[ℝ] ℝ) := inferInstance |
|
|
| def euclideanMetricForm : E →L[ℝ] E →L[ℝ] ℝ := innerSL ℝ |
|
|
| theorem exists_compact_chart_metric_extension (g : SmoothMetric E M) (p : M) |
| {K : Set E} (hK : IsCompact K) (hKO : K ⊆ (chartAt E p).target) : |
| ∃ ĝ : SmoothMetric E E, ∀ᶠ y in 𝓝ˢ K, ∀ v w : E, |
| ĝ.inner y v w = g.inner ((chartAt E p).symm y) |
| (mfderiv 𝓘(ℝ, E) 𝓘(ℝ, E) (chartAt E p).symm y v) |
| (mfderiv 𝓘(ℝ, E) 𝓘(ℝ, E) (chartAt E p).symm y w) := by |
| obtain ⟨V, hV, hKV, hVO⟩ := normal_exists_closure_subset hK.isClosed (chartAt E p).open_target hKO |
| obtain ⟨η, hηone, hηzero, hηrange⟩ := |
| exists_contMDiffMap_one_nhds_of_subset_interior (I := 𝓘(ℝ, E)) (n := ⊤) (t := V) |
| hK.isClosed (by simpa only [hV.interior_eq] using hKV) |
| have hη : ContDiff ℝ ∞ η := contMDiff_iff_contDiff.mp η.contMDiff |
| have hsη : tsupport η ⊆ (chartAt E p).target := by |
| apply (closure_mono (show Function.support η ⊆ V from ?_)).trans hVO |
| intro x hx |
| by_contra hn |
| exact hx (hηzero x hn) |
| have hηφ : ContDiff ℝ ∞ (fun y => η y • chartMetricForm g p y) := by |
| rw [← contMDiff_iff_contDiff] |
| apply contMDiff_of_tsupport |
| intro y hy |
| have hyO := hsη (tsupport_smul_subset_left η (chartMetricForm g p) hy) |
| rw [contMDiffAt_iff_contDiffAt] |
| exact hη.contDiffAt.smul (((chartMetricForm_contDiffOn g p) y hyO).contDiffAt |
| ((chartAt E p).open_target.mem_nhds hyO)) |
| let B : E → E →L[ℝ] E →L[ℝ] ℝ := |
| fun y => η y • chartMetricForm g p y + (1-η y) • (euclideanMetricForm (E := E)) |
| have hB : ContDiff ℝ ∞ B := hηφ.add ((contDiff_const.sub hη).smul contDiff_const) |
| have hsym : ∀ y v w, B y v w = B y w v := by |
| intro y v w |
| change η y * chartMetricForm g p y v w + (1-η y) * inner ℝ v w = |
| η y * chartMetricForm g p y w v + (1-η y) * inner ℝ w v |
| rw [chartMetricForm_symm, real_inner_comm] |
| have hpos : ∀ y v, v ≠ 0 → 0 < B y v v := by |
| intro y v hv |
| have hip : 0 < inner ℝ v v := real_inner_self_pos.mpr hv |
| change 0 < η y * chartMetricForm g p y v v + (1-η y) * inner ℝ v v |
| by_cases hy : y ∈ (chartAt E p).target |
| · have hp := chartMetricForm_pos g p hy hv |
| obtain ⟨hη0, hη1⟩ := hηrange y |
| by_cases hz : η y = 0 |
| · simpa only [hz, zero_mul, sub_zero, one_mul, zero_add] using hip |
| · exact add_pos_of_pos_of_nonneg (mul_pos (lt_of_le_of_ne hη0 (Ne.symm hz)) hp) |
| (mul_nonneg (sub_nonneg.mpr hη1) hip.le) |
| · have hnV : y ∉ V := fun hyV => hy (hVO (subset_closure hyV)) |
| simpa only [hηzero y hnV, zero_mul, sub_zero, one_mul, zero_add] using hip |
| refine ⟨flatMetricOfForm B hB hsym hpos, ?_⟩ |
| filter_upwards [hηone] with y hy |
| intro v w |
| change B y v w = _ |
| rw [← chartMetricForm_pairing] |
| simp [B, hy] |
| end YauCounterexamples |
|
|
| end |
|
|
| end OAI |
|
|
