| import Mathlib |
| import OAI.Geometry.CubicTorus.Regularity.SurfaceH1Tangency |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma restricted_L2_extension {F : Type*} [NormedAddCommGroup F] |
| {S : Set Plane} (hS : MeasurableSet S) (u : Lp F 2 (volume.restrict S)) : |
| ∃ U : Lp F 2 (volume : Measure Plane), |
| (U : Plane → F) =ᵐ[volume.restrict S] (u : Plane → F) := by |
| have hm : MemLp (S.indicator (u : Plane → F)) 2 (volume : Measure Plane) := |
| (memLp_indicator_iff_restrict hS).mpr (Lp.memLp u) |
| refine ⟨hm.toLp _, ?_⟩ |
| filter_upwards [ae_restrict_of_ae hm.coeFn_toLp, ae_restrict_mem hS] with y hy hys |
| exact hy.trans (indicator_of_mem hys _) |
|
|
| theorem PlaneWeakGradientOn.cmc_classical_near |
| {p : Plane → Plane} (hp : Continuous p) (hpb : ∀ y, ‖p y‖ ≤ 1) |
| {R α D : ℝ} (hR : 0 < R) (hα : 0 < α) (hα1 : α ≤ 1) (hD : 0 ≤ D) |
| (hHolder : ∀ x ∈ Metric.ball (0 : Plane) R, ∀ y ∈ Metric.ball (0 : Plane) R, |
| ‖cmcJacobian (p x) - cmcJacobian (p y)‖ ≤ D * (dist x y)^α) |
| {u : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| {G : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| (hw : PlaneWeakGradientOn (Metric.ball 0 R) u G) |
| (hd : ∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ Metric.ball 0 R → |
| (∫ y, fderiv ℝ ψ y (cmcJacobian (p y) (G y))) = 0) |
| {f : Plane → ℝ} (hf : ContinuousOn f (Metric.ball 0 R)) |
| (hfu : f =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] (u : Plane → ℝ)) : |
| HolderJet 1 f 0 ∧ LinearCMCEquationNear f p (fun _ => 0) := by |
| obtain ⟨U,hU⟩ := restricted_L2_extension measurableSet_ball u |
| obtain ⟨V,hV⟩ := restricted_L2_extension measurableSet_ball G |
| have hUV := hw.toWeakGradient.local_congr measurableSet_ball hU hV |
| have hVD : ∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ Metric.ball 0 R → |
| (∫ y, fderiv ℝ ψ y (cmcJacobian (p y) (V y))) = 0 := by |
| intro ψ hψ hc hs |
| rw [← hd ψ hψ hc hs] |
| apply integral_congr_ae |
| filter_upwards [(ae_restrict_iff' measurableSet_ball).mp hV] with y hy |
| by_cases hys : y ∈ Metric.ball (0 : Plane) R |
| · rw [hy hys] |
| · rw [fderiv_of_notMem_tsupport ℝ (fun hh => hys (hs hh))] |
| rfl |
| obtain ⟨H,s,K,β,hs,hsR,hK,hβ,hβ1,hHc,hHe,hHp⟩ := |
| cmc_linearized_holder_gradient hp hpb hR hα hα1 hD hHolder U V hUV hVD |
| have hsub : Metric.ball (0 : Plane) s ⊆ Metric.ball 0 R := |
| Metric.ball_subset_ball (by linarith) |
| have hfu' : f =ᵐ[volume.restrict (Metric.ball (0 : Plane) s)] (u : Plane → ℝ) := |
| ae_restrict_of_ae_restrict_of_subset hsub hfu |
| have hU' : (U : Plane → ℝ) =ᵐ[volume.restrict (Metric.ball (0 : Plane) s)] (u : Plane → ℝ) := |
| ae_restrict_of_ae_restrict_of_subset hsub hU |
| have hwf : PlaneWeakGradient (Metric.ball 0 s) f H := |
| (hUV.mono hsub).local_congr measurableSet_ball |
| (hfu'.trans hU'.symm) hHe |
| obtain ⟨hfc,hfd⟩ := hwf.holder_C1 hs (hf.mono hsub) hHc hHp |
| have hdr (y : Plane) (hy : y ∈ Metric.ball (0 : Plane) (s / 4)) : |
| gradient f y = H y := by |
| have ht := hwf.continuous_hasFDerivAt hs (hf.mono hsub) hHc hy |
| simp only [gradient, ht.fderiv, LinearIsometryEquiv.symm_apply_apply] |
| refine ⟨⟨hfc.contDiffAt (Metric.ball_mem_nhds _ (by positivity)), |
| s / 4, β, K, by positivity, hβ, hβ1, hK, hfd⟩, s / 4, by positivity, ?_⟩ |
| intro ψ hψ hc ht |
| have hss : Metric.ball (0 : Plane) (s/4) ⊆ Metric.ball 0 s := |
| Metric.ball_subset_ball (by linarith) |
| rw [← hVD ψ hψ hc (ht.trans (hss.trans hsub))] |
| apply integral_congr_ae |
| filter_upwards [(ae_restrict_iff' measurableSet_ball).mp hHe] with y hy |
| by_cases hys : y ∈ Metric.ball (0 : Plane) (s/4) |
| · simp only [hdr y hys, hy (hss hys), sub_zero] |
| · rw [fderiv_of_notMem_tsupport ℝ (fun hh => hys (ht hh))] |
| rfl |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem PlaneWeakGradientOn.cmc_smoothAt |
| {p : Plane → Plane} (hp : Continuous p) (hpb : ∀ y, ‖p y‖ ≤ 1) |
| {R α D : ℝ} (hR : 0 < R) (hα : 0 < α) (hα1 : α ≤ 1) (hD : 0 ≤ D) |
| (hHolder : ∀ x ∈ Metric.ball (0 : Plane) R, ∀ y ∈ Metric.ball (0 : Plane) R, |
| ‖cmcJacobian (p x) - cmcJacobian (p y)‖ ≤ D * (dist x y)^α) |
| (hpc : ContDiffAt ℝ (⊤ : ℕ∞) p 0) |
| {u : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| {G : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| (hw : PlaneWeakGradientOn (Metric.ball 0 R) u G) |
| (hd : ∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ Metric.ball 0 R → |
| (∫ y, fderiv ℝ ψ y (cmcJacobian (p y) (G y))) = 0) |
| {f : Plane → ℝ} (hf : ContinuousOn f (Metric.ball 0 R)) |
| (hfu : f =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] (u : Plane → ℝ)) : |
| ContDiffAt ℝ (⊤ : ℕ∞) f 0 := by |
| obtain ⟨hfc,heq⟩ := hw.cmc_classical_near hp hpb hR hα hα1 hD hHolder hd hf hfu |
| apply contDiffAt_infty.mpr |
| intro n |
| have hpJ : HolderJet (n+1) p 0 := ContDiffAt.holderJet (hpc.of_le (by |
| exact_mod_cast (show (n+2 : ℕ∞) ≤ ⊤ from le_top))) |
| exact (heq.schauder hpb hfc hpJ (HolderJet.const 0)).contDiffAt.of_le (by norm_cast; omega) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma continuous_memLp_top_ball {F : Type*} [NormedAddCommGroup F] |
| {f : Plane → F} (hf : Continuous f) (R : ℝ) : |
| MemLp f ∞ (volume.restrict (Metric.ball (0 : Plane) R)) := by |
| obtain ⟨C,hC⟩ := (isCompact_closedBall (0 : Plane) R).exists_bound_of_continuousOn hf.continuousOn |
| exact memLp_top_of_bound hf.aestronglyMeasurable C |
