On the geometric structure of currents tangent to smooth distributions
aa r X i v : . [ m a t h . DG ] A p r [version: March 31, 2020] to apper on J. Differential Geom. On the geometric structure of currentstangent to smooth distributions
Giovanni Alberti, Annalisa Massaccesi, Eugene Stepanov
Abstract.
It is well known that a k -dimensional smooth surface in a Euclideanspace cannot be tangent to a non-involutive distribution of k -dimensional planes. Inthis paper we discuss the extension of this statement to weaker notions of surfaces,namely integral and normal currents. We find out that integral currents behave tothis regard exactly as smooth surfaces, while the behavior of normal currents is rathermultifaceted. This issue is strictly related to a geometric property of the boundaryof currents, which is also discussed in details. Keywords: non-involutive distributions, Frobenius theorem, integral currents, nor-mal currents, geometric property of the boundary.
1. Introduction
The starting point of this paper is the following implication in Frobenius theorem:if V is a distribution of k -dimensional planes on an open set Ω in R n , and Σ is a k -dimensional smooth surface which is everywhere tangent to V , then V is involutiveat every point of Σ or, equivalently, Σ does not intersect the open set where V is non-involutive. In the following we refer to this statement simply as Frobeniustheorem.In the classical statement it is assumed that both the distribution V and thesurface Σ are sufficiently regular. In particular it suffices that V be of class C andΣ be a surface (submanifold) of class C , possibly with boundary. In this paper wediscuss the generalization of this result to weaker notions of surfaces, though notweakening the regularity assumption on V (see however § k -dimensional C -surface. More precisely, for every continuous distribution of k -planes V there exists a C -surface S such that the set Σ of all points of S where S is tangent to V has positive k -dimensional measure, regardless of the involutivityof V (this result was proved for a special V in [6], Theorem 1.4, and it can be easilyderived, at least for some V , by the main result in [1]; the general version can befound in [5]).We notice that Frobenius theorem holds if the boundary of Σ (relative to S ) isnot too large; for example, it suffices that H k − ( ∂ Σ) be finite, where H k − is the( k − § V . Thusone is naturally led to wonder what happens for the largest class of currents with“nice” boundary, namely normal currents. It turns out that this case is much more G. Alberti, A. Massaccesi, E. Stepanov interesting, and in particular the validity of Frobenius theorem depends also on how“diffuse” the current is (Theorem 1.3).Notice that our results are local in nature, and therefore, even if stated in theEuclidean space, they actually hold in Riemannian manifolds, and indeed even inFinsler manifolds.Some of the results in this paper were announced in [4].
Description of the results
Through the rest of this paper V is a C -distribution of k -planes on an open setΩ in R n , and N ( V ) is the open set of all points where V is non-involutive. Let T be an integral current in Ω which is tangent to V . Thenthe support of T does not intersect the non-involutivity set N ( V ) . A version of this statement was first proved in the second author’s dissertation([17], Theorem 2.2.6), following a completely different argument.The next step is to consider normal currents. We recall here that these currentsshare many properties with integral currents, including that of having a nice bound-ary, but differ in many regards. In particular integral k -dimensional currents aresupported on k -dimensional (rectifiable) sets, while k -dimensional normal currentscan be quite “sparse”, even absolutely continuous with respect to the Lebesguemeasure.The following example, proposed by M. Zworski in [20], shows that Frobeniustheorem does not hold in general for normal currents. Consider a simple k -vectorfield v = v ∧ · · · ∧ v k of class C on R n and let T be the k -current given by T = v L n . Then T is a normal current onevery bounded open set Ω in R n (see § V spanned by v , . . . , v k , regardless of its involutivity.It turns out that there is a general result behind this example, and more preciselya normal current T which is tangent to a distribution V must be sufficiently “sparse”on the non-involutivity set N ( V ), and the degree of “sparseness” depends on howmuch non-involutive the distribution V is.A precise statement requires some preparation. We let b V be the distributionspanned by the vectorfields tangent to V and their first commutators (see § d = k, . . . , n we set N ( V, d ) := (cid:8) x ∈ Ω : dim( b V ( x )) = d (cid:9) . (Thus N ( V ) is the union of all sets N ( V, d ) with d = k + 1 , . . . , n .)We then consider a normal k -current T on Ω, which we write as T = τ µ where µ is a finite positive measure and τ is a k -vectorfield which is nonzero µ -a.e. (cf. § T agrees with the support of µ .Along the same line we write the boundary of T as ∂T = τ ′ µ ′ .We say that T is tangent to V if span( τ ( x )) = V ( x ) for µ -a.e. x , Most of the terminology used in this introduction is properly defined in Section 2; the precisedefinition of N ( V ) is given in § The precise meaning of “ T is tangent to V ” is given in § The span of a (non necessarily simple) k -vector is defined in § eometric structure of currents 3 We define the degree of sparseness of a measure in terms of absolute continuitywith respect to the Hausdorff measure H d or the integral geometric measures I dt (the higher is d , the sparser is the measure). If T is a normal k -current and wetake µ and µ ′ as above, then µ is absolutely continuous with respect to I kt andtherefore also with respect to H k (see § µ ≪ I kt ≪ H k . Similarly µ ′ ≪ I k − t ≪ H k − .We can now state the main result for normal currents. Take V , T = τ µ and ∂T = τ ′ µ ′ as above, and assume that T betangent to V . Then (i) the restriction of µ to the set N ( V ) satisfies µ x N ( V ) ≪ µ ′ ; (ii) for d > k there holds µ x N ( V, d ) ≪ I dt ≪ H d ; (iii) for d > k there holds µ ′ x N ( V, d ) ≪ H d − . Using Theorem 1.3 we can show that Frobenius theorem holds for normal currentsthat satisfy certain additional conditions:
Take V , T = τ µ and ∂T = τ ′ µ ′ as above, and assume that T betangent to V . If any of the following conditions holds then the support of T does notintersect N ( V ) : (a) µ is concentrated on a I k +1 t -null Borel set; (b) µ is concentrated on a µ ′ -null Borel set; (c) T is a rectifiable current (possibly with non-integeral multiplicity); (d) ∂T = 0 . (i) Regarding condition (a) in Corollary 1.4, we recall that thefollowing implications hold for every Borel set E : I k +1 t ( E ) = 0 ⇐ H k +1 ( E ) = 0 ⇐ dim H ( E ) < k + 1 , where dim H ( E ) is the Hausdorff dimension of E .(ii) Under condition (c), Corollary 1.4 generalizes Theorem 1.1.(iii) Even though the measures µ and µ ′ in the representations of T and ∂T arenot unique, the statements of Theorem 1.3 and Corollary 1.4 do not depend on thechoice of these measures (cf. Remark 2.11(ii)).As already pointed out in [4], the validity of Frobenius theorem for normal currentsis strictly related to the following property of the boundary. Let T = τ µ be a normal k -currenton the open set Ω with boundary ∂T = τ ′ µ ′ . We say that T has the geometricproperty of the boundary if, up to a modification of τ in a µ -null set,(a) the map x span( τ ( x )) is continuous on the support of T ;(b) span( τ ′ ( x )) ⊂ span( τ ( x )) for µ ′ -a.e. x .It is easy to check that if T is tangent to the distribution V then these conditionsare equivalent to the inclusionspan( τ ′ ( x )) ⊂ V ( x ) for µ ′ -a.e. x . (1.1) We say that µ is concentrated on a Borel set E if µ (Ω \ E ) = 0; this implies that the supportof µ is contained in the closure of E (but not necessarily in E ). G. Alberti, A. Massaccesi, E. Stepanov
