import Mathlib import OAI.Geometry.CubicTorus.Estimates.CorrectedChain namespace OAI section section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus variable {X : Type*} [NormedAddCommGroup X] [NormedSpace ℝ X] namespace Periodic lemma simultaneousCofactor_second_deriv (l : List Variation) (x n : Cover) : deriv (deriv (fun t ↦ chainCofactor (simultaneousChain l t) x n)) 0=(normalJet l x n).2.2 := by induction l using List.reverseRecOn with | nil => simp [simultaneousChain,chainCofactor,normalJet] | append_singleton l a ih => have hm : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ chainMove (simultaneousChain l t) x) := (simultaneousMove_smooth l).comp (contDiff_id.prodMk contDiff_const) have hc : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ chainCofactor (simultaneousChain l t) x n) := (simultaneousCofactor_smooth l).comp (contDiff_id.prodMk (contDiff_const : ContDiff ℝ (⊤ : ℕ∞) (fun _ : ℝ ↦ (x,n)))) have hf : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ coordinateNormalVelocity a.coordinate a.function (chainMove (simultaneousChain l t) x) (chainCofactor (simultaneousChain l t) x n)) := (wallVelocity_contDiff a).comp (hm.prodMk hc) have he : (fun t ↦ chainCofactor (simultaneousChain (l++[a]) t) x n)= (fun t ↦ chainCofactor (simultaneousChain l t) x n+ t • coordinateNormalVelocity a.coordinate a.function (chainMove (simultaneousChain l t) x) (chainCofactor (simultaneousChain l t) x n)) := by funext t simp [simultaneousChain,List.map_append,chainCofactor_append,chainCofactor,coordinateCofactor_eq] rw [he,second_deriv_add_time_smul hc hf,ih,normalJet_append] dsimp only congr 2 have hdf : HasFDerivAt (fun p : Cover×Cover ↦ coordinateNormalVelocity a.coordinate a.function p.1 p.2) (fderiv ℝ (fun p : Cover×Cover ↦ coordinateNormalVelocity a.coordinate a.function p.1 p.2) (x,n)) (chainMove (simultaneousChain l 0) x,chainCofactor (simultaneousChain l 0) x n) := by simpa only [simultaneousMove_zero,simultaneousCofactor_zero] using ((wallVelocity_contDiff a).differentiable (by simp) (x,n)).hasFDerivAt have hd := hdf.comp_hasDerivAt 0 ((hm.differentiable (by simp) 0).hasDerivAt.prodMk (hc.differentiable (by simp) 0).hasDerivAt) simpa only [Function.comp_def,simultaneousMove_deriv,simultaneousCofactor_deriv, (normalJet_first l x n).1,(normalJet_first l x n).2] using hd.deriv theorem simultaneousArea_second_deriv (l : List Variation) (μ : FiniteMeasure NormalBundle) : deriv (deriv (fun t ↦ ∫ z,‖chainCofactor (simultaneousChain l t) z.1 z.2‖ ∂(μ : Measure NormalBundle))) 0 = ∫ z,‖chainNormalVelocity l z.1 z.2‖^2-⟪(z.2 : Cover),chainNormalVelocity l z.1 z.2⟫^2+ ⟪(z.2 : Cover),(normalJet l z.1 z.2).2.2⟫ ∂(μ : Measure NormalBundle) := by have h := normalArea_second_derivative (simultaneousCofactor_smooth l) (by intro p _; exact simultaneousCofactor_zero l p.1 p.2) μ simpa only [normalEmbedding,simultaneousCofactor_deriv,simultaneousCofactor_second_deriv] using h end Periodic end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus namespace Periodic def componentDivergence (a : Variation) (x : Cover) : ℝ := fderiv ℝ a.function x (EuclideanSpace.single a.coordinate 1) lemma componentDivergence_smooth (a : Variation) : ContDiff ℝ (⊤ : ℕ∞) (componentDivergence a) := (a.smooth.fderiv_right (by simp)).clm_apply contDiff_const def massJet (l : List Variation) (x : Cover) : Cover×ℝ×ℝ := l.foldl (fun j a ↦ (j.1+a.function x • EuclideanSpace.single a.coordinate 1, j.2.1+componentDivergence a x, j.2.2+2*(fderiv ℝ (componentDivergence a) x j.1+j.2.1*componentDivergence a x))) (0,0,0) lemma massJet_append (l : List Variation) (a : Variation) (x : Cover) : massJet (l++[a]) x = ((massJet l x).1+a.function x • EuclideanSpace.single a.coordinate 1, (massJet l x).2.1+componentDivergence a x, (massJet l x).2.2+2*(fderiv ℝ (componentDivergence a) x (massJet l x).1+ (massJet l x).2.1*componentDivergence a x)) := by simp only [massJet,List.foldl_append,List.foldl_cons,List.foldl_nil] lemma massJet_first (l : List Variation) (x : Cover) : (massJet l x).1=chainVelocity l x := by induction l using List.reverseRecOn with | nil => simp [massJet,chainVelocity] | append_singleton l a ih => simpa [massJet_append,chainVelocity,List.map_append] using congrArg (fun u ↦ u+a.function x • EuclideanSpace.single a.coordinate (1 : ℝ)) ih lemma simultaneousJacobian_append (l : List Variation) (a : Variation) (x : Cover) (t : ℝ) : chainJacobian (simultaneousChain (l++[a]) t) x=chainJacobian (simultaneousChain l t) x+ t*(chainJacobian (simultaneousChain l t) x*componentDivergence a (chainMove (simultaneousChain l t) x)) := by simp only [simultaneousChain,List.map_append,chainJacobian_append,List.map_cons,List.map_nil, chainJacobian,coordinateJacobian,mul_one,componentDivergence] ring lemma simultaneousJacobian_deriv (l : List Variation) (x : Cover) : deriv (fun t ↦ chainJacobian (simultaneousChain l t) x) 0=(massJet l x).2.1 := by induction l using List.reverseRecOn with | nil => simp [simultaneousChain,chainJacobian,massJet] | append_singleton l a ih => have hm : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ chainMove (simultaneousChain l t) x) := (simultaneousMove_smooth l).comp (contDiff_id.prodMk contDiff_const) have hj : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ chainJacobian (simultaneousChain l t) x) := (simultaneousJacobian_smooth l).comp (contDiff_id.prodMk contDiff_const) simp_rw [simultaneousJacobian_append l a x] simpa [massJet_append,ih] using deriv_add_time_smul hj (hj.mul ((componentDivergence_smooth a).comp hm)) lemma simultaneousJacobian_second_deriv (l : List Variation) (x : Cover) : deriv (deriv (fun t ↦ chainJacobian (simultaneousChain l t) x)) 0=(massJet l x).2.2 := by induction l using List.reverseRecOn with | nil => simp [simultaneousChain,chainJacobian,massJet] | append_singleton l a ih => have hm : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ chainMove (simultaneousChain l t) x) := (simultaneousMove_smooth l).comp (contDiff_id.prodMk contDiff_const) have hj : ContDiff ℝ (⊤ : ℕ∞) (fun t : ℝ ↦ chainJacobian (simultaneousChain l t) x) := (simultaneousJacobian_smooth l).comp (contDiff_id.prodMk contDiff_const) have hd : HasFDerivAt (componentDivergence a) (fderiv ℝ (componentDivergence a) x) (chainMove (simultaneousChain l 0) x) := by simpa only [simultaneousMove_zero] using ((componentDivergence_smooth a).differentiable (by simp) x).hasFDerivAt have hf := hd.comp_hasDerivAt 0 ((hm.differentiable (by simp) 0).hasDerivAt) have he := ((hj.differentiable (by simp) 0).hasDerivAt.mul hf).deriv simp only [Function.comp_def,simultaneousJacobian_zero,simultaneousMove_zero, simultaneousMove_deriv,simultaneousJacobian_deriv,one_mul] at he change deriv (fun t ↦ chainJacobian (simultaneousChain l t) x * componentDivergence a (chainMove (simultaneousChain l t) x)) 0 = (massJet l x).2.1*componentDivergence a x+fderiv ℝ (componentDivergence a) x (chainVelocity l x) at he simp_rw [simultaneousJacobian_append l a x] have hh := second_deriv_add_time_smul hj (hj.mul ((componentDivergence_smooth a).comp hm)) simp only [smul_eq_mul,Function.comp_def] at hh rw [hh,ih,he,massJet_append] simp only [massJet_first] ring lemma simultaneousMass_second_deriv (l : List Variation) (E : Set Torus) : deriv (deriv (fun t ↦ chainMass (simultaneousChain l t) E)) 0= ∫ x in fundamentalCube∩quotientMap ⁻¹' E,(massJet l x).2.2 := by simpa only [chainMass,simultaneousJacobian_second_deriv] using second_deriv_compact_parameter_integral isCompact_closedFundamentalCube (cubeIntersection_compact_carrier E) (simultaneousJacobian_smooth l) 0 end Periodic end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus namespace Periodic theorem exists_chain_of_field {F : Cover → Cover} (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hp : ∀ m x,F (x+latticeVector m)=F x) : ∃ l : List Variation,(∀ x,chainVelocity l x=F x) ∧ (∀ E,chainFlux l E=∫ x in fundamentalCube∩quotientMap ⁻¹' E,divergence F x) ∧ ∀ x n,⟪n,chainNormalVelocity l x n⟫=‖n‖^2*divergence F x-⟪n,fderiv ℝ F x n⟫ := by have hx (i : Fin 3) : ∃ a : Variation,a.coordinate=i ∧ a.function=(fun x ↦ F x i) := exists_variation_of_function i ((EuclideanSpace.proj i : Cover →L[ℝ] ℝ).contDiff.comp hF) (fun m x ↦ by change F (x+latticeVector m) i=F x i; rw [hp]) choose a ha hb using hx refine ⟨List.ofFn a,?_,?_,?