| ((ae_restrict_mem measurableSet_ball).mono fun y hy => hC y (Metric.ball_subset_closedBall hy)) |
|
|
| theorem PlaneWeakGradientOn.weighted_L2 {R : ℝ} |
| {u : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| {G : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| (hw : PlaneWeakGradientOn (Metric.ball 0 R) u G) |
| {q : Plane → ℝ} (hq : ContDiff ℝ (⊤ : ℕ∞) q) : |
| ∃ (v : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) R))) |
| (J : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) R))), |
| (v =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] fun y => q y * u y) ∧ |
| (J =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] |
| fun y => q y • G y + u y • gradient q y) ∧ |
| PlaneWeakGradientOn (Metric.ball 0 R) v J := by |
| have hqL := continuous_memLp_top_ball hq.continuous R |
| have hdL := continuous_memLp_top_ball (plane_gradient_contDiff hq).continuous R |
| have hv : MemLp (fun y => q y * u y) 2 (volume.restrict (Metric.ball (0 : Plane) R)) := |
| hqL.mul (Lp.memLp u) |
| have hJ : MemLp (fun y => q y • G y + u y • gradient q y) 2 |
| (volume.restrict (Metric.ball (0 : Plane) R)) := |
| (hqL.smul (Lp.memLp G)).add ((Lp.memLp u).smul hdL) |
| refine ⟨hv.toLp _,hJ.toLp _,hv.coeFn_toLp,hJ.coeFn_toLp,?_⟩ |
| exact (hw.mul_smooth hq).congr hv.coeFn_toLp hJ.coeFn_toLp |
|
|
| theorem PlaneWeakGradientOn.jacobi_smoothAt |
| {p : Plane → Plane} (hp : Continuous p) (hpb : ∀ y, ‖p y‖ ≤ 1) |
| {R α D : ℝ} (hR : 0 < R) (hα : 0 < α) (hα1 : α ≤ 1) (hD : 0 ≤ D) |
| (hHolder : ∀ x ∈ Metric.ball (0 : Plane) R, ∀ y ∈ Metric.ball (0 : Plane) R, |
| ‖cmcJacobian (p x) - cmcJacobian (p y)‖ ≤ D * (dist x y)^α) |
| (hpc : ContDiffAt ℝ (⊤ : ℕ∞) p 0) |
| {g : Plane → ℝ} (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hpe : ∀ y ∈ Metric.ball (0 : Plane) R, p y = gradient g y) |
| (hlin : ∀ i : Fin 2, LinearCMCEquation |
| (fun y => fderiv ℝ g y (EuclideanSpace.single i 1)) (gradient g) |
| (fun _ => 0) (Metric.ball 0 R)) |
| {u : Lp ℝ 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| {G : Lp Plane 2 (volume.restrict (Metric.ball (0 : Plane) R))} |
| (hw : PlaneWeakGradientOn (Metric.ball 0 R) u G) |
| (hJ : ∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ Metric.ball 0 R → |
| (∫ y in Metric.ball (0 : Plane) R, (cmcWeight (gradient g y))^2 * |
| ⟪G y, cmcJacobian (gradient g y) (gradient ψ y)⟫ - |
| cmcWeight (gradient g y) * graphSquaredSecondFundamentalForm g y * u y * ψ y) = 0) |
| {f : Plane → ℝ} (hf : ContinuousOn f (Metric.ball 0 R)) |
| (hfu : f =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] (u : Plane → ℝ)) : |
| ContDiffAt ℝ (⊤ : ℕ∞) f 0 := by |
| let q : Plane → ℝ := fun y => cmcWeight (gradient g y) |
| have hq : ContDiff ℝ (⊤ : ℕ∞) q := cmcWeight_contDiff.comp (plane_gradient_contDiff hg) |
| obtain ⟨v,J,hv,hJv,hwJ⟩ := hw.weighted_L2 hq |
| have hde := weak_cmc_graph_jacobi_linearization Metric.isOpen_ball hw hg hlin hJ |
| have hd : ∀ ψ : Plane → ℝ, ContDiff ℝ (⊤ : ℕ∞) ψ → HasCompactSupport ψ → |
| tsupport ψ ⊆ Metric.ball 0 R → |
| (∫ y, fderiv ℝ ψ y (cmcJacobian (p y) (J y))) = 0 := by |
| intro ψ hψ hc hs |
| rw [← hde ψ hψ hc hs] |
| apply integral_congr_ae |
| filter_upwards [(ae_restrict_iff' measurableSet_ball).mp hJv] with y hy |
| by_cases hys : y ∈ Metric.ball (0 : Plane) R |
| · rw [hy hys, hpe y hys] |
| · rw [fderiv_of_notMem_tsupport ℝ (fun hh => hys (hs hh))] |
| rfl |
| have hvf : (fun y => q y * f y) =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] |
| (v : Plane → ℝ) := by |
| filter_upwards [hv,hfu] with y hyv hyf |
| rw [hyv,hyf] |
| have hsm := hwJ.cmc_smoothAt hp hpb hR hα hα1 hD hHolder hpc hd |
| (hq.continuous.continuousOn.mul hf) hvf |
| have heq : f = fun y => (q y * f y)/q y := by |
| funext y |
| rw [mul_div_cancel_left₀ _ (cmcWeight_pos _).ne'] |
| rw [heq] |
| exact hsm.div hq.contDiffAt (cmcWeight_pos _).ne' |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem surfaceH1_jacobi_graph_smoothAt |
| (μ : FiniteMeasure NormalBundle) (q : Lp ℝ ∞ (μ : Measure NormalBundle)) |
| {u : SurfaceJet μ} (hu : u∈surfaceH1 μ) |
| (hhom : ∀ z∈surfaceH1 μ, surfaceJacobiB μ q u z=0) |
| (a : Cover) {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} {K : ℝ≥0} |
| (hgL : LipschitzWith K g) (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| {φ : Cover → ℝ} (hφ : Measurable φ) (hφb : ∀ x, φ x∈Icc 0 1) |
| (hφO : tsupport φ ⊆ {x | ∀ i, 0<x i ∧ x i<1}) |
| (hm : (μ : Measure NormalBundle).withDensity (fun z => ENNReal.ofReal (φ z.1)) = |
| Measure.map (actualNormalGraphChart a hn g) |
| (volume.withDensity (fun y => ENNReal.ofReal |
| (φ (a+normalGraph n g y)*‖graphAreaVector n g y‖)))) |
| {χ : Cover → ℝ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ) (hc : HasCompactSupport χ) |
| (hχO : tsupport χ ⊆ {x | ∀ i, 0<x i ∧ x i<1}) |
| (hφχ : ∀ x∈tsupport χ, φ x=1) |
| {R α D : ℝ} (hR : 0<R) (hα : 0<α) (hα1 : α≤1) (hD : 0≤D) |
| (hunit : ∀ y∈Metric.ball (0 : Plane) R, χ (a+normalGraph n g y)=1) |
| {p : Plane → Plane} (hp : Continuous p) (hpb : ∀ y, ‖p y‖≤1) |
| (hpc : ContDiffAt ℝ (⊤ : ℕ∞) p 0) |
| (hpe : ∀ y∈Metric.ball (0 : Plane) R, p y=gradient g y) |
| (hHolder : ∀ x∈Metric.ball (0 : Plane) R, ∀ y∈Metric.ball (0 : Plane) R, |
| ‖cmcJacobian (p x)-cmcJacobian (p y)‖≤D*(dist x y)^α) |
| (hlin : ∀ i : Fin 2, LinearCMCEquation |
| (fun y => fderiv ℝ g y (EuclideanSpace.single i 1)) (gradient g) |
| (fun _ => 0) (Metric.ball 0 R)) |