Take V and T = τ µ as above, and assume that T be tangentto V . Then the following assertions are equivalent: (i) T has the geometric property of the boundary, that is, (1.1) holds; (ii) the support of µ does not intersect N ( V ) . (i) The current associated to an oriented surface Σ of class C has the geometric property of the boundary; indeed condition (b) in § x ∈ ∂ Σ the tangent space T x ( ∂ Σ) is contained in T x Σ.(ii) Example 1.2 and Theorem 1.7 show that there are normal currents T whichare tangent to a distribution V of class C and do not have the geometric property ofthe boundary. In this case one may ask where the inclusion span( τ ′ ( x )) ⊂ V ( x ) holdsand where it does not; a detailed answer is given in Theorem 3.3 and Remark 3.4.(iii) Theorem 1.7 implies that the geometric property of the boundary holds if T is tangent to a distribution of k -planes of class C and satisfies one of the conditions(a)–(d) in Corollary 1.4, e.g., if T is an integral current. In [3] we give an example ofintegral current which is tangent to a continuous distribution of k -planes and doesnot have the geometric property of the boundary. Additional comments
We collect here furtherremarks on the property defined in § § T such that ∂T is singular with respect to T (that is, µ ′ is singular with respect to µ ) has thegeometric property of the boundary—the point is that the k -vectorfield τ is onlydetermined up to µ -null sets, and therefore it can be arbitrarily modified in a setwhich is µ ′ -full.(ii) The relation between the geometric property of the boundary and Frobeniustheorem for currents was first pointed out by the second author in her disserta-tion [17], where a version of Theorem 1.1 is obtained as a corollary of the geometricproperty of the boundary of integral currents ([17], Lemma 2.2.1).(iii) We point out that in [17] and [3] the sentence “the current T is tangent tothe distribution V ” has a stronger meaning than in this paper. Here it means thatthe tangent plane span( τ ( x )) is prescribed µ -a.e., while there it means that both thetangent plane span( τ ) and its orientation are prescribed µ -a.e.Under this stronger notion of tangency, in [3] it is proved that the geometricproperty of the boundary holds for integral currents that are tangent to a contin-uous distribution V of k -planes (while here we need that V be of class C , cf. Re-mark 1.8(iii)). The statement of Theorem 1.3 depends crucially onthe sets N ( V, d ), which are defined using the distribution b V spanned by the vector-fields tangent to V and their first commutators (see § V spanned by the Lie algebra generated by V ,that is, by the vectorfields tangent to V and their commutators of all orders. Clearlythe distribution V contains b V , and the inclusion may be strict. If this is the case,replacing the sets N ( V, d ) by N ( V, d ) := (cid:8) x ∈ Ω : dim( V ( x )) = d (cid:9) eometric structure of currents 5 in statements (ii) and (iii) of Theorem 1.3 yields a stronger results. We believe thatsuch results are true, but cannot be obtained by a modification of the present proof. Extensions of Frobenius theorem toweaker notions of surfaces have been studied by many authors. For instance, in [16],Theorem 2.1, it is proved that
Sobolev sets of dimension m in the (sub-Riemannian)Heisenberg group H n cannot be horizontal for m > n , that is, images of Sobolevmaps with derivative of rank m from open subsets of R m into R n +1 ≃ H n cannotbe tangent to the horizontal distribution. In the opposite direction, Theorem 1.14 in[7] shows that graphs of BV functions from R to R can be tangent to the horizontaldistribution in the Heisenberg group H ≃ R × R .Using Theorem 1.1 we partially recover Theorem 2.1 in [16] and extend it to amore general setting. Let Ω be an open subset in R n and V a C -distribution of k -planes on Ω . Let A be an open set in R k and let u : A → Ω be a continuous map ofclass W ,p loc with p > k such that, for a.e. z ∈ A , the image of the differential of u at z is V ( u ( z )) . Then u ( A ) does not intersect the non-involutivity set N ( V ) . Given a distribution of k -planes V and a k -dimensionalsurface S of class C , we say that a closed subset Σ of S is a tangency set of S and V if the tangent space T x S agrees with V ( x ) for every x ∈ Σ. In this context thestatement of Frobenius theorem reduces to H k (cid:0) Σ ∩ N ( V ) (cid:1) = 0 . (1.2)As already pointed out at the beginning of this introduction, if S is of class C then (1.2) does not hold for all tangency sets Σ, but holds if Σ has finite perimeterrelative to S . Note that this condition is implied by (but not equivalent to) anyof the following: a) the (topological) boundary of Σ relative to S is H k − -finite;b) the boundary of the canonical current associated to Σ has finite mass.On the other hand, if the surface S is of class C , or even of class C , , then (1.2)holds for every tangency set Σ, regardless of the regularity of its boundary (see forinstance [8], Theorem 1.3). This result is generalized in [5] by proving that if S is ofclass C ,α for some 0 < α < fractional regularity. This showsthat the validity of Frobenius theorem depends on a combination of the regularityof ∂ Σ and of the supporting surface S , and not just on the former. It would beinteresting to extend this result to more general currents. Through this paper we always assume thatthe distribution V is of class C , which is the minimal regularity required to defineinvolutivity in the classical sense (see § § V is less regular than C using a suitable distributionalformulation. However, in order to extend the results stated above to less regular V ,a major difficulty seems to be the correct definition of the non-involutivity set N ( V ). This claim follows from two results by S. Delladio: in [10], Corollary 4.1, he proves that H k -a.e. point x of a finite perimeter set Σ is a superdensity point, i.e., H k (cid:0) ( B ( x, r ) ∩ S ) \ Σ (cid:1) = o ( r k +1 ),and in [11], Corollary 1.1, he proves that the set of superdensity points of Σ does not intersect N ( V ),and therefore (1.2) holds. G. Alberti, A. Massaccesi, E. Stepanov
Structure of the paper.
Section 2 contains the notation and some preliminaryresults. The main result in Section 3 is the key identity (3.4), which allows us toestablish a very precise connection between Frobenius theorem for normal currentsand the geometric property of the boundary (Theorem 3.3). All statements givenin this introduction are more or less straightforward consequences of identity (3.4)and Theorem 3.3; the proofs are collected in Section 4.
Acknowledgements.