_⟩ · intro x simp only [chainVelocity,List.map_ofFn,List.sum_ofFn,Function.comp_apply,ha,hb] ext j simp [Finset.sum_apply,Pi.single_apply,smul_eq_mul] · intro E simp only [chainFlux,List.map_ofFn,List.sum_ofFn,Function.comp_apply,ha,hb] exact (flux_sum_components E hF).symm · intro x n simp only [chainNormalVelocity,List.map_ofFn,List.sum_ofFn,Function.comp_apply,ha,hb, inner_sum,coordinateNormalVelocity,inner_sub_right,real_inner_smul_right, real_inner_self_eq_norm_sq,inner_gradient_right,conj_trivial, Finset.sum_sub_distrib,← Finset.sum_mul] simp_rw [fderiv_component (hF.differentiable (by simp) x)] dsimp [divergence] congr 1 · ring · simp [PiLp.inner_apply,EuclideanSpace.single,mul_comm] theorem isMinimizerAt_stable_fields {V : ℝ} (hV : V∈Ioo 0 1) {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) : ∃ (μ : FiniteMeasure NormalBundle) (H : ℝ),IsNormalMeasure E μ ∧ (∀ F : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) F → (∀ n x,F (x+latticeVector n)=F x) → (∫ z,tangentialDivergence F z.1 z.2 ∂(μ : Measure NormalBundle))= H*(∫ z,⟪(z.2 : Cover),F z.1⟫ ∂(μ : Measure NormalBundle))) ∧ ∀ F : Cover → Cover,ContDiff ℝ (⊤ : ℕ∞) F → (∀ n x,F (x+latticeVector n)=F x) → (∫ z,⟪(z.2 : Cover),F z.1⟫ ∂(μ : Measure NormalBundle))=0 → ∃ l : List Variation,(∀ x,deriv (fun t ↦ chainMove (simultaneousChain l t) x) 0=F x) ∧ 0≤deriv (deriv (fun t ↦ ∫ z,‖chainCofactor (simultaneousChain l t) z.1 z.2‖ ∂(μ : Measure NormalBundle))) 0- H*deriv (deriv (fun t ↦ chainMass (simultaneousChain l t) E)) 0 := by obtain ⟨μ,H,hμ,hH,hs⟩ := isMinimizerAt_stable_periodic hV hE hmin refine ⟨μ,H,hμ,fun F hf hp ↦ weakCMC_identity hμ hH hf hp,?_⟩ intro F hf hp hz obtain ⟨l,hl,hflux,_⟩ := exists_chain_of_field hf hp refine ⟨l,fun x ↦ (simultaneousMove_deriv l x).trans (hl x),hs l ?_⟩ rw [simultaneousMass_deriv,hflux] obtain ⟨C,hC,hbound⟩ := periodic_bounded hf.continuous hp rw [← normalMeasure_flux hμ hf hp (by linarith : 0 rw [cover_decompose v,map_sum,map_sum] simp only [map_smul,smul_eq_mul] rfl @[simp] lemma coverTrace_zero : coverTrace 0=0 := by simp [coverTrace] @[simp] lemma coverTrace_add (L M : Cover →L[ℝ] Cover) : coverTrace (L+M)=coverTrace L+coverTrace M := by simp [coverTrace,Finset.sum_add_distrib] @[simp] lemma coverTrace_smul (a : ℝ) (L : Cover →L[ℝ] Cover) : coverTrace (a • L)=a*coverTrace L := by simp [coverTrace,Finset.mul_sum] def coverTraceCLM : (Cover →L[ℝ] Cover) →L[ℝ] ℝ := ({toFun := coverTrace, map_add' := coverTrace_add, map_smul' := by intro a L; exact coverTrace_smul a L} : (Cover →L[ℝ] Cover) →ₗ[ℝ] ℝ).toContinuousLinearMap @[simp] lemma coverTraceCLM_apply (L : Cover →L[ℝ] Cover) : coverTraceCLM L=coverTrace L := rfl lemma coverTrace_comp_comm (L M : Cover →L[ℝ] Cover) : coverTrace (L.comp M)=coverTrace (M.comp L) := by simp only [coverTrace,ContinuousLinearMap.comp_apply] conv_lhs => enter [2,i]; rw [cover_clm_component] conv_rhs => enter [2,i]; rw [cover_clm_component] rw [Finset.sum_comm] apply Finset.sum_congr rfl intro i _ apply Finset.sum_congr rfl intro j _ ring lemma coverTrace_smulRight (l : Cover →L[ℝ] ℝ) (v : Cover) : coverTrace (l.smulRight v)=l v := by conv_rhs => rw [cover_decompose v] simp only [coverTrace,ContinuousLinearMap.smulRight_apply,PiLp.smul_apply, smul_eq_mul,map_sum,map_smul] apply Finset.sum_congr rfl intro i _ ring lemma divergence_eq_coverTrace (F : Cover → Cover) (x : Cover) : divergence F x=coverTrace (fderiv ℝ F x) := rfl def cofactorLinear (L : Cover →L[ℝ] Cover) (n : Cover) : Cover := coverTrace L • n-L.adjoint n def cofactorQuadratic (L : Cover →L[ℝ] Cover) (n : Cover) : Cover := ((coverTrace L)^2-coverTrace (L.comp L)) • n- (2*coverTrace L) • L.adjoint n+2 • L.adjoint (L.adjoint n) lemma cofactorLinear_add (L M : Cover →L[ℝ] Cover) (n : Cover) : cofactorLinear (L+M) n=cofactorLinear L n+cofactorLinear M n := by simp only [cofactorLinear,coverTrace_add,map_add,add_apply] module lemma cofactorQuadratic_add (L M : Cover →L[ℝ] Cover) (n : Cover) : cofactorQuadratic (L+M) n=cofactorQuadratic L n+cofactorQuadratic M n+ 2 • (cofactorLinear M (cofactorLinear L n)-cofactorLinear (M.comp L) n) := by simp only [cofactorQuadratic,cofactorLinear,coverTrace_add,ContinuousLinearMap.add_comp, ContinuousLinearMap.comp_add,coverTrace_comp_comm L M,map_add,add_apply, ContinuousLinearMap.adjoint_comp,ContinuousLinearMap.comp_apply,map_sub,map_smul] module end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus theorem isMinimizerAt_halfspace_L1_tilt_small {V : ℝ} (hV : V ∈ Ioo 0 1) {E : Set Torus} (hE : MeasurableSet E) (hmin : IsMinimizerAt V E) {μ : FiniteMeasure NormalBundle} (hμ : IsNormalMeasure E μ) {H : ℝ} (hH : ∀ D : WallVariation, normalFirstVariation μ D = H*directionalFlux E D.coordinate D.function) {n : Cover} (hn : ‖n‖ = 1) {ε : ℝ} (hε : 0 < ε) : ∃ η : ℝ, 0 < η ∧ ∀ (c : MovingCenter.InteriorCenter) (r : ℝ), 0 < r → r < η → (∫ x in Metric.closedBall (0 : Cover) 1, |blowupCharacteristic E c r x-(coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) < η → (∫ z, blowupTilt n z ∂(blowupNormalMeasure μ c r 2 : Measure BlowupNormalBundle)) < ε := by by_contra h push Not at h let d : ℕ → ℝ := fun i ↦ 1/((i : ℝ)+1) have hd (i : ℕ) : 0 < d i := by dsimp [d]; positivity have hb (i : ℕ) := h (d i) (hd i) choose c r hr hrd he ht using hb have hdlim : Tendsto d atTop (𝓝 0) := tendsto_one_div_add_atTop_nhds_zero_nat have hrlim : Tendsto r atTop (𝓝 0) := squeeze_zero (fun i ↦ (hr i).le) (fun i ↦ (hrd i).le) hdlim have hconv : Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 1, |blowupCharacteristic E (c i) (r i) x- (coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) atTop (𝓝 0) := squeeze_zero (fun i ↦ integral_nonneg (fun _ ↦ abs_nonneg _)) (fun i ↦ (he i).le) hdlim obtain ⟨ν,k,U,hU,hν,_,hstat⟩ := isMinimizerAt_moving_paired_normal_limit hV hE hmin hμ hH c r hr hrlim (measurableSet_coverHalfspace n) hconv have hflux (F : Cover → Cover) (hF : ContDiff ℝ (⊤ : ℕ∞) F) (hs : tsupport F ⊆ Metric.closedBall (0 : Cover) 1) := (hstat F hF hs).2 have heq := moving_blowup_perimeter_eq_normal_ball hV hE hmin hμ (fun i ↦ c (i+k)) (fun i ↦ r (i+k)) (fun i ↦ hr _) ((hrlim.comp (tendsto_add_atTop_nat k)).mono_left hU) (measurableSet_coverHalfspace n) ((hconv.comp (tendsto_add_atTop_nat k)).mono_left hU) hν hflux (by norm_num : (0 : ℝ) < 1/2) (by norm_num : (1/2 : ℝ) ≤ 1) have hmodel : ∀ᵐ x : Cover ∂volume, x ∈ Metric.ball (0 : Cover) 1 → (x ∈ coverHalfspace n ↔ x ∈ coverHalfspace n) := ae_of_all _ (by simp) have hmass : (ν : Measure BlowupNormalBundle).real {z | z.1 ∈ Metric.ball (0 : Cover) (1/2)} = Real.pi*(1/2)^2 := by rw [Measure.real_def,← heq, halfspace_model_perimeter_ball (measurableSet_coverHalfspace n) hn hmodel (by norm_num) (by norm_num), ENNReal.toReal_ofReal (by positivity)] have hflat := flat_set_limit_normal (measurableSet_coverHalfspace n) hn hmodel hmass hflux have htilt := flat_moving_normal_tilt_limit (fun i ↦ (c (i+k) : Cover)) (fun i ↦ r (i+k)) hν hflat obtain ⟨i,hi⟩ := (htilt.eventually (Iio_mem_nhds hε)).exists exact (not_lt_of_ge (ht (i+k))) hi theorem isMinimizerAt_halfspace_L1_normal_charts {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) {H : ℝ} (hH : ∀ D : WallVariation, normalFirstVariation μ D = H*directionalFlux E D.coordinate D.function) : ∃ C K ρ δ : ℝ, 0 < C ∧ 0 < K ∧ 0 < ρ ∧ ρ ≤ 1/16 ∧ 0 < δ ∧ ∀ n : Cover, ‖n‖ = 1 → ∀ e : ℝ, 0 < e → e ≤ δ → ∃ η : ℝ, 0 < η ∧ ∀ (c : Cover) (r : ℝ), ‖c-cubeCenter‖ ≤ 1/16 → 0 < r → r < η → (∫ x in Metric.closedBall (0 : Cover) 1, |blowupCharacteristic E c r x-(coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) < η → BoundaryNormalChart E c n (r/272) e C K ρ := by obtain ⟨R,C,K,ρ,hR,hC,hK,hρ,hρs,hmod⟩ := MovingCenter.isMinimizerAt_moving_normal_modulus hV hE hmin hcenter obtain ⟨N,hN,hp,hu⟩ := normalMeasure_polar_unit hμ obtain ⟨δ,hδ,hcharts⟩ := hmod μ hμ N hN hp hu refine ⟨C,K,ρ,δ,hC,hK,hρ,hρs,hδ,?_⟩ intro n hn e he heδ obtain ⟨η,hη,hsmall⟩ := isMinimizerAt_halfspace_L1_tilt_small hV hE hmin hμ hH hn (show 0 < e^2/272^2 by positivity) refine ⟨min η (min (272*R) (min (272*δ*e^2) 1)),by positivity,?_⟩ intro c r hc hr hrs hL1 have hrη : r < η := hrs.trans_le (min_le_left _ _) have hrR : r/272 ≤ R := by have hh := hrs.trans_le ((min_le_right _ _).trans (min_le_left _ _)) linarith have hratio : (r/272)/e^2 ≤ δ := by apply (div_le_iff₀ (sq_pos_of_pos he)).mpr have hh := hrs.trans_le ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _))) linarith have hr1 : r < 1 := hrs.trans_le ((min_le_right _ _).trans ((min_le_right _ _).trans (min_le_right _ _))) let d : MovingCenter.InteriorCenter := ⟨c,by linarith⟩ have ht := hsmall d r hr hrη (hL1.trans_le (min_le_left _ _)) have hchart : Metric.ball c (r/8) ⊆ {x | ∀ j, 0 < x j ∧ x j < 1} := Metric.ball_subset_closedBall.trans (MovingCenter.closedBall_interior (c := d) (by linarith)) have htilt := spatial_tilt_le_blowupTilt hN hp hu c n hr hchart have hbound : (∫ x in Metric.closedBall (quotientMap c) (17*(r/272)), ‖N x-n‖^2 ∂(spatialNormalMeasure μ : Measure Torus)) ≤ (r/272)^2*e^2 := by rw [show 17*(r/272)=r/16 by ring] calc _ ≤ _ := htilt _ ≤ r^2*(e^2/272^2) := mul_le_mul_of_nonneg_left ht.le (sq_nonneg _) _ = _ := by ring exact hcharts c hc n (r/272) e hn (div_pos hr (by norm_num)) hrR he heδ hratio hbound end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma localL1_eventually_lt_model {g : ℕ → Cover → ℝ} (hg : ∀ i, Measurable (g i)) (hgb : ∀ i x, |g i x| ≤ 1) {f F : Cover → ℝ} (hf : Measurable f) (hfb : ∀ x, |f x| ≤ 1) (hF : Measurable F) (hFb : ∀ x, |F x| ≤ 1) {R : ℝ} {l : Filter ℕ} (hconv : Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) R, |g i x-f x|) l (𝓝 0)) {η : ℝ} (hclose : (∫ x in Metric.closedBall (0 : Cover) R, |f x-F x|) < η) : ∀ᶠ i in l, (∫ x in Metric.closedBall (0 : Cover) R, |g i x-F x|) < η := by have he := hconv.eventually (Iio_mem_nhds (sub_pos.mpr hclose)) filter_upwards [he] with i hi let m : Measure Cover := volume.restrict (Metric.closedBall (0 : Cover) R) have ht := dist_triangle (boundedFunctionL1 m (g i) (hg i) (hgb i)) (boundedFunctionL1 m f hf hfb) (boundedFunctionL1 m F hF hFb) simp only [boundedFunctionL1_dist] at ht change (∫ x in Metric.closedBall (0 : Cover) R, |g i x-F x|) ≤ (∫ x in Metric.closedBall (0 : Cover) R, |g i x-f x|) + (∫ x in Metric.closedBall (0 : Cover) R, |f x-F x|) at ht linarith theorem isMinimizerAt_close_limit_fixed_scale_graphs {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) {H : ℝ} (hH : ∀ D : WallVariation, normalFirstVariation μ D = H*directionalFlux E D.coordinate D.function) {n : Cover} (hn : ‖n‖ = 1) : ∃ η : ℝ, 0 < η ∧ ∀ (c : ℕ → MovingCenter.InteriorCenter) (r : ℕ → ℝ), (∀ i, 0 < r i) → Tendsto r atTop (𝓝 0) → (∀ᶠ i in atTop, ‖(c i : Cover)-cubeCenter‖ ≤ 1/32) → ∀ A : Set Cover, MeasurableSet A → Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 1, |blowupCharacteristic E (c i) (r i) x-A.indicator (fun _ ↦ (1 : ℝ)) x|) atTop (𝓝 0) → (∀ ρ : ℝ, 0 < ρ → ρ ≤ 1 → 0 < volume.real (A ∩ Metric.ball (0 : Cover) ρ) ∧ 0 < volume.real (Aᶜ ∩ Metric.ball (0 : Cover) ρ)) → (∫ x in Metric.closedBall (0 : Cover) 1, |A.indicator (fun _ ↦ (1 : ℝ)) x- (coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) < η → ∀ᶠ i in atTop, ∃ a : Cover, dist a (c i : Cover) < r i/(544*4096) ∧ ∃ g : Plane → ℝ, LipschitzWith (1/4) g ∧ ContDiffOn ℝ 1 g (Metric.ball 0 ((r i/544)/1024)) ∧ (∀ y ∈ Metric.ball (0 : Plane) ((r i/544)/1024), quotientMap a+quotientMap (normalGraph n g y) ∈ frontier (openRepresentative E) ∧ ‖normalGraph n g y‖ < (r i/544)/512) ∧ (∀ x ∈ quotientMap ⁻¹' frontier (openRepresentative E) ∩ Metric.ball a ((r i/544)/128), normalGraph n g (normalPlaneCoordinates n (x-a)) = x-a) := by obtain ⟨R,hR,hgraphs⟩ := MovingCenter.isMinimizerAt_moving_C1_graph hV hE hmin hcenter obtain ⟨N,hN,hp,hu⟩ := normalMeasure_polar_unit hμ obtain ⟨δ,hδ,hgraph⟩ := hgraphs μ hμ N hN hp hu let e := δ/2 have he : 0 < e := by dsimp [e]; positivity obtain ⟨η,hη,hsmall⟩ := isMinimizerAt_halfspace_L1_tilt_small hV hE hmin hμ hH hn (show 0 < e^2/544^2 by positivity) refine ⟨η,hη,?_⟩ intro c r hr hrlim hc A hA hconv hphases hclose have hpositive : 0 < min η (min (544*R) (min (544*δ*e^2) (1/32))) := by positivity have hs := hrlim.eventually (Iio_mem_nhds hpositive) have hL1 := localL1_eventually_lt_model (fun i ↦ measurable_blowupCharacteristic hE (c i) (r i)) (fun i ↦ blowupCharacteristic_bound E (c i) (r i)) (measurable_const.indicator hA) (cover_indicator_bound A) (measurable_const.indicator (measurableSet_coverHalfspace n)) (cover_indicator_bound (coverHalfspace n)) hconv hclose have hpoints := isMinimizerAt_moving_frontier_approximation hV hE hmin c r hr hrlim hA hconv (by norm_num : (0:ℝ) < 1/(544*4096)) (by norm_num : (1:ℝ)/(544*4096) ≤ 1) (hphases _ (by norm_num) (by norm_num)) filter_upwards [hc,hs,hL1,hpoints] with i hci hri hL1i hpoint have hrη : r i < η := hri.trans_le (min_le_left _ _) have hri' := hri.trans_le (min_le_right _ _) have hti := hsmall (c i) (r i) (hr i) hrη hL1i obtain ⟨a,ha,hbase⟩ := hpoint have had : dist a (c i : Cover) < r i/(544*4096) := by simpa only [Metric.mem_ball,div_eq_mul_inv,one_mul] using ha have hrR : r i/544 ≤ R := by have hh := hri'.trans_le (min_le_left _ _); linarith have hrsmall : r i < 1/32 := hri'.trans_le ((min_le_right _ _).trans (min_le_right _ _)) have hratio : (r i/544)/e^2 ≤ δ := by apply (div_le_iff₀ (sq_pos_of_pos he)).mpr have hh := hri'.trans_le ((min_le_right _ _).trans (min_le_left _ _)) linarith have