| (hqp : (fun y => q (actualNormalGraphChart a hn g y)) |
| =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] graphSquaredSecondFundamentalForm g) |
| {f : Plane → ℝ} (hf : ContinuousOn f (Metric.ball 0 R)) |
| (hfu : f =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] |
| fun y => u.1 (actualNormalGraphChart a hn g y)) : |
| ContDiffAt ℝ (⊤ : ℕ∞) f 0 := by |
| have hφunit (y : Plane) (hy : y∈Metric.ball (0 : Plane) R) : φ (a+normalGraph n g y)=1 := by |
| apply hφχ |
| apply subset_tsupport |
| rw [Function.mem_support,hunit y hy] |
| exact one_ne_zero |
| have hd := graph_chart_measure_le a hn hg.continuous hφb hm measurableSet_ball hφunit |
| have hcell (y : Plane) (hy : y∈Metric.ball (0 : Plane) R) (i : Fin 3) : |
| (a+normalGraph n g y) i∈Icc 0 1 := by |
| have hz : a+normalGraph n g y∈tsupport φ := subset_tsupport _ (by |
| rw [Function.mem_support,hφunit y hy]; exact one_ne_zero) |
| exact ⟨(hφO hz i).1.le,(hφO hz i).2.le⟩ |
| let hchart := actualNormalGraphChart_measurable a hn hg.continuous |
| let v := lpPullbackLE hchart hd u.1 |
| let w := lpPullbackLE hchart hd u.2 |
| let G := graphCovectorLp hn hgL w |
| have hv := lpPullbackLE_ae hchart hd u.1 |
| have hG : G =ᵐ[volume.restrict (Metric.ball (0 : Plane) R)] |
| fun y => graphCovector n g y (u.2 (actualNormalGraphChart a hn g y)) := by |
| filter_upwards [graphCovectorLp_ae hn hgL w,lpPullbackLE_ae hchart hd u.2] with y hy hz |
| change G y=graphCovector n g y (w y) at hy |
| change w y=u.2 (actualNormalGraphChart a hn g y) at hz |
| rw [hy,hz] |
| have hw := surfaceH1_chart_weak_gradient μ a hn hgL Metric.isOpen_ball |
| (hg.of_le (by simp)).contDiffOn hcell hd hu |
| apply hw.jacobi_smoothAt hp hpb hR hα hα1 hD hHolder hpc hg hpe hlin |
| · intro ψ hψ hψc hs |
| have he := surfaceJacobiB_chart_form μ q u a hn hg.continuous hφ |
| (fun x => (hφb x).1) hφO hm (fun y _ => hg.differentiable (by simp) y) |
| hχ hc hχO hφχ Metric.isOpen_ball hunit hψ hs |
| rw [hhom _ (subset_closure (LinearMap.mem_range_self _ _))] at he |
| refine Eq.trans ?_ he.symm |
| apply integral_congr_ae |
| filter_upwards [hv,hG,hqp] with y hy hz hqz |
| change v y=u.1 (actualNormalGraphChart a hn g y) at hy |
| rw [hy,hz,hqz] |
| · exact hf |
| · exact hfu.trans hv.symm |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| @[simp] lemma reverseNormal_reverseNormal (z : NormalBundle) : |
| reverseNormal (reverseNormal z)=z := by |
| apply Prod.ext |
| · rfl |
| · apply Subtype.ext; simp [reverseNormal] |
|
|
| lemma reverseNormal_measurePreserving (μ : FiniteMeasure NormalBundle) : |
| MeasurePreserving reverseNormal (μ : Measure NormalBundle) |
| (μ.map reverseNormal : Measure NormalBundle) := ⟨continuous_reverseNormal.measurable,rfl⟩ |
|
|
| lemma reverseNormal_measurePreserving_back (μ : FiniteMeasure NormalBundle) : |
| MeasurePreserving reverseNormal (μ.map reverseNormal : Measure NormalBundle) |
| (μ : Measure NormalBundle) := by |
| refine ⟨continuous_reverseNormal.measurable,?_⟩ |
| change Measure.map reverseNormal (Measure.map reverseNormal (μ : Measure NormalBundle))=_ |
| rw [Measure.map_map continuous_reverseNormal.measurable continuous_reverseNormal.measurable] |
| simp only [Function.comp_def,reverseNormal_reverseNormal,Measure.map_id'] |
|
|
| def normalReverseLp (μ : FiniteMeasure NormalBundle) (p : ℝ≥0∞) [Fact (1≤p)] |
| {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] : |
| Lp F p (μ : Measure NormalBundle) →ₗᵢ[ℝ] |
| Lp F p (μ.map reverseNormal : Measure NormalBundle) := |
| Lp.compMeasurePreservingₗᵢ ℝ reverseNormal (reverseNormal_measurePreserving_back μ) |
|
|
| lemma normalReverseLp_ae (μ : FiniteMeasure NormalBundle) (p : ℝ≥0∞) [Fact (1≤p)] |
| {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] |
| (u : Lp F p (μ : Measure NormalBundle)) : |
| normalReverseLp μ p u =ᵐ[(μ.map reverseNormal : Measure NormalBundle)] u ∘ reverseNormal := |
| Lp.coeFn_compMeasurePreserving u (reverseNormal_measurePreserving_back μ) |
|
|
| lemma smoothScalarValue_reverse (f : smoothPeriodicScalars) (z : NormalBundle) : |
| smoothScalarValue f (reverseNormal z)=smoothScalarValue f z := rfl |
|
|
| lemma smoothScalarGradient_reverse (f : smoothPeriodicScalars) (z : NormalBundle) : |
| smoothScalarGradient f (reverseNormal z)=smoothScalarGradient f z := by |
| change gradient f.val z.1-⟪-(z.2 : Cover),gradient f.val z.1⟫ • -(z.2 : Cover)=_ |
| rw [inner_neg_left,neg_smul,smul_neg,neg_neg] |
| rfl |
|
|
| def surfaceJetReverse (μ : FiniteMeasure NormalBundle) : |
| SurfaceJet μ →L[ℝ] SurfaceJet (μ.map reverseNormal) := |
| (normalReverseLp μ 2).toContinuousLinearMap.prodMap |
| (normalReverseLp μ 2).toContinuousLinearMap |
|
|
| lemma surfaceJetReverse_smooth (μ : FiniteMeasure NormalBundle) (f : smoothPeriodicScalars) : |
| surfaceJetReverse μ (surfaceSmoothJet μ f)=surfaceSmoothJet (μ.map reverseNormal) f := by |
| apply Prod.ext |
| · apply Lp.ext |
| have ha := (reverseNormal_measurePreserving_back μ).quasiMeasurePreserving.ae_eq_comp |
| ((smoothScalarValue f).coeFn_toLp (p := 2) (𝕜 := ℝ) (μ : Measure NormalBundle)) |
| filter_upwards [normalReverseLp_ae μ 2 (surfaceSmoothJet μ f).1,ha, |
| (smoothScalarValue f).coeFn_toLp (p := 2) (𝕜 := ℝ) |
| (μ.map reverseNormal : Measure NormalBundle)] with z hz ha hb |
| change (surfaceJetReverse μ (surfaceSmoothJet μ f)).1 z=_ at hz |
| change (surfaceSmoothJet μ f).1 (reverseNormal z)=smoothScalarValue f (reverseNormal z) at ha |