This research was partly carried out during several visits of theauthors: A.M. at the Mathematics Department in Pisa (supported by the University ofPisa through the 2015 PRA Grant “Variational methods for geometric problems”); E.S. atthe Mathematics Department in Pisa (supported by the 2018 INdAM-GNAMPA project“Geometric Measure Theoretical approaches to Optimal Networks”); G.A. and A.M. atCIRM in Trento (supported by the CIRM “Research in Pairs” program).The research of G.A. has been partially supported by the Italian Ministry of Universityand Research (MIUR) through PRIN project 2010A2TFX2 and by the European ResearchCouncil (ERC) through project 291497. A.M. has been partially supported by ERC throughproject 306247 and by the European Union’s Horizon 2020 programme through project752018. E.S. has been partially supported by the Russian Foundation for Basic Research(RFBR) through grant
2. Notation and preliminary results
We assume that the reader is somewhat familiar with the theory of currents.Therefore in this section we only briefly recall the basic notions of multilinear algebraand of the theory of currents, mainly to fix the notation, and describe in more detailsonly those notions that are of less common use.Through this paper we tacitly assume that sets and functions are Borel measurableand measures are defined on the Borel σ -algebra, and are real-valued and finite (withthe notable exception of Lebesgue, Hausdorff and integral geometric measures).Here is a list of frequently used notations: µ x F restriction of a measure µ to a Borel set F , that is, [ µ x F ]( E ) := µ ( E ∩ F )for every Borel set E in X ; ρµ measure associated to a measure µ on X and a Borel density ρ , that is,[ ρµ ]( E ) := R E ρ dµ for every Borel set E in X ; f µ pushforward of a measure µ on X according to a Borel map f : X → Y ,that is, [ f µ ]( E ) := µ ( f − ( E )) for every Borel set E in Y ; f T pushforward of a current T according to a map f (see, e.g., [14], § | µ | variation measure associated to a real- or vector-valued measure µ ; µ ≪ λ the measure µ is absolutely continuous with respect to the measure λ ; µ a , µ s absolutely continuous and singular part of a measure µ with respect to agiven measure λ . L n , H d Lebesgue measure on R n and d -dimensional Hausdorff measure; I dt d -dimensional integral geometric measures ( § I ( n, k ) set of all multi-indices i := ( i , . . . , i k ) with 1 ≤ i < · · · < i k ≤ n ; eometric structure of currents 7 ∧ k ( V ) space of k -vectors in a linear space V ; the canonical basis of ∧ k ( R n ) isformed by the simple k -vectors e i := e i ∧ · · · ∧ e i k with i ∈ I ( n, k ), where { e , . . . , e n } is the canonical basis of R n ; ∧ k ( R n ) is endowed with theEuclidean norm | · | associated to this basis; ∧ k ( V ) space of k -covectors on a linear space V ; the canonical basis of ∧ k ( R n ) isformed by the simple k -covectors d x i := d x i ∧ · · · ∧ d x i k with i ∈ I ( n, k ),where { d x , . . . , d x n } is the canonical basis of the dual of R n ; ∧ k ( R n ) isendowed with Euclidean norm | · | associated to this basis;d x := d x ∧ · · · ∧ d x n ; ∧ exterior product of k -vectors, or of h -covectors; y , x interior products of a k -vector and a h -covector ( § ⋆ Hodge-type operator on k -vectors and k -covectors ( § v ) span of a k -vector v ( § k -form ( § k -vectorfield ( § v, v ′ ] Lie bracket of vectorfields v and v ′ ( § W ( µ, · ) decomposability bundle of a measure µ ( § N ( V ) non-involutivity set of a distribution of k -planes V ( § b V and N ( V, d ), see § Given d = 1 , . . . , n and t ∈ [1 , + ∞ ], wedenote by I dt the d -dimensional integral geometric measure of exponent t on R n .The precise definition of this measure can be found in [12], § § I dt is invariant under isometries of R n , it agrees withthe Hausdorff measure H d on regular d -dimensional surfaces of R n , and in generalsatisfies I dt ≤ H d . Moreover, and this is essential to this paper, a Borel set E is I dt -null if and only if H d (cid:0) p V ( E ) (cid:1) = 0 for a.e. d -plane V in R n ,where p V stands for the orthogonal projection on V and “a.e.” refers to the Haarmeasure on the Grassmannian of d -planes in R n .Note that the class of I dt -null Borel sets is the same for all t and is strictly largerthan the class of H d -null sets (indeed by the Besicovitch-Federer projection theoremthe first class contains all sets which are H d -finite and purely d -unrectifiable).In particular the fact that a measure µ is absolutely continuous with respect to I dt does not depend on the exponent t and implies that µ is absolutely continuous withrespect to H d (but the converse does not hold). Multilinear algebra
In this subsection we review the basic notions of multilinear algebra; we considermultivectors and multicovectors in a general linear space V . For a thorough treatiseof this topic, we refer the reader to [12], § However, none of the results in this paper depend on the specific choice of the norm.
G. Alberti, A. Massaccesi, E. Stepanov
Given a k -vector v in V and an h -covector α on V with h ≤ k , the interior product v x α is the ( k − h )-vector in V defined by h v x α ; β i := h v ; α ∧ β i for every ( k − h )-covector β ;if k ≤ h , the interior product v y α is the ( h − k )-covector defined by h w ; v y α i := h w ∧ v ; α i for every ( h − k )-vector w .Note that given a k -vector v , an h -covector α and an h ′ -covector α ′ with h + h ′ ≤ k ,then v x ( α ∧ α ′ ) = ( v x α ) x α ′ . Similarly, given a k -vector v , a k ′ -vector v ′ and an h -covector α with k + k ′ ≤ h ,then ( v ∧ v ′ ) y α = v y ( v ′ y α ) . k -vector. Given a k -vector v in V , we denote by span( v ) thesmallest of all linear subspaces W of V such that v belongs to ∧ k ( W ). This def-inition is well-posed because every k -vector in W is canonically identified with a k -vector in V via the inclusion map i : W → V , and assuming this identification wehave ∧ k ( W ) ∩ ∧ k ( W ′ ) = ∧ k ( W ∩ W ′ ) for every W, W ′ subspaces of V . We have thefollowing properties (see [2], Proposition 5.9):(i) if v = 0 then span( v ) = { } ;(ii) if v = 0 then dim(span( v )) ≥ k ;(iii) if v is simple and non-trivial, that is, v = v ∧ · · · ∧ v k with v , . . . , v k linearlyindependent vectors in V , then span( v ) is the subspace of V spanned by v , . . . , v k and dim(span( v )) = k ;(iv) conversely, if dim(span( v )) = k then v is simple and non-trivial;(v) span( v ) consists of all vectors of the form v x α with α ∈ ∧ k − ( V ).The next lemma will be used in the proofs later. Let v be a k -vector in V and let W be a d -dimensional subspaceof V . Then span( v ) ⊂ W if any of the following conditions hold: (a) there exist a subspace W ′ of W with d ′ := dim( W ′ ) ≥ k , and an integer h with ≤ h ≤ d ′ − k + 1 such that span( v ∧ w ) ⊂ W for every w ∈ ∧ h ( W ′ ) ; (b) v ∧ w = 0 for every w ∈ ∧ d − k +1 ( W ) ; (c) k = 1 and there exist an integer ≤ h ≤ d and a simple h -vector w ∈ ∧ h ( W ) with w = 0 such that span( v ∧ w ) ⊂ W ; Proof.
The proof is divided in three steps, one for each condition.
Step 1: if condition (a) holds then span( v ) ⊂ W . We argue by contradiction, andprove that if span( v ) W then there exists w ∈ ∧ h ( W ′ ) such that span( v ∧ w ) W .To this aim we choose vectors e , . . . , e n in V so that e , . . . , e d ′ form a basis of W ′ , e , . . . , e d form a basis of W , and e , . . . , e n form a basis of V . Then we write v as v = X i ∈ I ( n,k ) v i e i . For example, given linearly independent vectors v , . . . , v , then span( v ∧ v + v ∧ v ) is thelinear subspace spanned by v , . . . , v . eometric structure of currents 9 Since span( v ) is not contained in W there exists a multi-index j = ( j , . . . , j k ) in I ( n, k ) such that v j = 0 and j k > d . This means that at most k − j belong to { , . . . , d ′ } ; thus there are at least d ′ − k + 1 indices in { , . . . , d ′ } that arenot in j , and since h ≤ d ′ − k + 1 we can find a multi-index j ′ ∈ I ( d ′ , h ) whose indicesare all different from those of j (with a slight abuse of notation we write j ′ ∩ j = ∅ ).We now set w := ˆ e j ′ . Then v ∧ w = X i : j ′ ∩ i = ∅ v i ˆ e i ∧ ˆ e j ′ . Let j ∪ j ′ denote the multi-index in I ( n, k + h ) that contains the indices in j andin j ′ . The formula above shows that the coordinate ( v ∧ w ) j ∪ j ′ is equal to ± v j andin particular it does not vanish; since j ∪ j ′ contains j k and j k > d , we deduce that v ∧ w is not a ( k + h )-vector in W , that is, span( v ∧ w ) W , as claimed. Step 2: condition (b) implies condition (a) . More precisely, condition (a) holdswith W ′ := W and h := d ′ − k + 1 = d − k + 1. Step 3: condition (c) implies condition (a) . More precisely, condition (a) holdswith W ′ := span( w ): notice indeed that since w is simple then d ′ := dim( W ′ ) = h ,and therefore the h -vectors in ∧ h ( W ′ ) are just multiples of w . (cid:3) We consider the operator ⋆ that acts on allvectors and covectors of R n , and more precisely maps k -vectors into ( n − k )-covectorsand vice versa, and is defined by the following property: for every v ∈ ∧ k ( R n ) andevery α ∈ ∧ k ( R n ) there holds ⋆ v := v y d x , ⋆ α := e x α , where d x := d x ∧ · · · ∧ d x n and e := e ∧ · · · ∧ e n . Note that the definition of the interior products (see § h w ; ⋆ v i = h w ∧ v ; d x i , h ⋆ α ; β i = h e ; α ∧ β i , for every ( n − k )-vector w and every ( n − k )-covector β .Moreover for every i ∈ I ( n, k ) one has ⋆ e i = sign( j , i ) d x j , ⋆ d x i = sign( i , j ) e j , (2.1)where j is the multi-index in I ( n, n − k ) consisting of all indices which are not in i , and sign( j , i ) is the sign of the permutation that reorders the sequence of indices j , . . . , j n − k , i , . . . , i k . The identities in (2.1) show that ⋆ is an involution, that is, ⋆ ( ⋆ v ) = v and ⋆ ( ⋆ α ) = α .Among the many identities relating ⋆ and the various products, we will use thefollowing one: for every k -vector v and every h -covector α with h ≤ k one has ⋆ ( v x α ) = ( ⋆ v ) ∧ α . (2.2) Forms, vectorfields, currents
Here we review the basic definitions and results concerning differential forms,vectorfields and currents. These objects will be defined on a general open set Ω in R n , n ≥ This operator is similar to the standard Hodge star operator but not the same; it is defined in[12], § We view k -forms as maps ω : Ω → ∧ k ( R n ), whichwe sometime write in terms of the canonical basis of ∧ k ( R n ), that is, ω ( x ) = X i ∈ I ( n,k ) ω i ( x ) d x i . Similarly we view a k -vectorfield as a map v : Ω → ∧ k ( R n ), which we write as v ( x ) = X i ∈ I ( n,k ) v i ( x ) e i . If ω is a k -form of class C , the exterior derivative d ω is the ( k +1)-form defined in coordinates by the usual formula:d ω ( x ) := X i ∈ I ( n,k ) n X j =1 ∂ω i ∂x j ( x ) d x j ∧ d x i = n X j =1 d x j ∧ ∂ω∂x j ( x ) . (2.3)If v is a k -vectorfield of class C , the divergence div v is the ( k − v ( x ) := X i ∈ I ( n,k ) n X j =1 ∂v i ∂x j ( x ) e i x d x j = n X j =1 ∂v∂x j ( x ) x d x j . (2.4)The latter definition cannot be considered as standard as (2.3): we refer to [12], § v := ( − n − k ⋆ (d( ⋆ v )) . (2.5)Finally, for k = 1, formula (2.4) reduces to the usual definition of divergence of avectorfield (recall that e i x d x j = h e i ; d x j i = δ ij ). The exterior derivative satisfies a Leibniz rule with respectto the exterior product: given a k -form ω and a k ′ -form ω ′ on Ω, both of class C ,one has d (cid:0) ω ∧ ω ′ (cid:1) = d ω ∧ ω ′ + ( − k ω ∧ d ω ′ . (2.6)The divergence satisfies a Leibniz rule with respect to the interior product: givena k -vector v and an h -form ω on Ω, both of class C and with h ≤ k , one hasdiv( v x ω ) = ( − h ((div v ) x ω + v x d ω ) , (2.7)which follows from (2.6) using (2.2) and (2.5). Given two vectorfields v , v ′ on Ω ofclass C , the Lie bracket [ v, v ′ ] is the vectorfield on Ω defined by[ v, v ′ ]( x ) := ∂v∂v ′ ( x ) − ∂v ′ ∂v ( x ) = d x v ( v ′ ( x )) − d x v ′ ( v ( x )) , where d x v and d x v ′ stand for the differentials of v and v ′ at the point x , viewed aslinear maps from R n into itself. Formulas relating divergence and exterior product (or exterior derivative and interior product)are more complicated, see § eometric structure of currents 11 Consider now a simple k -vectorfield v = v ∧ · · · ∧ v k with k ≥ v i isa vectorfield of class C on Ω. Then the divergence of v can be computed using thefollowing version of Cartan’s formula:div v = k X i =1 ( − i − div v i (cid:16) ^ j = i v j (cid:17) + X ≤ i
The proof of (2.12) is divided in several steps.