hnear : ‖a-cubeCenter‖ ≤ 1/16 := by have hh := dist_triangle a (c i : Cover) cubeCenter simp only [dist_eq_norm] at hh had linarith have hchart : Metric.ball (c i : Cover) (r i/8) ⊆ {x | ∀ j, 0 < x j ∧ x j < 1} := Metric.ball_subset_closedBall.trans (MovingCenter.closedBall_interior (by linarith : r i/8 < 3/8)) have htilt := spatial_tilt_le_blowupTilt hN hp hu (c i) n (hr i) hchart have hadq : dist (quotientMap a) (quotientMap (c i)) < r i/(544*4096) := by rw [dist_eq_norm,← quotientMap_sub] exact (norm_quotientMap_le _).trans_lt (by simpa only [dist_eq_norm] using had) have hsub : Metric.closedBall (quotientMap a) (17*(r i/544)) ⊆ Metric.closedBall (quotientMap (c i)) (r i/16) := by intro x hx have hx' : dist x (quotientMap a) ≤ 17*(r i/544) := hx have hh := dist_triangle x (quotientMap a) (quotientMap (c i)) change dist x (quotientMap (c i)) ≤ r i/16 linarith [hr i] have hbound : (∫ x in Metric.closedBall (quotientMap a) (17*(r i/544)), ‖N x-n‖^2 ∂(spatialNormalMeasure μ : Measure Torus)) ≤ (r i/544)^2*e^2 := by calc _ ≤ (∫ x in Metric.closedBall (quotientMap (c i)) (r i/16), ‖N x-n‖^2 ∂(spatialNormalMeasure μ : Measure Torus)) := setIntegral_mono_set (polar_tilt_integrable hN hu n).integrableOn (Eventually.of_forall fun _ ↦ sq_nonneg _) hsub.eventuallyLE _ ≤ _ := htilt _ ≤ (r i)^2*(e^2/544^2) := mul_le_mul_of_nonneg_left hti.le (sq_nonneg _) _ = _ := by ring exact ⟨a,had,hgraph a hnear hbase n (r i/544) e hn (div_pos (hr i) (by norm_num)) hrR he (by dsimp [e]; linarith) hratio hbound⟩ end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus theorem isMinimizerAt_close_limit_fixed_scale_normals {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) {H : ℝ} (hH : ∀ D : WallVariation, normalFirstVariation μ D = H*directionalFlux E D.coordinate D.function) : ∃ C K ρ δ : ℝ, 0 < C ∧ 0 < K ∧ 0 < ρ ∧ ρ ≤ 1/16 ∧ 0 < δ ∧ ∀ n : Cover, ‖n‖ = 1 → ∀ e : ℝ, 0 < e → e ≤ δ → ∃ η : ℝ, 0 < η ∧ ∀ (c : ℕ → Cover) (r : ℕ → ℝ), (∀ i, 0 < r i) → Tendsto r atTop (𝓝 0) → (∀ᶠ i in atTop, ‖c i-cubeCenter‖ ≤ 1/16) → ∀ A : Set Cover, MeasurableSet A → Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 1, |blowupCharacteristic E (c i) (r i) x-A.indicator (fun _ ↦ (1 : ℝ)) x|) atTop (𝓝 0) → (∫ x in Metric.closedBall (0 : Cover) 1, |A.indicator (fun _ ↦ (1 : ℝ)) x- (coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) < η → ∀ᶠ i in atTop, BoundaryNormalChart E (c i) n (r i/272) e C K ρ := by obtain ⟨C,K,ρ,δ,hC,hK,hρ,hρs,hδ,hcharts⟩ := isMinimizerAt_halfspace_L1_normal_charts hV hE hmin hcenter hμ hH refine ⟨C,K,ρ,δ,hC,hK,hρ,hρs,hδ,?_⟩ intro n hn e he heδ obtain ⟨η,hη,hsmall⟩ := hcharts n hn e he heδ refine ⟨η,hη,?_⟩ intro c r hr hrlim hc A hA hconv hclose have hs := hrlim.eventually (Iio_mem_nhds hη) have hL1 := localL1_eventually_lt_model (fun i ↦ measurable_blowupCharacteristic hE (c i) (r i)) (fun i ↦ blowupCharacteristic_bound E (c i) (r i)) (measurable_const.indicator hA) (cover_indicator_bound A) (measurable_const.indicator (measurableSet_coverHalfspace n)) (cover_indicator_bound (coverHalfspace n)) hconv hclose filter_upwards [hc,hs,hL1] with i hci hri hL1i exact hsmall (c i) (r i) hci (hr i) hri hL1i lemma ae_closedBall_imp_of_ae_ball {P : Cover → Prop} {a : Cover} {r : ℝ} (h : ∀ᵐ x ∂(volume : Measure Cover), x ∈ Metric.ball a r → P x) : ∀ᵐ x ∂(volume : Measure Cover), x ∈ Metric.closedBall a r → P x := by have hz : ∀ᵐ x ∂(volume : Measure Cover), x ∉ Metric.sphere a r := by rw [ae_iff] simpa only [not_not,Metric.sphere,dist_eq_norm,Set.mem_ofPred_eq] using Measure.addHaar_sphere (volume : Measure Cover) a r filter_upwards [h,hz] with x hx hxz hxc apply hx have hh : dist x a ≤ r := hxc have hn : dist x a ≠ r := hxz exact lt_of_le_of_ne hh hn lemma localL1_limit_congr_ae_ball {g : ℕ → Cover → ℝ} {f F : Cover → ℝ} {R : ℝ} {l : Filter ℕ} (heq : ∀ᵐ x ∂(volume : Measure Cover), x ∈ Metric.ball (0 : Cover) R → f x = F x) (hconv : Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) R, |g i x-f x|) l (𝓝 0)) : Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) R, |g i x-F x|) l (𝓝 0) := by have hh := ae_closedBall_imp_of_ae_ball heq have he (i : ℕ) : (∫ x in Metric.closedBall (0 : Cover) R, |g i x-f x|) = ∫ x in Metric.closedBall (0 : Cover) R, |g i x-F x| := by apply setIntegral_congr_ae Metric.isClosed_closedBall.measurableSet filter_upwards [hh] with x hx hxb rw [hx