| change (surfaceSmoothJet (μ.map reverseNormal) f).1 z=smoothScalarValue f z at hb |
| exact hz.trans (ha.trans hb.symm) |
| · apply Lp.ext |
| have ha := (reverseNormal_measurePreserving_back μ).quasiMeasurePreserving.ae_eq_comp |
| ((smoothScalarGradient f).coeFn_toLp (p := 2) (𝕜 := ℝ) (μ : Measure NormalBundle)) |
| filter_upwards [normalReverseLp_ae μ 2 (surfaceSmoothJet μ f).2,ha, |
| (smoothScalarGradient f).coeFn_toLp (p := 2) (𝕜 := ℝ) |
| (μ.map reverseNormal : Measure NormalBundle)] with z hz ha hb |
| change (surfaceJetReverse μ (surfaceSmoothJet μ f)).2 z=_ at hz |
| change (surfaceSmoothJet μ f).2 (reverseNormal z)=smoothScalarGradient f (reverseNormal z) at ha |
| change (surfaceSmoothJet (μ.map reverseNormal) f).2 z=smoothScalarGradient f z at hb |
| exact hz.trans (ha.trans ((smoothScalarGradient_reverse f z).trans hb.symm)) |
|
|
| theorem surfaceH1_reverse (μ : FiniteMeasure NormalBundle) {u : SurfaceJet μ} |
| (hu : u∈surfaceH1 μ) : surfaceJetReverse μ u∈surfaceH1 (μ.map reverseNormal) := by |
| apply map_mem_closure (surfaceJetReverse μ).continuous hu |
| rintro _ ⟨f,rfl⟩ |
| exact ⟨f,(surfaceJetReverse_smooth μ f).symm⟩ |
|
|
| lemma surfaceMultiplier_reverse (μ : FiniteMeasure NormalBundle) |
| (q : Lp ℝ ∞ (μ : Measure NormalBundle)) (u : Lp ℝ 2 (μ : Measure NormalBundle)) : |
| surfaceMultiplier (μ.map reverseNormal) (normalReverseLp μ ∞ q) (normalReverseLp μ 2 u)= |
| normalReverseLp μ 2 (surfaceMultiplier μ q u) := by |
| apply Lp.ext |
| have hm := (reverseNormal_measurePreserving_back μ).quasiMeasurePreserving.ae_eq_comp |
| (Lp.coeFn_lpSMul (r := 2) q u) |
| filter_upwards [Lp.coeFn_lpSMul (r := 2) (normalReverseLp μ ∞ q) (normalReverseLp μ 2 u), |
| normalReverseLp_ae μ ∞ q,normalReverseLp_ae μ 2 u, |
| normalReverseLp_ae μ 2 (surfaceMultiplier μ q u),hm] with z hz hq hu hv hm |
| change (surfaceMultiplier (μ.map reverseNormal) (normalReverseLp μ ∞ q) |
| (normalReverseLp μ 2 u)) z=_ at hz |
| change surfaceMultiplier μ q u (reverseNormal z)=_ at hm |
| simp only [Function.comp_apply,smul_eq_mul,Pi.mul_apply] at hz hq hu hv hm |
| rw [hz,hq,hu,hv,hm] |
|
|
| lemma surfaceJacobiB_reverse (μ : FiniteMeasure NormalBundle) |
| (q : Lp ℝ ∞ (μ : Measure NormalBundle)) (u v : SurfaceJet μ) : |
| surfaceJacobiB (μ.map reverseNormal) (normalReverseLp μ ∞ q) |
| (surfaceJetReverse μ u) (surfaceJetReverse μ v)=surfaceJacobiB μ q u v := by |
| change ⟪normalReverseLp μ 2 u.2,normalReverseLp μ 2 v.2⟫- |
| ⟪surfaceMultiplier (μ.map reverseNormal) (normalReverseLp μ ∞ q) |
| (normalReverseLp μ 2 u.1),normalReverseLp μ 2 v.1⟫=_ |
| rw [surfaceMultiplier_reverse, |
| (normalReverseLp μ 2 (F := Cover)).inner_map_map u.2 v.2, |
| (normalReverseLp μ 2 (F := ℝ)).inner_map_map (surfaceMultiplier μ q u.1) v.1] |
| rfl |
|
|
| lemma surfaceJacobi_homogeneous_of_smooth (μ : FiniteMeasure NormalBundle) |
| (q : Lp ℝ ∞ (μ : Measure NormalBundle)) (u : SurfaceJet μ) |
| (h : ∀ f : smoothPeriodicScalars,surfaceJacobiB μ q u (surfaceSmoothJet μ f)=0) : |
| ∀ v∈surfaceH1 μ,surfaceJacobiB μ q u v=0 := by |
| have hc : Continuous (fun v : SurfaceJet μ => surfaceJacobiB μ q u v) := |
| (surfaceJacobiB_continuous μ q).comp (continuous_const.prodMk continuous_id) |
| have hr : Set.range (surfaceSmoothJet μ) ⊆ {v | surfaceJacobiB μ q u v=0} := by |
| rintro _ ⟨f,rfl⟩ |
| exact h f |
| exact fun v hv => closure_minimal hr (isClosed_eq hc continuous_const) hv |
|
|
| theorem surfaceJacobi_homogeneous_reverse (μ : FiniteMeasure NormalBundle) |
| (q : Lp ℝ ∞ (μ : Measure NormalBundle)) {u : SurfaceJet μ} |
| (hh : ∀ v∈surfaceH1 μ,surfaceJacobiB μ q u v=0) : |
| ∀ v∈surfaceH1 (μ.map reverseNormal), |
| surfaceJacobiB (μ.map reverseNormal) (normalReverseLp μ ∞ q) |
| (surfaceJetReverse μ u) v=0 := by |
| apply surfaceJacobi_homogeneous_of_smooth |
| intro f |
| calc |
| _ = surfaceJacobiB (μ.map reverseNormal) (normalReverseLp μ ∞ q) |
| (surfaceJetReverse μ u) (surfaceJetReverse μ (surfaceSmoothJet μ f)) := |
| congrArg (surfaceJacobiB (μ.map reverseNormal) (normalReverseLp μ ∞ q) |
| (surfaceJetReverse μ u)) (surfaceJetReverse_smooth μ f).symm |
| _ = surfaceJacobiB μ q u (surfaceSmoothJet μ f) := surfaceJacobiB_reverse μ q u _ |
| _ = 0 := hh _ (subset_closure (LinearMap.mem_range_self _ _)) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| lemma smooth_graph_extension_compact {g : Plane → ℝ} {R : ℝ} (hR : 0<R) |
| (hg : ContDiffOn ℝ (⊤ : ℕ∞) g (Metric.ball 0 R)) : |
| ∃ (f : Plane → ℝ) (K : ℝ≥0),ContDiff ℝ (⊤ : ℕ∞) f ∧ |
| HasCompactSupport f ∧ LipschitzWith K f ∧ EqOn f g (Metric.ball 0 (R/2)) := by |
| let b : ContDiffBump (0 : Plane) := ⟨R/2,3*R/4,by linarith,by linarith⟩ |
| have hs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| let f : Plane → ℝ := fun y => b y*g y |
| have hf : ContDiff ℝ (⊤ : ℕ∞) f := |
| contDiff_local_vector_cutoff Metric.isOpen_ball hg b.contDiff hs |
| have hfc : HasCompactSupport f := b.hasCompactSupport.mul_right |
| obtain ⟨K,hK⟩ := compact_C1_lipschitz (hf.of_le (by simp)) hfc |
| refine ⟨f,K,hf,hfc,hK,?_⟩ |
| intro y hy |
| change b y*g y=g y |
| rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall hy),one_mul] |
|
|
| lemma smooth_graph_symbol_extension {g : Plane → ℝ} {K : ℝ≥0} |
| (hgL : LipschitzWith K g) (hK : (K : ℝ)≤1) {R : ℝ} (hR : 0<R) |
| (hg : ContDiffOn ℝ (⊤ : ℕ∞) g (Metric.ball 0 R)) : |