Step 1: V + V = V + V and V + V = b V . These equalities follow from theinclusion div( w ∧ w ′ ) − [ w, w ′ ] ∈ V , which holds for every pair of 1-vectorfields w, w ′ tangent to V , and is a consequencesof formula (2.9). Step 2: V ⊂ V + V . Every 2-vectorfield w of class C tangent to V can bewritten as w = X ≤ i The equivalence of (i), (ii) and (iii) follows immediatelyfrom Proposition 2.17.Let us prove the implication (iii) ⇒ (iv). Assertion (iii) means that div v is a( k − V . Thus (div v ) x d x i is a 2-vector in V and v ∧ ((div v ) x d x i ) is a( k + 1)-vector in V , and it must vanish because V has dimension k .Finally, let us prove the implication (iv) ⇒ (iii). Every vector in span(div v ) canbe written as (div v ) x α for some ( k − α (see § v ∧ ((div v ) x α ) = 0, which in turn implies that (div v ) x α belongs to the spanof v , which is V (here we use that v is simple and nontrivial). (cid:3) Corollary 2.18(iv) shows that the in-volutivity of a distribution V spanned by a k -vectorfield v is characterized by theequation v ∧ ((div v ) x d x i ) = 0 for every i ∈ I ( n, k − k = 2 reduces to v ∧ div v = 0.We point out that equation (2.13) makes sense even if v is less regular than C .More precisely, the right-hand side of (2.13) is a well-defined distribution if v anddiv v belong, locally, to function spaces which are in duality (and are closed undermultiplication by functions of class C ∞ c ) and therefore one can define involutivityfor such classes of vectorfields.For example, it suffices that v be continuous and div v be a locally finite measure,or that v belong to the Sobolev class H s loc for some s ≥ v ∈ H − s loc . Inparticular it suffices that v ∈ H / loc (in this case div v ∈ H − / loc because the divergenceis a first-order differential operator). Decomposability bundle and sparseness of a measure Here we briefly recall the notion of decomposability bundle of a measure µ , andshow that the dimension of this bundle gives a lower bound on the degree of sparse-ness of µ , expressed in terms of absolute continuity with respect to integral geometricmeasures I dt . eometric structure of currents 15 Here we briefly sketch the def-inition of the decomposability bundle of a measure, introduced in [2], § µ on the open set Ω, we denote by F ( µ ) the class of allfamilies { F t : t ∈ I } parametrized by I := [0 , 1] such that: • each F t is a 1-dimensional rectifiable set in Ω; • the measure λ := R I ( H x F t ) dt satisfies λ ≪ µ . The decomposability bundle of µ is a map that to every x ∈ Ω associates a (possiblytrivial) linear subspace of R n , denoted in this paper by W ( µ, x ), which is uniquelydetermined up to µ -null sets by the following properties:(i) for every { F t } ∈ F ( µ ) there holds T x F t ⊂ W ( µ, x ) for H -a.e. x ∈ F t anda.e. t ∈ I , where T x F t is the approximate tangent line to the set F t at x ;(ii) W ( µ, · ) is µ -minimal among all bundles W ( · ) that satisfy property (i), in thesense that W ( µ, x ) ⊂ W ( x ) for µ -a.e. x .Besides some results already contained in [2], we will need the following statement,which is a consequence of a remarkable theorem by G. De Philippis and F. Rindler [9]. Let µ and W ( µ, · ) be as above, d be an integer, and E be aBorel set such that dim( W ( µ, x )) ≥ d for µ -a.e. x ∈ E . Then µ x E ≪ I dt ≪ H d . For the proof we need the following two lemmas. Let µ and W ( µ, · ) be as above, let f : Ω → R m be a map of class C , and let f µ be the pushforward of µ according to f . Then d x f ( W ( µ, x )) ⊂ W ( f µ, f ( x )) for µ -a.e. x ∈ Ω , (2.14) where d x f : R n → R m is the differential of f at x . Sketch of proof. We argue by contradiction and assume that there exists a Borelset F with µ ( F ) > τ on Ω such that(a) d x f ( τ ( x )) / ∈ W ( f µ, f ( x )) and | τ ( x ) | = 1 for µ -a.e. x ∈ F ;(b) τ ( x ) ∈ W ( µ, x ) for µ -a.e. Step 1. Using property (b) and Corollary 6.5 in [2] we find a normal 1-current T such that τ is the Radon-Nikod´ym density of T with respect to µ , that is, T = τ µ + T s with T s singular with respect to µ . Possibly removing from F a µ -null subset where T s is concentrated, we can assume that T x F = τ µ x F . Step 2. Using Theorem 5.5 in [2] we find a family of 1-dimensional rectifiablesets { F t : t ∈ I } with I := [0 , 1] such that for every t ∈ I and H -a.e. x ∈ F t theapproximate tangent space T x F t is spanned by τ ( x ), and R I ( H x F t ) dt = µ x F . Step 3. For every t ∈ I the set E t := f ( F t ) is rectifiable, the approximate tangentspace T f ( x ) E t is spanned by d x f ( τ ( x )) for H -a.e. x ∈ F t , the measures H x E t and f ( H x F t ) are absolutely continuous with respect to each other, and Z I ( H x E t ) dt ≪ Z I f ( H x F t ) dt = f ( µ x B ) ≤ f µ . Recall that λ is given by λ ( E ) := R I H ( F t ∩ E ) dt for every Borel set E ⊂ Ω, where dt is theLebesgue measure; we implicitly require that this integral is well-defined and finite. Thus the family { E t : t ∈ I } belongs to F ( f µ ) and therefore property (i) in § t ∈ I and H -a.e. x ∈ F t ,d x f ( τ ( x )) ∈ T f ( x ) E t ⊂ W ( f µ, f ( x )) , which means that d x f ( τ ( x )) ∈ W ( f µ, f ( x )) for µ -a.e. x ∈ F , in contradiction withproperty (a) above. (cid:3) Let µ and W ( µ, · ) be as above, and assume that dim( W ( µ, x )) ≥ d for µ -a.e. x . Then µ ≪ I dt . Proof. We first introduce some notation: • λ d is the Haar measure on the Grassmannian Gr( d, n ); • for every V ∈ Gr( d, n ), p V : R n → V is the orthogonal projection onto V and µ V is the pushforward of the measure µ according to p V .Using the characterization of I dt -null sets given in § µ ≪ I dt is implied by the assertion µ V ≪ H d for λ d -a.e. V . The proof ofthe latter is divided in four steps. Step 1: if W ∈ Gr( d ′ , n ) with d ′ ≥ d then p V ( W ) = V for λ d -a.e. V . Possi-bly replacing W with a subspace, we can assume that W has dimension d . Sinceker( p V ) = V ⊥ , we have that p V ( W ) = V if and only if dim( W ∩ V ⊥ ) = 0 . Therefore, taking into account that the map V V ⊥ is a bijection from Gr( d, n )to Gr( n − d, n ) that preserves the respective Haar measures, we can reformulate theclaim as follows: dim( W ∩ Z ) = 0 for λ n − d -a.e. Z ∈ Gr( n − d, n ).This is equivalent to saying that the set S k := (cid:8) Z ∈ Gr( n − d, n ) : dim( W ∩ Z ) = k (cid:9) is λ n − d -null for every k > 0, which is a consequence of the fact that S k is actually asmooth submanifold of Gr( n − d, n ) with dimension strictly lower than Gr( n − d, n ). Step 2: for λ d -a.e. V one has p V ( W ( µ, x )) = V for µ -a.e. x ∈ Ω . By assumptionwe have dim( W ( µ, x )) ≥ d for µ -a.e. x , and then it suffices to use Step 1. Step 3: for λ d -a.e. V one has W ( µ V , y ) = V for µ V -a.e. y ∈ V . (2.15)By applying Lemma 2.22 to the map f := p V we obtain that, for every d -plane V , W ( µ V , p V ( x )) ⊃ p V ( W ( µ, x )) for µ -a.e. x ∈ Ω,and recalling Step 2 we obtain that, for λ d -a.e. V , W ( µ V , p V ( x )) ⊃ p V ( W ( µ, x )) = V for µ -a.e. x ∈ Ω,which implies W ( µ V , y ) ⊃ V for µ V -a.e. y ∈ V because µ V is the pushforward of µ through p V . To obtain (2.15) it is enough torecall that W ( µ V , y ) ⊂ V for µ V -a.e. y ∈ V because µ V is a measure on V . Step 4: µ V ≪ H d for λ d -a.e. V . Identity (2.15) means the following: if we iden-tify the d -plane V with R d (isometrically), then µ V is a measure on R d whose decom-posability bundle is a.e. equal to R d , and therefore Corollary 1.12 and Lemma 3.1 eometric structure of currents 17 in [9] imply that µ V is absolutely continuous with respect to the Lebesgue measureon R d , that is, the restriction of H d to V . (cid:3) Proof of Proposition 2.21. Let ¯ µ be the restriction of µ to the set E . By Propo-sition 2.9(i) in [3] we have that W (¯ µ, x ) = W ( µ, x ) for ¯ µ -a.e. x , which implies thatdim( W (¯ µ, x )) ≥ d for ¯ µ -a.e. x . We conclude the proof by applying Lemma 2.23. (cid:3) Sparseness of a normal current and of its boundary In this subsection we establish a relation between the degree of sparseness of anormal current T and that of its boundary ∂T , both expressed in terms of absolutecontinuity with respect to Hausdorff measures. Let T = τ µ be a normal k -current on the open set Ω in R n with boundary ∂T = τ ′ µ ′ such that (a) there exists a real number α ∈ [ k, n ] such that µ ≪ H α . (b) there exists a C -vectorfield v on Ω such that v ∧ τ = 0 µ -a.e.Let E := (cid:8) x ∈ Ω : v ( x ) ∧ τ ′ ( x ) = 0 (cid:9) Then µ ′ x E ≪ H α − . (i) If τ is simple, the condition v ∧ τ = 0 µ -a.e. in assumption (b)is equivalent to v ( x ) ∈ span( τ ( x )) for µ -a.e. x , that is, v is tangent to T .(ii) If k > τ ′ is simple, then E = { x : v ( x ) / ∈ span( τ ′ ( x )) } .(iii) If k = 1 then τ ′ is a real function with τ ′ = 0 µ ′ -a.e., and therefore E can beequivalently defined as E = { x : v ( x ) = 0 } .Before the proof we present two examples that illustrate the optimality of thisstatement: the first one shows that the vectorfield v cannot be just continuous, andthe second one shows that the Hausdorff measures cannot be replaced by the integralgeometric measures. For every a ∈ [0 , 1] let T a be the integral 1-current in R asso-ciated to the (oriented) curve parametrized by γ a ( t ) := ( t, at ) with t ∈ [0 , T be the normal 1-current given by the superposition of all T a , that is, h T ; ω i := Z h T a ; ω i da for every 1-form ω of class C ∞ c .Then T = (cid:16) x , x x (cid:17) L x F , ∂T = H x I − δ , where δ is the Dirac mass at 0, and F, I are the sets in R defined by F := (cid:8) x : 0 ≤ x ≤ , ≤ x ≤ x (cid:9) , I := (cid:8) x : x = 1 , ≤ x ≤ (cid:9) . Thus µ ≪ L = H but µ ′ H .Let now v : F → R be the vectorfield given by v ( x ) := (1 , x /x ) if x = 0 and v (0) := (1 , v is of class C , / on F and can be extendedto the entire R with the same regularity. Moreover v is tangent to T and nevervanishes on the set F , which contains the support of µ ′ ; therefore the set E containsthe support of µ ′ (see Remark 2.25(iii)) and thus µ ′ x E = µ ′ H , which means that Proposition 2.24 fails for this choice of v . (On the other hand, every vectorfield v of class C tangent to T must vanish at 0, thus for such v the set E does notcontain 0 and µ ′ x E ≪ H , in accordance with Proposition 2.24.) Take a Borel function g : [0 , → R whose graph Γ is purelyunrectifiable and H -finite, and for every r ∈ R consider the map f r : [0 , → R given by f r ( s ) := ( s, g ( s ) + r ). For every a ∈ [0 , 1] we denote by T a the 1-current in R associated to the (oriented) vertical segment I a := [ f ( a ) , f ( a )], and by T thesuperposition of all such T a (defined as in the previous example). Then T = e L x F , ∂T = λ − λ , where e := (0 , F is the union of the segments I a with 0 ≤ a ≤ 1, and λ r is thepushforward of the Lebesgue measure on [0 , 1] according to the map f r .Thus µ ≪ L = H = I t . Moreover the constant vectorfield v ( x ) := e istangent to T and never vanishes, and then Remark 2.25(iii) yields µ ′ x E = µ ′ .On the other hand µ ′ is supported on the set Γ ∪ (Γ + e ), which is H -finiteand purely unrectifiable, and therefore I t -null, which implies that µ ′ x E = µ ′ issingular with respect to I t . (However µ ′ ≪ H , as predicted by Proposition 2.24.)We now pass to the proof of Proposition 2.24. Using a localization argument we reduce to the case where Ω = R n and T and v have compact support. Moreover we can assume that | τ ′ | = 1 µ ′ -a.e. In the proof we use the flow associated to v , namely the map Φ : R × R n → R n defined by Φ(0 , x ) = x , ∂ Φ ∂t ( t, x ) = v (Φ( t, x )) for every t ∈ R , x ∈ R n ,and we write Φ t ( x ) for Φ( t, x ). Since v is of class C and compactly supported, themap Φ is well-defined and of class C , and each map Φ t is bi-Lipschitz and agreeswith the identity out of a compact set which does not depend on t . Let T and v be as in Proposition 2.24, and let Φ be as above.Given t , t ∈ R , let [[ t , t ]] be the -current in R associated to the oriented interval [ t , t ] . Then the pushforward of the product current [[ t , t ]] × ∂T on R × R n accordingto the map Φ satisfies Φ (cid:0) [[ t , t ]] × ∂T (cid:1) = (Φ t ) T − (Φ t ) T . (2.16) Proof. The homotopy formula (see for instance [14], § ∂ (cid:0) Φ ([[ t , t ]] × T ) (cid:1) = (Φ t ) T − (Φ t ) T − Φ (cid:0) [[ t , t ]] × ∂T (cid:1) , and then (2.16) reduces to Φ (cid:0) [[ t , t ]] × T (cid:1) = 0 . (2.17)The proof of (2.17) is divided in two steps. To this end we denote by d ( t,x ) Φ thedifferential of Φ at the point ( t, x ), viewed as a linear map from R × R n to R n , andlet e := (1 , ∈ R × R n . Moreover we tacitly identify v ∈ R n with (0 , v ) ∈ R × R n ,which yields an identification of k -vectors in R n with k -vectors in R × R n . Step 1: for every t and µ -a.e. x the pushforward of the ( k + 1) -vector e ∧ τ ( x ) according to the linear map d ( t,x ) Φ is null, that is (cid:0) d ( t,x ) Φ (cid:1) ( e ∧ τ ( x )) = 0 . (2.18) eometric structure of currents 19 By differentiating the semigroup identity Φ( t, Φ( s, x )) = Φ( t + s, x ) with respect to s at s = 0 we get d ( t,x ) Φ( e ) = d ( t,x ) Φ( v ( x )) , and then (d ( t,x ) Φ) ( e ∧ τ ( x )) = (d ( t,x ) Φ) ( v ( x ) ∧ τ ( x )) , and using assumption (b) in Proposition 2.24 