hxb] simpa only [he] using hconv lemma volume_real_affine_preimage_ball {A : Set Cover} (hA : MeasurableSet A) (p : Cover) {s : ℝ} (hs : 0 < s) (ρ : ℝ) : volume.real (((fun x ↦ p+s • x) ⁻¹' A) ∩ Metric.ball (0 : Cover) ρ) = s⁻¹^3*volume.real (A ∩ Metric.ball p (s*ρ)) := by have he (D : Set Cover) (hD : MeasurableSet D) (a : Cover) (r : ℝ) : (∫ x in Metric.ball a r, D.indicator (fun _ ↦ (1 : ℝ)) x) = volume.real (D ∩ Metric.ball a r) := by rw [integral_indicator hD,Measure.restrict_restrict hD,integral_const,smul_eq_mul,mul_one, Measure.real_def,Measure.restrict_apply_univ] rfl have hD : MeasurableSet ((fun x : Cover ↦ p+s • x) ⁻¹' A) := hA.preimage (by fun_prop) rw [← he A hA p,← he _ hD 0] rw [integral_ball_affine_scaling _ p hs ρ] have hh (x : Cover) : ((fun x ↦ p+s • x) ⁻¹' A).indicator (fun _ ↦ (1 : ℝ)) x = A.indicator (fun _ ↦ (1 : ℝ)) (p+s • x) := rfl simp only [hh] field_simp end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus def ActualBoundaryGraph (E : Set Torus) (c n : Cover) (q : ℝ) : Prop := ∃ a : Cover, dist a c < q/(544*4096) ∧ ∃ g : Plane → ℝ, LipschitzWith (1/4) g ∧ ContDiffOn ℝ 1 g (Metric.ball 0 ((q/544)/1024)) ∧ (∀ y ∈ Metric.ball (0 : Plane) ((q/544)/1024), quotientMap a+quotientMap (normalGraph n g y) ∈ frontier (openRepresentative E) ∧ ‖normalGraph n g y‖ < (q/544)/512) ∧ (∀ x ∈ quotientMap ⁻¹' frontier (openRepresentative E) ∩ Metric.ball a ((q/544)/128), normalGraph n g (normalPlaneCoordinates n (x-a)) = x-a) theorem actual_primary_fixed_scale_packets {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) {H : ℝ} (hH : ∀ D : WallVariation, normalFirstVariation μ D = H*directionalFlux E D.coordinate D.function) : ∃ C K ρ δ : ℝ, 0 < C ∧ 0 < K ∧ 0 < ρ ∧ ρ ≤ 1/16 ∧ 0 < δ ∧ ∀ n : Cover, ‖n‖ = 1 → ∀ e : ℝ, 0 < e → e ≤ δ → ∃ η : ℝ, 0 < η ∧ ∀ (r : ℕ → ℝ), (∀ i, 0 < r i) → Tendsto r atTop (𝓝 0) → ∀ A : Set Cover, MeasurableSet A → Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 2, |blowupCharacteristic E cubeCenter (r i) x-A.indicator (fun _ ↦ (1 : ℝ)) x|) atTop (𝓝 0) → ∀ p : Cover, ‖p‖ < 1 → (∀ t : ℝ, 0 < t → 0 < volume.real (A ∩ Metric.ball p t) ∧ 0 < volume.real (Aᶜ ∩ Metric.ball p t)) → ∀ s : ℝ, 0 < s → s ≤ 1 → (∫ x in Metric.closedBall (0 : Cover) 1, |A.indicator (fun _ ↦ (1 : ℝ)) (p+s • x)- (coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) < η → ∀ᶠ i in atTop, ActualBoundaryGraph E (cubeCenter+r i • p) n (r i*s) ∧ BoundaryNormalChart E (cubeCenter+r i • p) n ((r i*s)/272) e C K ρ := by obtain ⟨C,K,ρ,δ,hC,hK,hρ,hρs,hδ,hnormals⟩ := isMinimizerAt_close_limit_fixed_scale_normals hV hE hmin hcenter hμ hH refine ⟨C,K,ρ,δ,hC,hK,hρ,hρs,hδ,?_⟩ intro n hn e he heδ obtain ⟨ηN,hηN,hN⟩ := hnormals n hn e he heδ obtain ⟨ηG,hηG,hG⟩ := isMinimizerAt_close_limit_fixed_scale_graphs hV hE hmin hcenter hμ hH hn refine ⟨min ηN ηG,lt_min hηN hηG,?_⟩ intro r hr hrlim A hA hconv p hp hboundary s hs hs1 hclose obtain ⟨k,hk⟩ := eventually_atTop.mp (hrlim.eventually (Iio_mem_nhds (by norm_num : (0:ℝ)<1/32))) let c : ℕ → MovingCenter.InteriorCenter := fun i ↦ ⟨cubeCenter+r (i+k) • p,by simp only [add_sub_cancel_left,norm_smul,Real.norm_of_nonneg (hr _).le] have hh := hk (i+k) (by omega) nlinarith [hr (i+k)]⟩ let q : ℕ → ℝ := fun i ↦ r (i+k)*s have hq : ∀ i, 0 < q i := fun i ↦ mul_pos (hr _) hs have hqlim : Tendsto q atTop (𝓝 0) := by simpa only [q,Function.comp_def,zero_mul] using (hrlim.comp (tendsto_add_atTop_nat k)).mul_const s have hc (i : ℕ) : ‖(c i : Cover)-cubeCenter‖ ≤ 1/32 := by change ‖cubeCenter+r (i+k) • p-cubeCenter‖ ≤ 1/32 rw [add_sub_cancel_left,norm_smul,Real.norm_of_nonneg (hr _).le] have hh := hk (i+k) (by omega) nlinarith [hr (i+k)] let D : Set Cover := (fun x ↦ p+s • x) ⁻¹' A have hD : MeasurableSet D := hA.preimage (by fun_prop) have hsub : Metric.closedBall p (s*1) ⊆ Metric.closedBall (0 : Cover) 2 := by intro x hx have hx' : ‖x-p‖ ≤ s := by simpa only [Metric.mem_closedBall,dist_eq_norm,mul_one] using hx have hh := norm_add_le (x-p) p rw [sub_add_cancel] at hh simpa only [Metric.mem_closedBall,dist_zero_right] using (by linarith : ‖x‖ ≤ 2) have hshift := localL1_affine_tendsto (fun i ↦ measurable_blowupCharacteristic hE cubeCenter (r i)) (fun i ↦ blowupCharacteristic_bound E cubeCenter (r i)) (measurable_const.indicator hA) (cover_indicator_bound A) hconv p hs hsub have hlim : Tendsto (fun i ↦ ∫ x in Metric.closedBall (0 : Cover) 1, |blowupCharacteristic E (c i) (q i) x-D.indicator (fun _ ↦ (1 : ℝ)) x|) atTop (𝓝 0) := by have hh := hshift.comp (tendsto_add_atTop_nat k) convert hh using 2 rename_i i apply integral_congr_ae apply ae_of_all intro x change |blowupCharacteristic E (cubeCenter+r (i+k) • p) (r (i+k)*s) x- D.indicator (fun _ ↦ (1 : ℝ)) x| = |blowupCharacteristic E cubeCenter (r (i+k)) (p+s • x)- A.indicator (fun _ ↦ (1 : ℝ)) (p+s • x)| rw [blowupCharacteristic_affine] rfl have hphases (t : ℝ) (ht : 0 < t) (_ : t ≤ 1) : 0 < volume.real (D ∩ Metric.ball (0 : Cover) t) ∧ 0 < volume.real (Dᶜ ∩ Metric.ball (0 : Cover) t) := by have hh := hboundary (s*t) (mul_pos hs ht) change 0 < volume.real (((fun x ↦ p+s • x) ⁻¹' A) ∩ Metric.ball (0 : Cover) t) ∧ 0 < volume.real (((fun x ↦ p+s • x) ⁻¹' Aᶜ) ∩ Metric.ball (0 : Cover) t) rw [volume_real_affine_preimage_ball hA p hs,volume_real_affine_preimage_ball hA.compl p hs] exact ⟨mul_pos (by positivity) hh.1,mul_pos (by positivity) hh.2⟩ have hclose' : (∫ x in Metric.closedBall (0 : Cover) 1, |D.indicator (fun _ ↦ (1 : ℝ)) x-(coverHalfspace n).indicator (fun _ ↦ (1 : ℝ)) x|) < min ηN ηG := hclose have hgg := hG c q hq hqlim (Eventually.of_forall hc) D hD hlim hphases (hclose'.trans_le (min_le_right _ _)) have hnn := hN (fun i ↦ (c i : Cover)) q hq hqlim (Eventually.of_forall (fun i ↦ (hc i).trans (by norm_num : (1:ℝ)/32 ≤ 1/16))) D hD hlim (hclose'.trans_le (min_le_left _ _)) have hall : ∀ᶠ i in atTop, ActualBoundaryGraph E (cubeCenter+r (i+k) • p) n (r (i+k)*s) ∧ BoundaryNormalChart E (cubeCenter+r (i+k) • p) n ((r (i+k)*s)/272) e C K ρ := by filter_upwards [hgg,hnn] with i hgi hni exact ⟨hgi,hni⟩ obtain ⟨j,hj⟩ := eventually_atTop.mp hall refine eventually_atTop.mpr ⟨j+k,?_⟩ intro i hi have hh := hj (i-k) (by omega) simpa only [Nat.sub_add_cancel (by omega : k ≤ i)] using hh end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus lemma exists_uniform_subseq_of_bounded_modulus {X Y : Type*} [PseudoMetricSpace X] [CompactSpace X] [MetricSpace Y] [ProperSpace Y] (f : ℕ → X → Y) (hc : ∀ i, Continuous (f i)) (y₀ : Y) (M : ℝ) (hb : ∀ i x, dist (f i x) y₀ ≤ M) (hm : ∀ ε : ℝ, 0 < ε → ∃ η : ℝ, 0 < η ∧ ∀ i x y, dist x y < η → dist (f i x) (f i y) < ε) : ∃ (g : X → Y) (φ : ℕ → ℕ), Continuous g ∧ StrictMono φ ∧ TendstoUniformly (fun i ↦ f (φ i)) g atTop := by let F : ℕ → BoundedContinuousFunction X Y := fun i ↦ BoundedContinuousFunction.mkOfCompact ⟨f i,hc i⟩ have heq : Equicontinuous ((↑) : range F → X → Y) := by apply UniformEquicontinuous.equicontinuous rw [Metric.uniformEquicontinuous_iff] intro ε hε obtain ⟨η,hη,hηm⟩ := hm ε hε refine ⟨η,hη,?_⟩ intro x y hxy rintro ⟨_,i,rfl⟩ exact hηm i x y hxy have hcpt := BoundedContinuousFunction.arzela_ascoli (Metric.closedBall y₀ M) (isCompact_closedBall y₀ M) (range F) (by rintro _ x ⟨i,rfl⟩; exact hb i x) heq obtain ⟨g,_,φ,hφ,hconv⟩ := hcpt.tendsto_subseq (fun i ↦ subset_closure (mem_range_self i)) exact ⟨g,φ,g.continuous,hφ,BoundedContinuousFunction.tendsto_iff_tendstoUniformly.mp hconv⟩ def normalGraphDerivative (n ν : Cover) : Plane →L[ℝ] ℝ := (⟪ν,n⟫)⁻¹ • (-((innerSL ℝ ν).comp (normalPlaneLinearIsometry n).toContinuousLinearMap)) lemma transverse_inner_lower {n ν : Cover} (hn : ‖n‖ = 1) (hν : ‖n-ν‖ ≤ 1/2) : 1/2 ≤ ⟪ν,n⟫ := by have hh := abs_real_inner_le_norm (n-ν) n rw [inner_sub_left,real_inner_self_eq_norm_sq,hn,mul_one] at hh have hl := (abs_le.mp hh).2 norm_num at hl linarith lemma normalGraphDerivative_continuousOn {n : Cover} (hn : ‖n‖ = 1) : ContinuousOn (normalGraphDerivative n) (Metric.closedBall n (1/2)) := by have hi : ContinuousOn (fun ν : Cover ↦ (⟪ν,n⟫)⁻¹) (Metric.closedBall n (1/2)) := by apply (continuous_id.inner continuous_const).continuousOn.inv₀ intro ν hν have hnorm : ‖n-ν‖ ≤ 1/2 := by simpa only [Metric.mem_closedBall,dist_eq_norm,norm_sub_rev] using hν exact ne_of_gt (lt_of_lt_of_le (by norm_num : (0:ℝ)<1/2) (transverse_inner_lower hn hnorm)) exact hi.smul ((((innerSL ℝ).continuous.continuousOn).clm_comp continuousOn_const).neg) end CubicTorus end end section noncomputable section open MeasureTheory Set Filter open scoped ENNReal NNReal Topology RealInnerProductSpace namespace CubicTorus def rescaledActualBoundary (E : Set Torus) (c : Cover) (q : ℝ) : Set Cover := (fun x ↦ quotientMap (c+q • x)) ⁻¹' frontier (openRepresentative E) end CubicTorus end end end end OAI