| ∃ (p : Plane → Plane) (D : ℝ),ContDiff ℝ (⊤ : ℕ∞) p ∧ |
| (∀ y,‖p y‖≤1) ∧ 0≤D ∧ EqOn p (gradient g) (Metric.ball 0 (R/2)) ∧ |
| ∀ x y,‖cmcJacobian (p x)-cmcJacobian (p y)‖≤D*dist x y := by |
| let b : ContDiffBump (0 : Plane) := ⟨R/2,3*R/4,by linarith,by linarith⟩ |
| have hs : tsupport b⊆Metric.ball 0 R := by |
| rw [b.tsupport_eq] |
| exact Metric.closedBall_subset_ball (by dsimp [b]; linarith) |
| have hgD : ContDiffOn ℝ (⊤ : ℕ∞) (gradient g) (Metric.ball 0 R) := by |
| intro y hy |
| have hd := (hg.contDiffAt (Metric.isOpen_ball.mem_nhds hy)).fderiv_right |
| (m := (⊤ : ℕ∞)) (by simp) |
| exact ((InnerProductSpace.toDual ℝ Plane).symm.toContinuousLinearEquiv.contDiff.contDiffAt.comp |
| y hd).contDiffWithinAt |
| let p : Plane → Plane := fun y => b y • gradient g y |
| have hp : ContDiff ℝ (⊤ : ℕ∞) p := |
| contDiff_local_vector_cutoff Metric.isOpen_ball hgD b.contDiff hs |
| have hpc : HasCompactSupport p := b.hasCompactSupport.smul_right |
| obtain ⟨C,hC⟩ := compact_C1_lipschitz (hp.of_le (by simp)) hpc |
| have hpb (y : Plane) : ‖p y‖≤1 := by |
| have hgrad : ‖gradient g y‖≤K := by |
| simpa only [gradient,LinearIsometryEquiv.norm_map] using |
| norm_fderiv_le_of_lipschitz ℝ hgL (x₀ := y) |
| change ‖b y • gradient g y‖≤1 |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (b.nonneg (x := y))] |
| exact ((mul_le_mul_of_nonneg_right (b.le_one (x := y)) (norm_nonneg _)).trans |
| (by simpa using hgrad)).trans hK |
| obtain ⟨L,hL,hLip⟩ := cmcJacobian_uniform_lipschitz |
| refine ⟨p,L*C,hp,hpb,mul_nonneg hL C.coe_nonneg,?_,?_⟩ |
| · intro y hy |
| change b y • gradient g y=gradient g y |
| rw [b.one_of_mem_closedBall (Metric.ball_subset_closedBall hy),one_smul] |
| · intro x y |
| calc |
| _ ≤ L*‖p x-p y‖ := hLip _ _ (hpb x) (hpb y) |
| _ ≤ L*(C*dist x y) := mul_le_mul_of_nonneg_left |
| (by simpa only [← dist_eq_norm] using hC.dist_le_mul x y) hL |
| _ = _ := by ring |
|
|
| theorem WeakCMCGraph.smooth_jacobi_data {g : Plane → ℝ} {K : ℝ≥0} |
| (hgL : LipschitzWith K g) (hK : (K : ℝ)≤1) {R H : ℝ} (hR : 0<R) |
| (hg : ContDiffOn ℝ (⊤ : ℕ∞) g (Metric.ball 0 R)) |
| (hw : WeakCMCGraph g H (Metric.ball 0 R)) : |
| ∃ (f : Plane → ℝ) (Kf : ℝ≥0) (p : Plane → Plane) (r D : ℝ), |
| 0<r ∧ r≤R/2 ∧ ContDiff ℝ (⊤ : ℕ∞) f ∧ HasCompactSupport f ∧ |
| LipschitzWith Kf f ∧ EqOn f g (Metric.ball 0 (R/2)) ∧ |
| ContDiff ℝ (⊤ : ℕ∞) p ∧ (∀ y,‖p y‖≤1) ∧ 0≤D ∧ |
| EqOn p (gradient f) (Metric.ball 0 r) ∧ |
| (∀ x y,‖cmcJacobian (p x)-cmcJacobian (p y)‖≤D*dist x y) ∧ |
| ∀ i : Fin 2,LinearCMCEquation |
| (fun y => fderiv ℝ f y (EuclideanSpace.single i 1)) (gradient f) |
| (fun _ => 0) (Metric.ball 0 r) := by |
| obtain ⟨f,Kf,hf,hfc,hfL,hfg⟩ := smooth_graph_extension_compact hR hg |
| obtain ⟨p,D,hp,hpb,hD,hpg,hpD⟩ := smooth_graph_symbol_extension hgL hK hR hg |
| obtain ⟨f₀,r₀,hr₀,_,hf₀,hf₀g,hlin,_⟩ := hw.coordinate_jacobi_chart hgL hK hR hg |
| let r := min (R/2) r₀ |
| have hr : 0<r := lt_min (half_pos hR) hr₀ |
| have hrR : r≤R/2 := min_le_left _ _ |
| have hr0 : r≤r₀ := min_le_right _ _ |
| have he (y : Plane) (hy : y∈Metric.ball (0 : Plane) r) : f=ᶠ[𝓝 y]g := by |
| filter_upwards [Metric.isOpen_ball.mem_nhds (Metric.ball_subset_ball hrR hy)] with z hz |
| exact hfg hz |
| have he0 (y : Plane) (hy : y∈Metric.ball (0 : Plane) r) : f=ᶠ[𝓝 y]f₀ := by |
| filter_upwards [Metric.isOpen_ball.mem_nhds hy] with z hz |
| exact (hfg (Metric.ball_subset_ball hrR hz)).trans |
| (hf₀g (Metric.ball_subset_ball hr0 hz)).symm |
| refine ⟨f,Kf,p,r,D,hr,hrR,hf,hfc,hfL,hfg,hp,hpb,hD,?_,hpD,?_⟩ |
| · intro y hy |
| rw [hpg (Metric.ball_subset_ball hrR hy)] |
| simp only [gradient,(he y hy).fderiv_eq] |
| · intro i |
| apply LinearCMCEquation.congr Metric.isOpen_ball |
| ((hlin i).mono (Metric.ball_subset_ball hr0)) |
| · intro y hy |
| exact congrArg (fun L : Plane →L[ℝ] ℝ => L (EuclideanSpace.single i 1)) (he0 y hy).fderiv_eq |
| · intro y hy |
| simp only [gradient,(he0 y hy).fderiv_eq] |
| · intro y hy; rfl |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem surfaceCurvatureLp_graph_ae (μ : FiniteMeasure NormalBundle) |
| (N : Cover → Cover) (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| {σ : ℝ} (hσ : σ=1 ∨ σ= -1) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=σ • (z.2 : Cover)) |
| (a : Cover) {n : Cover} (hn : ‖n‖=1) {g : Plane → ℝ} |
| (hg : ContDiff ℝ (⊤ : ℕ∞) g) {S : Set Plane} (hS : IsOpen S) |
| (hcell : ∀ y∈S,∀ i,(a+normalGraph n g y) i∈Icc 0 1) |
| (hd : Measure.map (actualNormalGraphChart a hn g) (volume.restrict S)≤μ) |
| (hu : ∀ y∈S,∀ᶠ x in 𝓝 (a+normalGraph n g y),‖N x‖=1) : |
| (fun y => surfaceCurvatureLp μ N hN (actualNormalGraphChart a hn g y)) |
| =ᵐ[volume.restrict S] graphSquaredSecondFundamentalForm g := by |
| let Φ : Plane → Cover := fun y => a+normalGraph n g y |
| have hΦ : Continuous Φ := continuous_const.add |
| ((normalPlaneLinearIsometry n).continuous.add (hg.continuous.smul continuous_const)) |
| have hmp : Measure.QuasiMeasurePreserving (actualNormalGraphChart a hn g) |
| (volume.restrict S) (μ : Measure NormalBundle) := |
| ⟨actualNormalGraphChart_measurable a hn hg.continuous,Measure.absolutelyContinuous_of_le hd⟩ |
| have hmatchae : (fun y => N (Φ y)) =ᵐ[volume.restrict S] |
| fun y => σ • graphSlopeNormal n (gradient g y) := by |
| filter_upwards [hmp.ae_eq_comp hae,ae_restrict_mem hS.measurableSet] with y hy hys |
| change N (actualNormalGraphChart a hn g y).1=σ • |
