In this note we announce some results, due to appear in [2], [3], on the
structure of integral and normal currents, and their relation to Frobenius
theorem. In particular we show that an integral current cannot be tangent to a
distribution of planes which is nowhere involutive (Theorem 3.6), and that a
normal current which is tangent to an involutive distribution of planes can be
locally foliated in terms of integral currents (Theorem 4.3). This statement
gives a partial answer to a question raised by Frank Morgan in [1].Comment: 8 page

Alberti, Giovanni

Massaccesi, Annalisa

English

arXiv

ALBERTI, GIOVANNI

Massaccesi A.

Archivio istituzionale della ricerca - Università di Padova

Original Citation:On some geometric properties of currents and Frobenius theoremEuropean Mathematical Society Publishing HousePublisher:Published version:DOI:Terms of use:Open Access(Article begins on next page)This article is made available under terms and conditions applicable to Open Access Guidelines, asdescribed at http://www.unipd.it/download/file/fid/55401 (Italian only)Availability:This version is available at: 11577/3317120 since: 2021-02-19T10:54:30Z10.4171/RLM/788Università degli Studi di PadovaPadua Research Archive - Institutional RepositoryA NOTE ON GEOMETRIC PROPERTIES FOR CURRENTSGIOVANNI ALBERTI AND ANNALISA MASSACCESI1. IntroductionThe celebrated Frobenius Theorem states the equivalence between the completeintegrability of a k-dimensional simple vectorfield ξ = τ1 ∧ . . .∧ τk ∈ C1(Rd; Λk(Rd))in Rd – that is, the existence of a local foliation of Rd by k-dimensional tangentsubmanifolds of class C2 – and the involutivity condition for ξ, that is[τm, τn](x) ∈ span{τ1(x) . . . , τk(x)} x ∈ Rd , ∀m,n = 1, . . . , k,where [⋅, ⋅] denotes the Lie bracket (commutator) between vector fields in Rd.In this paper, we consider the following natural question: given a k-vectorfieldξ, is it possible to generalize the Frobenius Theorem to k-dimensional surface of aweaker type? In Section 4 we prove the following version of Frobenius Theorem forintegral currents, where we use the standard notation R = ⟦Σ, ξ, θ⟧ to denote thecurrent R in Rd defined byR(ω) = ∫Σ⟨ω(x), ξ(x)⟩θ(x)dHk(x), ∀ω ∈ C∞c (Rd; Λk(Rd))for Σ ⊂ Rd a k-rectifiable set, ξ a k-vectorfield and θ an integer multiplicity.Theorem 1.1. Let ξ = τ1 ∧ . . . ∧ τk be a k-dimensional simple vector field in Rd,with τ1, . . . , τk ∈ C1(Rd), and let R = ⟦Σ, ξ, θ⟧ be a k-dimensional integral currentin Rd. Then, for every pair m,n = 1, . . . , k and for every x in the closure of the setof points of positive density of Σ, one has[τm, τn](x) ∈ span{τ1(x), . . . , τk(x)} .The proof of Theorem 1.1 builds upon two results. The first one, which isprobably the main result of the paper, establishes an intuitive geometric propertyfor the boundary of an integral current.Theorem 1.2. Let R be a k-dimensional integral current in Rd and let ξ ∈ C 1(Rd; Λk(Rd))be a continuous k-vectorfield which is tangent to R. Then, the orientation η(x) ∈Λk−1(Rd) of the boundary ∂R is a subspace of ξ(x) for H k−1-almost every x in thesupport of ∂R.This result is studied in Section 3 and it is obtained by a blow-up technique.The second main tool in the proof of Theorem 1.1 is the following DifferentialGeometry result.Lemma 1.3. If ξ is a smooth non-involutive simple k-vectorfield, then there existsa (k − 1)-form α with the following properties:(i) ⟨ξ,dα⟩ ≠ 0 on a suitable open set;(ii) if η is a simple (k−1)-vector field representing a linear subspace of ξ, then⟨η,α⟩ = 