| ((actualNormalGraphChart a hn g y).2 : Cover) at hy |
| rw [actualNormalGraphChart_position a hn g (hcell y hys),actualNormalGraphChart_normal] at hy |
| exact hy |
| have hmatch : EqOn (fun y => N (Φ y)) |
| (fun y => σ • graphSlopeNormal n (gradient g y)) S := |
| Measure.eqOn_open_of_ae_eq hmatchae hS (hN.continuous.comp hΦ).continuousOn |
| (((graphSlopeNormal_contDiff hn).continuous.comp |
| (plane_gradient_contDiff hg).continuous).const_smul σ).continuousOn |
| have hq := hmp.ae_eq_comp |
| ((show C(NormalBundle,ℝ) from ⟨_, |
| (normal_curvature_continuous hN).comp (continuous_subtype_val.comp continuous_fst)⟩).coeFn_toLp |
| (p := ∞) (𝕜 := ℝ) (μ : Measure NormalBundle)) |
| filter_upwards [hq,ae_restrict_mem hS.measurableSet] with y hy hys |
| change surfaceCurvatureLp μ N hN (actualNormalGraphChart a hn g y)= |
| coverTrace ((fderiv ℝ N (actualNormalGraphChart a hn g y).1).comp |
| (fderiv ℝ N (actualNormalGraphChart a hn g y).1)) at hy |
| rw [hy,actualNormalGraphChart_position a hn g (hcell y hys)] |
| apply graph_normal_extension_curvature a hn hg y hσ hN (hu y hys) |
| filter_upwards [hS.mem_nhds hys] with z hz |
| exact hmatch hz |
|
|
| lemma surfaceCurvatureLp_reverse (μ : FiniteMeasure NormalBundle) |
| (N : Cover → Cover) (hN : ContDiff ℝ (⊤ : ℕ∞) N) : |
| normalReverseLp μ ∞ (surfaceCurvatureLp μ N hN)= |
| surfaceCurvatureLp (μ.map reverseNormal) N hN := by |
| apply Lp.ext |
| let Q : C(NormalBundle,ℝ) := ⟨_, |
| (normal_curvature_continuous hN).comp (continuous_subtype_val.comp continuous_fst)⟩ |
| have hq := (reverseNormal_measurePreserving_back μ).quasiMeasurePreserving.ae_eq_comp |
| (Q.coeFn_toLp (p := ∞) (𝕜 := ℝ) (μ : Measure NormalBundle)) |
| filter_upwards [normalReverseLp_ae μ ∞ (surfaceCurvatureLp μ N hN),hq, |
| Q.coeFn_toLp (p := ∞) (𝕜 := ℝ) (μ.map reverseNormal : Measure NormalBundle)] with z hz hq hr |
| exact hz.trans (hq.trans hr.symm) |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem normalMeasure_H1_jacobi_graph_smoothAt {E : Set Torus} (hE : MeasurableSet E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| (N : Cover → Cover) (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| {σ : ℝ} (hσ : σ=1 ∨ σ= -1) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=σ • (z.2 : Cover)) |
| {u : SurfaceJet μ} (hu : u∈surfaceH1 μ) |
| (hhom : ∀ z∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) u z=0) |
| {U : Set Cover} (a : Cover) {n : Cover} (hn : ‖n‖=1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hgL : LipschitzWith K g) (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hgraph : ∀ᵐ x : Cover ∂volume,x∈U → |
| (quotientMap x∈E ↔ ⟪n,x-a⟫<graphHeightExtension a n g x)) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ⊆U) (hφO : tsupport φ⊆{x | ∀ i,0<x i ∧ x i<1}) |
| {R : ℝ} (hφR : tsupport φ⊆Metric.closedBall a R) (hφb : ∀ x,φ x∈Icc 0 1) |
| {χ : Cover → ℝ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ) (hsχ : HasCompactSupport χ) |
| (hχO : tsupport χ⊆{x | ∀ i,0<x i ∧ x i<1}) |
| (hφχ : ∀ x∈tsupport χ,φ x=1) |
| {s D : ℝ} (hs : 0<s) (hD : 0≤D) |
| (hunit : ∀ y∈Metric.ball (0 : Plane) s,χ (a+normalGraph n g y)=1) |
| {p : Plane → Plane} (hp : ContDiff ℝ (⊤ : ℕ∞) p) (hpb : ∀ y,‖p y‖≤1) |
| (hpe : EqOn p (gradient g) (Metric.ball 0 s)) |
| (hpD : ∀ x y,‖cmcJacobian (p x)-cmcJacobian (p y)‖≤D*dist x y) |
| (hlin : ∀ i : Fin 2,LinearCMCEquation |
| (fun y => fderiv ℝ g y (EuclideanSpace.single i 1)) (gradient g) |
| (fun _ => 0) (Metric.ball 0 s)) |
| (hNu : ∀ y∈Metric.ball (0 : Plane) s,∀ᶠ x in 𝓝 (a+normalGraph n g y),‖N x‖=1) |
| {S : Set Torus} {F : Torus → ℝ} (hF : ContinuousOn F S) |
| (hS : ∀ y∈Metric.ball (0 : Plane) s,quotientMap (a+normalGraph n g y)∈S) |
| (hFu : (u.1 : NormalBundle → ℝ)=ᵐ[(μ : Measure NormalBundle)] F ∘ normalProjection) : |
| ContDiffAt ℝ (⊤ : ℕ∞) (fun y => F (quotientMap (a+normalGraph n g y))) 0 := by |
| have hm := normalMeasure_graph_area_measure hE hμ a hn hgL |
| (fun y _ => ((hg.fderiv_right (m := (⊤ : ℕ∞)) (by simp)).continuous.continuousAt (x := y))) |
| hgraph hφ hsφ hφU hφO hφR hφb |
| have hφunit (y : Plane) (hy : y∈Metric.ball (0 : Plane) s) : φ (a+normalGraph n g y)=1 := by |
| apply hφχ |
| apply subset_tsupport |
| rw [Function.mem_support,hunit y hy] |
| exact one_ne_zero |
| have hd := graph_chart_measure_le a hn hg.continuous hφb hm measurableSet_ball hφunit |
| have hcell (y : Plane) (hy : y∈Metric.ball (0 : Plane) s) (i : Fin 3) : |
| (a+normalGraph n g y) i∈Icc 0 1 := by |
| have hz : a+normalGraph n g y∈tsupport φ := subset_tsupport _ (by |
| rw [Function.mem_support,hφunit y hy]; exact one_ne_zero) |
| exact ⟨(hφO hz i).1.le,(hφO hz i).2.le⟩ |
| have hq := surfaceCurvatureLp_graph_ae μ N hN hσ hae a hn hg Metric.isOpen_ball hcell hd hNu |
| apply surfaceH1_jacobi_graph_smoothAt μ (surfaceCurvatureLp μ N hN) hu hhom a hn hgL hg |
| hφ.continuous.measurable hφb hφO hm hχ hsχ hχO hφχ hs |
| (α := 1) (by norm_num) (by norm_num) hD hunit hp.continuous hpb hp.contDiffAt |
| (fun y hy => hpe hy) (fun x _ y _ => by simpa only [Real.rpow_one] using hpD x y) hlin hq |
| · apply hF.comp |
| · exact (continuous_quotientMap.comp |
| (continuous_const.add ((normalPlaneLinearIsometry n).continuous.add |
| (hg.continuous.smul continuous_const)))).continuousOn |
| · exact hS |
| · have hmp : Measure.QuasiMeasurePreserving (actualNormalGraphChart a hn g) |
| (volume.restrict (Metric.ball (0 : Plane) s)) (μ : Measure NormalBundle) := |
| ⟨actualNormalGraphChart_measurable a hn hg.continuous,Measure.absolutelyContinuous_of_le hd⟩ |
| filter_upwards [hmp.ae_eq_comp hFu,ae_restrict_mem measurableSet_ball] with y hy hys |
| change u.1 (actualNormalGraphChart a hn g y)= |
| F (quotientMap (actualNormalGraphChart a hn g y).1) at hy |
| rw [actualNormalGraphChart_position a hn g (hcell y hys)] at hy |
| exact hy.symm |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem normalMeasure_H1_jacobi_signed_graph_smoothAt {E : Set Torus} (hE : MeasurableSet E) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| (N : Cover → Cover) (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| {u : SurfaceJet μ} (hu : u∈surfaceH1 μ) |
| (hhom : ∀ z∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) u z=0) |
| {U : Set Cover} (a : Cover) {n : Cover} (hn : ‖n‖=1) |
| {g : Plane → ℝ} {K : ℝ≥0} (hgL : LipschitzWith K g) (hg : ContDiff ℝ (⊤ : ℕ∞) g) |
| (hgraph : (∀ᵐ x : Cover ∂volume,x∈U → |
| (quotientMap x∈E ↔ graphSignedHeight a n g x<0)) ∨ |
| (∀ᵐ x : Cover ∂volume,x∈U → |
| (quotientMap x∈E ↔ 0<graphSignedHeight a n g x))) |
| {φ : Cover → ℝ} (hφ : ContDiff ℝ (⊤ : ℕ∞) φ) (hsφ : HasCompactSupport φ) |
| (hφU : tsupport φ⊆U) (hφO : tsupport φ⊆{x | ∀ i,0<x i ∧ x i<1}) |
| {R : ℝ} (hφR : tsupport φ⊆Metric.closedBall a R) (hφb : ∀ x,φ x∈Icc 0 1) |
| {χ : Cover → ℝ} (hχ : ContDiff ℝ (⊤ : ℕ∞) χ) (hsχ : HasCompactSupport χ) |
| (hχO : tsupport χ⊆{x | ∀ i,0<x i ∧ x i<1}) |
| (hφχ : ∀ x∈tsupport χ,φ x=1) |
| {s D : ℝ} (hs : 0<s) (hD : 0≤D) |
| (hunit : ∀ y∈Metric.ball (0 : Plane) s,χ (a+normalGraph n g y)=1) |
| {p : Plane → Plane} (hp : ContDiff ℝ (⊤ : ℕ∞) p) (hpb : ∀ y,‖p y‖≤1) |
| (hpe : EqOn p (gradient g) (Metric.ball 0 s)) |
| (hpD : ∀ x y,‖cmcJacobian (p x)-cmcJacobian (p y)‖≤D*dist x y) |
| (hlin : ∀ i : Fin 2,LinearCMCEquation |
| (fun y => fderiv ℝ g y (EuclideanSpace.single i 1)) (gradient g) |
| (fun _ => 0) (Metric.ball 0 s)) |
| (hNu : ∀ y∈Metric.ball (0 : Plane) s,∀ᶠ x in 𝓝 (a+normalGraph n g y),‖N x‖=1) |
| {S : Set Torus} {F : Torus → ℝ} (hF : ContinuousOn F S) |
| (hS : ∀ y∈Metric.ball (0 : Plane) s,quotientMap (a+normalGraph n g y)∈S) |
| (hFu : (u.1 : NormalBundle → ℝ)=ᵐ[(μ : Measure NormalBundle)] F ∘ normalProjection) : |
| ContDiffAt ℝ (⊤ : ℕ∞) (fun y => F (quotientMap (a+normalGraph n g y))) 0 := by |
| rcases hgraph with hphase | hphase |
| · have hgph : ∀ᵐ x : Cover ∂volume,x∈U → |
| (quotientMap x∈E ↔ ⟪n,x-a⟫<graphHeightExtension a n g x) := by |
| simpa only [graphSignedHeight,sub_neg] using hphase |
| have hn' : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(1 : ℝ) • (z.2 : Cover) := by |
| simpa only [one_smul] using hae |
| exact normalMeasure_H1_jacobi_graph_smoothAt hE hμ N hN (Or.inl rfl) hn' hu hhom |
| a hn hgL hg hgph hφ hsφ hφU hφO hφR hφb hχ hsχ hχO hφχ hs hD hunit |
| hp hpb hpe hpD hlin hNu hF hS hFu |
| · have hgph : ∀ᵐ x : Cover ∂volume,x∈U → |
| (x∈quotientMap ⁻¹' E ↔ graphHeightExtension a n g x<⟪n,x-a⟫) := by |
| simpa only [graphSignedHeight,sub_pos,mem_preimage] using hphase |
| have hcph := graph_superlevel_compl_ae a hn hg.continuous hgph |
| have hn' : ∀ᵐ z ∂(μ.map reverseNormal : Measure NormalBundle), |
| N z.1=(-1 : ℝ) • (z.2 : Cover) := by |
| have hh := (reverseNormal_measurePreserving_back μ).quasiMeasurePreserving.ae_eq_comp hae |
| filter_upwards [hh] with z hz |
| simpa only [Function.comp_apply,reverseNormal,neg_one_smul] using hz |
| have hf' : ((surfaceJetReverse μ u).1 : NormalBundle → ℝ) |
| =ᵐ[(μ.map reverseNormal : Measure NormalBundle)] F ∘ normalProjection := by |
| have hh := (reverseNormal_measurePreserving_back μ).quasiMeasurePreserving.ae_eq_comp hFu |
| filter_upwards [normalReverseLp_ae μ 2 u.1,hh] with z hz hh |
| exact hz.trans hh |
| have hh := surfaceJacobi_homogeneous_reverse μ (surfaceCurvatureLp μ N hN) hhom |
| rw [surfaceCurvatureLp_reverse] at hh |
| exact normalMeasure_H1_jacobi_graph_smoothAt hE.compl (isNormalMeasure_compl hE hμ) |
| N hN (Or.inr rfl) hn' (surfaceH1_reverse μ hu) hh |
| a hn hgL hg hcph hφ hsφ hφU hφO hφR hφb hχ hsχ hχO hφχ hs hD hunit |
| hp hpb hpe hpD hlin hNu hF hS hf' |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal NNReal Topology RealInnerProductSpace |
| namespace CubicTorus |
|
|
| theorem isMinimizerAt_centered_H1_jacobi_smooth {V : ℝ} (hV : V∈Ioo 0 1) |
| {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) |
| (hcenter : quotientMap cubeCenter∈frontier (openRepresentative E)) |
| {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) |
| (N : Cover → Cover) (hN : ContDiff ℝ (⊤ : ℕ∞) N) |
| (hae : ∀ᵐ z ∂(μ : Measure NormalBundle),N z.1=(z.2 : Cover)) |
| (hNu : ∀ x : Cover,quotientMap x∈frontier (openRepresentative E) → |
| ∀ᶠ y in 𝓝 x,‖N y‖=1) |
| {u : SurfaceJet μ} (hu : u∈surfaceH1 μ) |
| (hhom : ∀ z∈surfaceH1 μ,surfaceJacobiB μ (surfaceCurvatureLp μ N hN) u z=0) |
| {F : Torus → ℝ} (hF : ContinuousOn F (frontier (openRepresentative E))) |
| (hFu : (u.1 : NormalBundle → ℝ)=ᵐ[(μ : Measure NormalBundle)] F ∘ normalProjection) : |
| ∃ (n : Cover) (f : Plane → ℝ) (s : ℝ),‖n‖=1 ∧ 0<s ∧ f 0=0 ∧ |
| ContDiff ℝ (⊤ : ℕ∞) f ∧ |
| (∀ y∈Metric.ball (0 : Plane) s, |
| quotientMap (cubeCenter+normalGraph n f y)∈frontier (openRepresentative E)) ∧ |
| (∀ x∈Metric.ball (0 : Cover) s, |
| quotientMap (cubeCenter+x)∈frontier (openRepresentative E) ↔ |
| normalGraph n f (normalPlaneCoordinates n x)=x) ∧ |