0.12 GIOVANNI ALBERTI AND ANNALISA MASSACCESIThe proof of Theorem 1.1 follows quite readily once Theorem 1.2 and Lemma1.3 are available. Indeed, assuming by contradiction that ξ is non-involutive andrepresenting ∂R = ⟦Σ′, η, θ′⟧, up to a localization argument one obtains0 = ∫Σ′⟨η,α⟩θ′ dH k−1 = ∂R(α) = R(dα) = ∫Σ⟨ξ,dα⟩θ dH k ≠ 0 .Let us point out the Theorem 1.1 concerns only one of the two implicationsin Frobenius Theorem. One might wonder whether the converse is true in thefollowing version: given a normal current T = ξµ, where µ is a finite measure onRd and ξ ∈ C 1(Rd; Λk(Rd)) is an involutive vectorfield, is it possible to “foliate”T with a family of integral currents in a proper way (that is, without wasting anymass)? This question was formulated in a broader version by F. Morgan who askedin [1] whether, given a k-dimensional normal current T in Rd, it is possible to finda family of integral currents (Rλ)λ∈L (where L is a suitable measure space) suchthat(i) T = ∫LRλ dλ;(ii) M(T ) = ∫LM(Rλ)dλ;(iii) M(∂T ) = ∫LM(∂Rλ)dλ.In Section 5 we study this problem and we show that the normal current T = ξL ddoes not admit such a decomposition if ξ is a non-involutive vectorfield. Indeed,conditions (i) and (ii) imply that Rλ = ⟦Σλ, ξ, θλ⟧ for almost every λ ∈ L, thusλ-almost every Rλ should violate Theorem 1.1.2. NotationIn the following, we work with a fixed orthonormal basis {e1, . . . , ed} of Rd anda dual basis {dx1, . . . ,dxd} such that⟨ei; dxj⟩ = δij .A basis for the space of alternating vectors Λk(Rd) is given by{ei1 ∧ . . . ∧ eik ∶ 1 ≤ i1 < . . . < ik ≤ d} .Correspondingly, a basis for the space of alternating covectors Λh(Rd) is given by{dxi1 ∧ . . .dxih ∶ 1 ≤ i1 < . . . < ih ≤ d} .Definition 2.1. Given a h-covector w ∈ Λh(Rd) and a k-vector v ∈ Λk(Rd), withh ≥ k, the interior product is defined as⟨v̂; v ⌟w⟩ ∶= ⟨v̂ ∧ v;w⟩ ∀ v̂ ∈ Λh−k(Rd) .Vice versa, if h ≤ k, we can define⟨v ⌞w; ŵ⟩ ∶= ⟨v;w ∧ ŵ⟩ ∀ ŵ ∈ Λk−h(Rd) .Definition 2.2. Given a k-vector v ∈ Λk(Rd), we define the span of v as thesmallest linear subspace V of Rd such that v ∈ Λk(V ), that isspan(v) ∶= ⋂V ⊂Rd l.s.v∈Λk(V )V .Sometimes, when we have v ∈ Λk(Rd), v′ ∈ Λk′(Rd) and span(v′) ⊂ span(v), weabbreviate v′ ⊂ v.A NOTE ON GEOMETRIC PROPERTIES FOR CURRENTS 3Finally we recall the definition of the Lie bracket in a convenient coordinateversion.Definition 2.3. Given a vectorfield X ∶ U ⊂ Rd → Rd of class C 1, we represent itsaction on a smooth function f ∶ U → R as X(f) = ∇Xf ∶= ⟨∇f ;X⟩ = ∑di=1∂f∂xiXi.If X ≡ ei for some i ∈ {1, . . . , d}, then we also write ∇if =∂f∂xi. Given two C 1-vectorfields X,Y ∶ U → Rd their Lie bracket is [X,Y ] ∶=XY −Y X and [X,Y ](f) =∇X∇Y f −∇Y∇Xf . In coordinates[X,Y ]h = ⟨∇Yh;X⟩ − ⟨∇Xh;Y ⟩ =d∑i=1(∂Yh∂xiXi −∂Xh∂xiYi) , with h = 1, . . . , d .3. Geometric structure of the boundaryDefinition 3.1. Given a k-dimensional normal current T = ξµ, with orientationξ ∈ C(Rd; Λk(Rd)) and µ Radon measure in Rd, and its boundary ∂T = ηµ′, withorientation η and µ′ Radon measure in Rd, we say that T has the geometric boundaryproperty if(3.1) spanη(x) ⊂ span ξ(x) µ′-a.e. x .Remark 3.2. If ξ is a simple vectorfield, then the geometric boundary property isequivalent to the existence, µ′-a.e. x, of a w ∈ Λ1(Rd) such that(3.2) η(x) = ξ(x) ⌞w .In general, (3.2) is stronger than (3.1).Remark 3.3. The geometric boundary property does not hold for every normalcurrent T . For instance, if T = ξL d, where ξ is a compactly supported non-involutive vectorfield in the sense of Theorem 4.2 below, then ∂T has orientationdivξ and the geometric boundary property does not hold.Theorem 3.4. Let ξ be a continuous k-dimensional vectorfield on Rd and let R =⟦Σ, ξ, θ⟧ be a k-dimensional integral current, then T has the geometric boundaryproperty.The proof of Theorem 3.4 is based on the blow up technique and can be essentiallysplit in two parts:(1) for an integral current R with boundary ∂R = ⟦Σ′, η, θ′⟧ it is possible todo a blow up at H k−1-a.e. point of Σ′ and this blow up commutes withthe boundary;(2) if x0 ∈ Σ′ is one of the points where we could perform such a blow up andξ(x0) is the constant orientation of the blown up current and η(x0) is theconstant orientation of the blow up of the boundary of the current, thenspanη(x0) ⊂ span ξ(x0), because∂R0 ⌞ ν(ϕ) = ∂R0(ν) = −R0(ν ∧ dϕ) = −R0 ⌞ ν(dϕ) = 0for every ν ∈ Λ1(Rd) such that ξ(x0) ⌞ ν = 0.4 GIOVANNI ALBERTI AND ANNALISA MASSACCESI4. Frobenius-type result for integral currentsOne of the most important theorems in Differential Geometry fully answersto this question: under which conditions a given k-dimensional simple vectorfieldξ = τ1 ∧ . . . ∧ τk ∈ C∞(Rd; Λk(Rd)) represents the tangent vectorfield of a smoothmanifold M?Definition 4.1. A k-dimensional simple vectorfield ξ = τ1∧. . .∧τk, with τ1, . . . , τk ∈C 1(Rd; Λk(Rd)), is integrable if, for every point x0 ∈ Rd, there exist an open neigh-borhood U ∋ x0 and a k-dimensional submanifold M ∋ x0 such thatTxM = span{τ1(x), . . . , τk(x)}for every x ∈ U ∩M . We will say that ξ is completely integrable if, for every x0 ∈ Rd,there exist an open neighborhood U ∋ x0 and a C2-function1 F ∶ U → Rd−k such thatits level sets {F = p} are k-submanifolds with span{τ1, . . . , τk} as tangent space.Roughly speaking, a completely integrable vectorfield is a tangent field for a localfoliation of Rd in C 2-submanifolds. Obviously, a completely integrable vectorfieldis integrable.The proof of the following theorem characterizing integrable vectorfields can befound in [4] or in any other book about the basics of smooth manifolds.Theorem 4.2 (Frobenius Theorem). A k-dimensional simple vectorfield ξ = τ1 ∧. . . ∧ τk, with τ1, . . . , τk ∈ C1(Rd; Λk(Rd)), is completely integrable if and only if(4.1) [τm, τn](x) ∈ span{τ1(x), . . . , τk(x)}for every x ∈ Rd and for every m,n = 1, . . . , k.Let us remark that condition (4.1) does not depend on the choice of τ1, . . . , τk,but only on their product ξ = τ1 ∧ . . . ∧ τk.Definition 4.3. Consider a k-dimensional simple vectorfield ξ = τ1 ∧ . . . ∧ τk, suchthat τ1, . . . , τk ∈ C1(Rd; Λk(Rd)). We say that ξ is involutive2 if (4.1) holds.Lemma 4.4. Given a non-involutive simple k-vectorfield ξ = τ1 ∧ . . . ∧ τk, withτ1, . . . , τk ∈ C1(Rd; Λk(Rd)), there exist an open subset U ⊂ Rd and a (k − 1)-formα such that we have(4.2) ⟨ξ(x),dα(x)⟩ ≠ 0 ∀x ∈ Uand(4.3) ⟨η(x), α(x)⟩ = 0 ∀x ∈ Rdwhenever η(x) is a simple (k−1)-vectorfield representing a linear subspace of ξ(x).Consider a non-involutive k-vectorfield ξ in Rd, as above. We may wonder if thenon-involutivity property is strong enough to prevent not only the existence of asurface with tangent field ξ, but also the existence of an integral current with such1From this perspective, it is clear that this problem and the problem of the existence of apotential F ∶ U ⊂ Rd → R for a given map f ∶ U → Rd, with ∇F = f , are related and we refer tothem as integrability problems.2Actually, a more general definition of involutivity can be given, after noticing that divξ∧τm∧τn = (−1)k−1ξ ∧ [τm, τn]. Indeed, one can say that ξ = τ1 ∧ . . . ∧ τk is involutive if and only ifdivξ ∧ τm ∧ τn = 0 for every m,n = 1, . . . , d. This definition applies when τ1, . . . , τk are barelycontinuous, provided divξ is a measure.A NOTE ON GEOMETRIC PROPERTIES FOR CURRENTS 5a tangent field. The answer is affirmative, as we state in Theorem 4.5. We beginwith the key theorem.Theorem 4.5. Let ξ = τ1 ∧ . . . ∧ τk be a k-dimensional simple vectorfield on Rd,with τ1, . . . , τk ∈ C1(Rd), and let T ∈ Ik(Rd) be a k-dimensional integral currentwith R = ⟦Σ, ξ, θ⟧, then(4.4) [τm, τn](x) ∈ span{τ1(x), . . . , τk(x)}for every pair m,n = 1, . . . , k and for every x in the closure of the set of points ofpositive density of Σ.Proof. Choose a (k − 1)-form α satisfying (4.2) and (4.3) of Lemma 4.4 for someopen set U ⊂ Rd. Therefore,0 ≠ ⟨T ⌞U ; dα⟩ = ⟨∂T ⌞U ;α⟩ ,because of (4.2). But then Theorem 3.4 and condition (4.3) imply⟨∂T ⌞U ;α⟩ = 0and this is a contradiction. 5. Decomposition of normal currentsAn interesting problem in the theory of currents concerns the decomposition ofa normal current by means of a family of integral currents. This problem firstlyappeared in [1], formulated by F. Morgan. More precisely, given a normal currentT ∈ Nk(Rd), we ask whether there exists a family of integral currents (Rλ)λ∈L,where L is a suitable measure space, such that(i) T = ∫LRλ dλ, i.e., for every ω ∈ Dk(Rd), we can writeT (ω) = ∫LRλ(ω)dλ ;(ii) M(T ) = ∫LM(Rλ)dλ;(iii) M(∂T ) = ∫LM(∂Rλ)dλ.Condition (ii) and (iii) express the requirement of a decomposition where no mass iswasted. In the analysis below, we will discuss also weaker versions of the problem:we can drop condition (iii), and we can also change the type of “decomposing”currents, saying we are satisfied with a family of rectifiable currents, instead ofthe family of integral ones. We agree that, when we do not specify the type ofcurrents to which the “decomposing” family belongs, we will always be looking fora decomposition into integral currents; otherwise, we will always specify the typeof the decomposing currents.Let us briefly sketch the state of the art for this problem.(1) When the dimension of the normal current is 1, it is known that thereexists a decomposition satisfying (i) and (ii): see Proposition 4.4 in [5]and [8] or [7] for the proof (while a decomposition satisfying (i), (ii) and(iii) may not always exist).(2) In the special case of codimension 1 with an integer rectifiable bounda-ry, there exists a decomposition satisfying (i), (ii) and (iii), thanks to anobservation by M. Zworski in [9]. The core of this argument is the Hardt-Pitts decomposition proved in [3].Moreover, one can prove the following theorems.6 GIOVANNI ALBERTI AND ANNALISA MASSACCESITheorem 5.1. When the codimension of the normal current is 1, there exists adecomposition in rectifiable currents satisfying (i) and (ii).Theorem 5.2. If T = ξµ is a k-dimensional normal current and ξ ∈ C1(Λk(Rd))is involutive, then there exists a decomposition in rectifiable currents satisfying (i),(ii) and (iii).The existence of a decomposition for normal currents under suitable assumptionsis an isolated result. Indeed, in general the search for a decomposition by meansof integral currents is actually too strong: we claim that any normal current ofthe form ξL d cannot be decomposed into integral currents satisfying (i) and (ii),provided ξ is non-involutive. This claim follows from Lemma 5.3 and Theorem 4.5.In [9], M. Zworski exhibited this very same counterexample, claiming that, ingeneral, a normal current has no decomposition satisfying (i) and (ii), even if weallow the decomposing currents to be rectifiable only. However the proof in [9] doesnot work, as pointed out by Alberti (see Section 4.5 of [6]). There is a gap in theargument, possibly due to a misunderstanding when referring to the Federer Flat-ness Theorem 4.1.15 in [2]. We propose the same counterexample for the problemof decomposing a normal current with a family of integral currents satisfying (i)and (ii) (at page 66) and the results we got in Section 4 fill the aforementioned gap.Lemma 5.3. Consider a vector-valued measure µ = ξ∣µ∣, whereξ ∈ C∞(Rd; Λk(Rd))is a smooth k-vectorfield and assume that(i) µ = ∫L µλ dλ, where (L,λ) is a measure space;(ii) ∥µ∥ = ∫L ∥µλ∥dλ.Then, for λ-a.e. µλ, we have thatµλ = ξ∣µλ∣ .Theorem 5.4. Consider the normal current T ∶= ξL d ∈ Nk(Rd) given by thesmooth non-involutive vectorfield ξ ∈ C∞(Rd; Λk(Rd)). Then there exist no measurespace L and no family of integral currents (Rλ)λ∈L such that (i) and (ii) at page 66hold.The proof of this theorem is a consequence of Lemma 5.3 and Theorem 4.5.References[1] Some open problems in geometric measure theory and its applications suggested by participantsof the 1984 AMS summer institute. In J. E. Brothers, editor, Geometric measure theory andthe calculus of variations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math.,pages 441–464. Amer. Math. Soc., Providence, RI, 1986.[2] Herbert Federer. Geometric measure theory. Die Grundlehren der mathematischen Wis-senschaften, Band 153. Springer-Verlag New York Inc., New York, 1969.[3] Robert M. Hardt and Jon T. Pitts. Solving Plateau’s problem for hypersurfaces without thecompactness theorem for integral currents. In Geometric measure theory and the calculus ofvariations (Arcata, Calif., 1984), volume 44 of Proc. Sympos. Pure Math., pages 255–259.Amer. Math. Soc., Providence, RI, 1986.[4] John M. Lee. Introduction to smooth manifolds, volume 218 of Graduate Texts in Mathematics.Springer, New York, second edition, 2013.[5] Andrea Marchese and Annalisa Massaccesi. The steiner tree problem revisited through recti-fiable g-currents. Preprint, 2013.A NOTE ON GEOMETRIC PROPERTIES FOR CURRENTS 7[6] Frank Morgan. Geometric measure theory. Elsevier/Academic Press, Amsterdam, fourth edi-tion, 2009. A beginner’s guide.[7] E. Paolini and E. Stepanov. Decomposition of acyclic normal currents in a metric space.Preprint, 2012.[8] S. K. Smirnov. Decomposition of solenoidal vector charges into elementary solenoids, and thestructure of normal one-dimensional flows. Algebra i Analiz, 5(4):206–238, 1993.[9] Maciej Zworski. Decomposition of normal currents. Proc. Amer. Math. Soc., 102(4):831–839,1988.Università di PisaE-mail address: galberti1@dm.unipi.itMathematik Institut der Universität ZürichE-mail address: annalisa.massaccesi@math.uzh.ch

On some geometric properties of currents and Frobenius theorem

In this note we announce some results, due to appear in [2], [3], on the structure of integral and normal currents, and their relation to Frobenius theorem. In particular we show that an integral current cannot be tangent to a distribution of planes which is nowhere involutive (Theorem 3.6), and that a normal current which is tangent to an involutive distribution of planes can be locally foliated in terms of integral currents (Theorem 4.3). This statement gives a partial answer to a question raised by Frank Morgan in [1]

On some geometric properties of currents and Frobenius theorem

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