| ContDiffAt ℝ (⊤ : ℕ∞) (fun y => F (quotientMap (cubeCenter+normalGraph n f y))) 0 := by |
| obtain ⟨n,g,r,H,hn,hr,hg,hg0,hgc,hi,hcover,hw,_⟩ := |
| isMinimizerAt_smooth_CMC hV hE hmin (quotientMap cubeCenter) hcenter |
| obtain ⟨f,Kf,p,rj,D,hrj,hrjr,hf,hfc,hfL,hfg,hp,hpb,hD,hpf,hpD,hlin⟩ := |
| hw.smooth_jacobi_data hg (by norm_num) hr hgc |
| have hf0 : f 0=0 := (hfg (Metric.mem_ball_self (half_pos hr))).trans hg0 |
| have hcover' (x : Cover) (hx : x∈Metric.ball (0 : Cover) (r/2)) : |
| quotientMap (cubeCenter+x)∈frontier (openRepresentative E) ↔ |
| normalGraph n g (normalPlaneCoordinates n x)=x := by |
| rw [quotientMap_add] |
| exact hcover x (Metric.ball_subset_ball (by linarith) hx) |
| obtain ⟨R,ρ,U,hRr,_,_,_,hR,hρ,hgC,hU,hρU,hUR,hUO,_,hphase⟩ := |
| actual_centered_graph_phase_patch hV hE hmin hcenter hn (half_pos hr) |
| hg hg0 ((hgc.of_le (by simp)).mono (Metric.ball_subset_ball (by linarith))) hcover' |
| have hext (x : Cover) (hx : x∈U) : |
| graphSignedHeight cubeCenter n f x=graphSignedHeight cubeCenter n g x := by |
| have hy : normalPlaneCoordinates n (x-cubeCenter)∈Metric.ball (0 : Plane) (r/2) := by |
| have hz : ‖x-cubeCenter‖<R := by |
| simpa only [Metric.mem_ball,dist_eq_norm] using hUR hx |
| simpa only [Metric.mem_ball,dist_zero_right] using |
| (normalPlaneCoordinates_norm_le n (x-cubeCenter)).trans_lt (hz.trans hRr) |
| simp only [graphSignedHeight,graphHeightExtension,hfg hy] |
| have hphasef : (∀ᵐ x : Cover ∂volume,x∈U → |
| (quotientMap x∈E ↔ graphSignedHeight cubeCenter n f x<0)) ∨ |
| (∀ᵐ x : Cover ∂volume,x∈U → |
| (quotientMap x∈E ↔ 0<graphSignedHeight cubeCenter n f x)) := by |
| rcases hphase with hphase|hphase |
| · left; filter_upwards [hphase] with x hx hxu |
| rw [hext x hxu]; exact hx hxu |
| · right; filter_upwards [hphase] with x hx hxu |
| rw [hext x hxu]; exact hx hxu |
| let φ := radialHeightCutoff cubeCenter ρ |
| have hφ : ContDiff ℝ (⊤ : ℕ∞) φ := radialDensityWeight_contDiff _ _ _ |
| have hsφ : HasCompactSupport φ := radialDensityWeight_hasCompactSupport (by positivity) (by positivity) |
| have hφs : tsupport φ⊆Metric.closedBall cubeCenter (2*ρ) := |
| radialDensityWeight_tsupport (by positivity) (by positivity) |
| have hφU := hφs.trans hρU |
| have hφO := hφU.trans hUO |
| have hφR := hφU.trans (hUR.trans Metric.ball_subset_closedBall) |
| have hφb (x : Cover) : φ x∈Icc 0 1 := |
| ⟨radialDensityWeight_nonneg _ _ _ _,radialDensityWeight_le_one _ _ _ _⟩ |
| let χ := radialHeightCutoff cubeCenter (ρ/4) |
| have hχ : ContDiff ℝ (⊤ : ℕ∞) χ := radialDensityWeight_contDiff _ _ _ |
| have hsχ : HasCompactSupport χ := radialDensityWeight_hasCompactSupport (by positivity) (by positivity) |
| have hχs : tsupport χ⊆Metric.closedBall cubeCenter (ρ/2) := by |
| have hh := radialDensityWeight_tsupport (a := cubeCenter) |
| (by positivity : 0≤2*(ρ/4)) (by positivity : 0<3*(ρ/4)^2) |
| convert hh using 1 <;> congr 1 |
| ring |
| have hφχ (x : Cover) (hx : x∈tsupport χ) : φ x=1 := |
| radialHeightCutoff_one hρ (Metric.closedBall_subset_closedBall (by linarith) (hχs hx)) |
| have hχO : tsupport χ⊆{x | ∀ i,0<x i ∧ x i<1} := |
| hχs.trans ((Metric.closedBall_subset_closedBall (by linarith)).trans (hρU.trans hUO)) |
| have hΦ : Continuous (fun y => cubeCenter+normalGraph n f y) := |
| continuous_const.add ((normalPlaneLinearIsometry n).continuous.add |
| (hf.continuous.smul continuous_const)) |
| have hΦ0 : cubeCenter+normalGraph n f 0=cubeCenter := by |
| simp only [normalGraph,hf0,zero_smul,add_zero] |
| change cubeCenter+(normalPlaneLinearIsometry n) 0=cubeCenter |
| rw [map_zero,add_zero] |
| have hnb : ∀ᶠ y in 𝓝 (0 : Plane), |
| cubeCenter+normalGraph n f y∈Metric.ball cubeCenter (ρ/4) := by |
| apply hΦ.continuousAt |
| change Metric.ball cubeCenter (ρ/4)∈𝓝 (cubeCenter+normalGraph n f 0) |
| rw [hΦ0] |
| exact Metric.ball_mem_nhds _ (by positivity) |
| obtain ⟨t,ht,htb⟩ := Metric.mem_nhds_iff.mp hnb |
| let s := min rj t |
| have hs : 0<s := lt_min hrj ht |
| have hsj : s≤rj := min_le_left _ _ |
| have hsr : s≤r/2 := hsj.trans hrjr |
| have hunit (y : Plane) (hy : y∈Metric.ball (0 : Plane) s) : |
| χ (cubeCenter+normalGraph n f y)=1 := |
| radialHeightCutoff_one (by positivity) |
| (Metric.ball_subset_closedBall (htb (Metric.ball_subset_ball (min_le_right _ _) hy))) |
| have hin (y : Plane) (hy : y∈Metric.ball (0 : Plane) s) : |
| quotientMap (cubeCenter+normalGraph n f y)∈frontier (openRepresentative E) := by |
| have hy' := Metric.ball_subset_ball hsr hy |
| simp only [normalGraph,hfg hy',quotientMap_add] |
| exact hi y (Metric.ball_subset_ball (by linarith) hy') |
| refine ⟨n,f,s,hn,hs,hf0,hf,hin,?_,?_⟩ |
| · intro x hx |
| have hx' := Metric.ball_subset_ball hsr hx |
| have hp' : normalPlaneCoordinates n x∈Metric.ball (0 : Plane) (r/2) := by |
| simpa only [Metric.mem_ball,dist_zero_right] using |
| (normalPlaneCoordinates_norm_le n x).trans_lt |
| (show ‖x‖<r/2 from by simpa only [Metric.mem_ball,dist_zero_right] using hx') |
| simpa only [normalGraph,hfg hp'] using hcover' x hx' |
| · exact normalMeasure_H1_jacobi_signed_graph_smoothAt hE hμ N hN hae hu hhom |
| cubeCenter hn hfL hf hphasef hφ hsφ hφU hφO hφR hφb hχ hsχ hχO hφχ |
| hs hD hunit hp hpb (hpf.mono (Metric.ball_subset_ball hsj)) hpD |
| (fun i => (hlin i).mono (Metric.ball_subset_ball hsj)) |
| (fun y hy => hNu _ (hin y hy)) hF hin hFu |
|
|
| end CubicTorus |
|
|
| end |
| end |
|
|
| end OAI |
|
|