we get (2.18). Step 2: proof of (2.17) . Using (2.18), for every test ( k + 1)-form ω on R × R n weobtain (cid:10) Φ (cid:0) [[ t , t ]] × T (cid:1) ; ω (cid:11) = (cid:10) [[ t , t ]] × T ; Φ ( ω ) (cid:11) = Z t t Z R n (cid:10) ω (Φ( t, x )); (cid:0) d ( t,x ) Φ (cid:1) ( e ∧ τ ( x )) (cid:11) dµ ( x ) dt = 0 , and (2.17) is proved. (cid:3) Proof of Proposition 2.24. We argue by contradiction and assume that thereexists a compact set E ′ ⊂ E such that H α − ( E ′ ) = 0 , µ ′ ( E ′ ) > . Next we choose a point x ∈ E ′ where the map v ∧ τ ′ is approximately continuousand the set E ′ has density 1 (in both cases the underlying measure is µ ′ ). We fixfor the time being δ, r > E ′′ := E ′ ∩ B ( x , r ) , F := [0 , δ ] × E ′′ , G := Φ( F ) , and the k -current S := Φ (cid:0) [[0 , δ ]] × ∂T (cid:1) . We claim that(i) H α ( G ) = 0;(ii) S , viewed as a vector-vauled measure, satisfies S ≪ H α ;(iii) S ( G ) = 0 for δ and r suitably chosen.Note that (i) and (ii) imply S ( G ) = 0, which contradicts (iii). To conclude the proofit remains to prove claims (i)–(iii). Step 1: proof of (i) . We have the following chain of implications: H α − ( E ′ ) = 0 ⇒ H α − ( E ′′ ) = 0 ⇒ H α ( F ) = 0 ⇒ H α ( G ) = 0 (the second implication followsfrom [12], Theorem 2.10.45; the third one from the fact that Φ is Lipschitz). Step 2: proof of (ii) . Using the definition of the current S and Lemma 2.28 weobtain S = (Φ δ ) T − T ; to conclude we recall that T ≪ µ ≪ H α by assumption,and observe that (Φ δ ) T ≪ (Φ δ ) µ ≪ H α because the map Φ δ is bi-Lipschitz. Step 3: proof of (iii) . To prove this claim we set ρ ( r ) := µ ′ (cid:0) B ( x , r ) (cid:1) and showthat lim r → lim δ → S ( G ) δρ ( r ) = v ( x ) ∧ τ ′ ( x ) = 0 . (2.19)Let λ be the product measure on R × R n given by λ := ( L x [0 , δ ]) × µ ′ , and let g : R × R n → ∧ k ( R n ) be the map given by g ( t, x ) := (d ( t,x ) Φ) ( e ∧ τ ′ ( x )). Usingthe definition of pushforward of currents we obtain S ( G ) = Z Φ − ( G ) g dλ = A + A + A (2.20) where A := Z F g (0 , x ) dλ ( t, x ) = δ Z E ′′ g (0 , x ) dµ ′ ( x ) ,A := Z F (cid:0) g ( t, x ) − g (0 , x ) (cid:1) dλ ( t, x ) , A := Z Φ − ( G ) \ F g ( t, x ) dλ ( t, x ) . The definitions of g and Φ yield g (0 , x ) = v ( x ) ∧ τ ′ ( x ) . Using this identity, the definitions of ρ ( r ) and E ′′ , and the choice of x we obtainthat A δρ ( r ) = 1 ρ ( r ) Z B ( x ,r ) ∩ E ′ v ∧ τ ′ dµ ′ −→ r → v ( x ) ∧ τ ′ ( x ) . (2.21)Using the dominated convergence theorem and the fact that g is continuous in t anduniformly bounded we obtain that, for every fixed r , | A | δ ≤ Z B ( x ,r ) (cid:16) sup ≤ t ≤ δ | g ( t, x ) − g (0 , x ) | (cid:17) dµ ′ ( x ) −→ δ → . (2.22)Finally we let E δ be the projection of Φ − ( G ) ∩ (cid:0) [0 , δ ] × R n (cid:1) onto R n and noticethat this is a closed set that contains E ′′ and for every fixed r converges to E ′′ inthe Hausdorff distance as δ → 0. Since | g | is bounded by some constant m (becauseso is τ ′ ) we obtain that | A | δ ≤ mδ λ (cid:0) Φ − ( G ) \ F (cid:1) ≤ m µ ′ (cid:0) E δ \ E ′′ (cid:1) −→ δ → . (2.23)Putting together (2.20), (2.21), (2.22) and (2.23) we obtain (2.19). (cid:3) 3. The key identity The main result in this section is identity (3.4) in Proposition 3.2. Using thisidentity we obtain the fundamental relation between the boundary of a normal k -current tangent to a distribution of k -planes V and the distribution b V associated to V (Theorem 3.3).Through this section, k and n are integers that satisfy 2 ≤ k < n , Ω is an openset in R n , V is a distribution of k -planes V on Ω spanned by vectorfields v , . . . , v k of class C on Ω and v := v ∧ · · · ∧ v k , as usual.Moreover T is a normal k -current on Ω which is tangent to V , which we write as T = vµ where µ is a suitable signed measure (see Remark 2.13(ii)). In the sequel itis important to remember that µ is not necessarily positive.As usual write ∂T = τ ′ µ ′ where µ ′ is a positive measure and τ ′ is a density withvalues in ( k − V is globally spanned by k vectorfields,and not just locally (cf. § Take v and V as above and consider the ( k − -form α := ⋆ ( w ∧ u ) (3.1) where u = u ∧ · · · ∧ u n − k − is a simple ( n − k − -vector and w = w ∧ w is asimple -vectorfield on Ω with w , w vectorfields of class C tangent to V . Then eometric structure of currents 21 (i) v x α = 0 on Ω ; (ii) h v ; d α i = h v ∧ div w ∧ u ; d x i on Ω ; (iii) h v ; d α i 6 = 0 at every point of Ω where v ∧ div w ∧ u = 0 . Proof. To prove (i) we show that h v x α ; β i = 0 for every 1-covector β . Indeed h v x α ; β i = h v ; α ∧ β i = ( − k − h v ; β ∧ α i = ( − k − h v x β ; α i = ( − k − h v x β ; ⋆ ( w ∧ u ) i = ( − k − h ( v x β ) ∧ w ∧ u ; d x i = 0 , where the last equality holds because ( v x β ) ∧ w is a ( k + 1)-vectorfield tangent tothe distribution of k -planes V , and therefore it is everywhere null.Let us prove (ii). Using (2.5) we get h v ; d α i = h v ; d( ⋆ ( w ∧ u )) i = h v ; ⋆ (div( w ∧ u )) i = h v ∧ (div( w ∧ u )); d x i . (3.2)Since both w and u are simple we can use formula (2.8) to compute the divergenceof w ∧ u , obtaining div( w ∧ u ) = [ w , w ] ∧ u + w ′ , (3.3)where w ′ = (div w ) w ∧ u − (div w ) w ∧ u + n − k − X i =1 ( − i (cid:0) [ w , u i ] ∧ w − [ w , u i ] ∧ w (cid:1) ∧ (cid:16) ^ j = i u j (cid:17) . Now, each w i belongs to V = span( v ) by assumption, hence v ∧ w i = 0, which impliesthat v ∧ w ′ = 0. Therefore using (3.3) and (2.9) we get v ∧ div( w ∧ u ) = v ∧ [ w , w ] ∧ u = v ∧ div w ∧ u . Plugging this formula into (3.2) proves (ii).Finally, (iii) is an immediate consequence of (ii). (cid:3) Take v , V , T = vµ and ∂T = τ ′ µ ′ as above, and let w be a -vectorfield of class C on Ω which is tangent to V . Then the following identity ofmeasures (with values in ( k + 1) -vectors) holds: ( τ ′ ∧ w ) µ ′ = ( v ∧ div w ) µ . (3.4) Proof. The proof is divided in two cases. Case 1: w = w ∧ w with w , w vectorfields of class C tangent to V . Fix asimple ( n − k − u and let α be the ( k − T x α = ( v x α ) µ = 0.Therefore formula (2.11) yields0 = ( − k − ∂ ( T x α ) = ∂T x α − T x d α = ( τ ′ x α ) µ ′ − ( v x d α ) µ = h τ ′ ∧ w ∧ u ; d x i µ ′ − h v ∧ div w ∧ u ; d x i µ (in the last equality we used Lemma 3.1(ii)). Hence( τ ′ ∧ w ∧ u ) µ ′ = ( v ∧ div w ∧ u ) µ , which implies (3.4) by the arbitrariness of u . Case 2: w is arbitrary. Then w can be written in the form w = X ≤ i Through this proof we denote by X ( V ) the space of all 2-vectorfield of class C on Ω tangent to V .Let w ∈ X ( V ) and write µ ′ a = ρµ for a suitable density ρ . Then equation (3.4)can be rewritten as τ ′ ∧ w = 0 for µ ′ s -a.e. x , (3.5) τ ′ ∧ w = ρ v ∧ div w for µ ′ a -a.e. x . (3.6)The proof is now divided in three steps. The first one contains the proof ofstatement (i), while the others give statement (ii). Step 1: proof of statement (i) . Equation (3.5) implies that for every w ∈ X ( V )there exists a µ ′ s -null set N w such that τ ′ ( x ) ∧ w ( x ) = 0 for every x / ∈ N w . Takenow a countable dense family X ′ ⊂ X ( V ), and let N be the union of N w over all w ∈ X ′ . Then N is µ ′ s -null and it is easy to check that for every x / ∈ N and every w ∈ X ( V ) there holds τ ′ ( x ) ∧ w ( x ) = 0, which means that τ ′ ( x ) ∧ w = 0 for every 2-vector w in V ( x ),and therefore span( τ ′ ( x )) ⊂ V ( x ) (apply Lemma 2.4 with assumption (b)). Step 2: span( τ ′ ( x )) ⊂ b V ( x ) for µ ′ a -a.e. x . Let w ∈ X ( V ). Using equation (3.6)and the inclusion span( v ∧ div w ) ⊂ b V (Proposition 2.17) we obtainspan (cid:0) τ ′ ( x ) ∧ w ( x ) (cid:1) ⊂ b V ( x ) for µ ′ a -a.e. x ,and proceeding as in Step 1 we find a µ ′ a -null set N such that, for every x / ∈ N ,span (cid:0) τ ′ ( x ) ∧ w (cid:1) ⊂ b V ( x ) for every 2-vector w in V ( x ),and then span( τ ′ ( x )) ⊂ b V ( x ) (apply Lemma 2.4 with assumption (a)). Step 3: b V ( x ) ⊂ V ( x ) + span( τ ′ ( x )) for µ ′ a -a.e. x . Let w ∈ X ( V ). Using equa-tion (3.6) and the inclusion span( τ ′ ∧ w ) ⊂ span( τ ′ ) + V , we obtain thatspan (cid:0) v ( x ) ∧ div w ( x ) (cid:1) ⊂ span( τ ′ ( x )) + V ( x ) for µ ′ a -a.e. x . eometric structure of currents 23 Proceeding as in Step 1 we find a µ ′ a -null set N such that, for every x / ∈ N and every w ∈ X ( V ), span (cid:0) v ( x ) ∧ div w ( x ) (cid:1) ⊂ V ( x ) + span( τ ′ ( x )) , and using Lemma 2.4 with assumption (c) we obtain that, for every x / ∈ N ,div w ( x ) ∈ V ( x ) + span( τ ′ ( x )) . Then the claim follows using Proposition 2.17. (cid:3) Recall that b V ( x ) agrees with V ( x ) for every x in the involutivityset N ( V, k ) = Ω \ N ( V ), and strictly contains V ( x ) for every x in the non-involutivityset N ( V ). Then Theorem 3.3 implies that the inclusion that defines the geometricproperty of the boundary for T , namelyspan( τ ′ ( x )) ⊂ V ( x ) , holds for µ ′ s -a.e. x ∈ Ω and for µ ′ a -a.e. x ∈ Ω \ N ( V ), and does not hold for µ ′ a -a.e. x ∈ N ( V ). The case k = 2 and n = 3 of Proposition 3.2 is especially signifi-cant. In this case a form α with properties (i)-(iii) in Lemma 3.1 is simply given by α := ⋆ v , and equation (3.4) in Proposition 3.2 reduces to( τ ′ ∧ v ) µ ′ = ( v ∧ div v ) µ . 4. Proofs of the results in Section 1 In this section we prove Theorem 1.3, Corollary 1.4, and Theorems 1.1, 1.7 and1.12 (in this order).We follow the notation of Section 3. In particular V is spanned by v , . . . , v k , v := v ∧ , · · · ∧ v k , and T = τ µ where µ is a suitable signed measure. Note that it issufficient to prove the statements above for this measure µ , despite the fact that itmay be not positive (cf. Remark 2.11(ii)). Proof of Theorem 1.3. Throughout this proof we denote by X ( V ) the space ofall 2-vectorfield of class C on Ω tangent to V . The proof is divided in several steps. Step 1: proof of statement (i) . Let w ∈ X ( V ) and let µ = µ a + µ s be the Lebesguedecomposition of µ with respect to µ ′ . Then identity (3.4) yields ( v ∧ div w ) µ s = 0,and therefore we can find a | µ s | -null set N w such that, for every x / ∈ N w , v ( x ) ∧ div w ( x ) = 0 . Proceeding as in Step 1 of the proof of Theorem 3.3 we find a | µ s | -null set N suchthat the previous equation holds for every x / ∈ N and every w ∈ X ( V ), and applyingLemma 2.4 with assumption (c) we obtaindiv w ( x ) ∈ V ( x ) . By Proposition 2.17 this means V ( x ) = b V ( x ) at every x / ∈ N , that is, V is involutiveat every x / ∈ N or, in other words, the non-involutivity set N ( V ) is | µ s | -null, whichfinally implies µ x N ( V ) ≪ µ ≪ µ ′ .For the next step we denote by ¯ µ the restriction of | µ | to N ( V ). Recall that W (¯ µ, · ) is the decomposability bundle of ¯ µ (see § Step 2: W (¯ µ, x ) ⊃ b V ( x ) for ¯ µ -a.e. x . Indeed, since T = vµ and ∂T = τ ′ µ ′ arenormal currents, Theorem 5.10 in [2] implies that the decomposability bundles ofthe measures | µ | and µ ′ contain the span of τ and τ ′ respectively, that is, W ( | µ | , x ) ⊃ span( v ( x )) = V ( x ) for | µ | -a.e. x , (4.1) W ( µ ′ , x ) ⊃ span( τ ′ ( x )) for µ ′ -a.e. x . (4.2)On the other hand ¯ µ is absolutely continuous with respect to | µ | and also withrespect to µ ′ (by statement (i)) and therefore Proposition 2.9(i) in [2] yields W (¯ µ, x ) = W ( | µ | , x ) = W ( µ ′ , x ) for ¯ µ -a.e. x . (4.3)Putting together (4.1), (4.2) and (4.3) we obtain W (¯ µ, x ) ⊃ V ( x ) + span( τ ′ ( x )) for ¯ µ -a.e. x ,and we conclude the proof of the claim recalling that V + span( τ ′ ) = b V by Propo-sition 2.17. Step 3: proof of statement (ii) . For every d = k + 1 , . . . , n let µ d := | µ | x N ( V, d ).Since µ d is absolutely continuous with respect to ¯ µ , using Proposition 2.9(i) in [2]and Step 2 we obtain that W ( µ d , x ) = W (¯ µ, x ) ⊃ b V ( x ) for µ d -a.e. x ,and in particular dim( W ( µ d , x )) ≥ d for µ d -a.e. x . We now conclude using Proposi-tion 2.21 or Lemma 2.23.For the rest of the proof we fix d = k + 1 , . . . , n and consider the following sets:Ω d := (cid:8) x ∈ Ω : dim( b V ( x )) ≥ d (cid:9) = N ( V, d ) ∪ · · · ∪ N ( V, n ) ,F := (cid:8) x ∈ Ω d : V ( x ) ⊂ span( τ ′ ( x )) (cid:9) ,E i := (cid:8) x ∈ Ω d : v i ( x ) / ∈ span( τ ′ ( x )) (cid:9) with i = 1 , . . . , k . Step 4: µ ′ x F ≪ H d ≪ H d − . Using the identity b V = V + span( τ ′ ) (Proposi-tion 2.17), we have that for x ∈ F the linear space span( τ ′ ( x )) contains b V ( x ) andtherefore has dimension at least d . Thus (4.2) implies that W ( µ ′ , x ) has dimensionat least d for µ ′ -a.e. x ∈ F , and the claim follows from Proposition 2.21. Step 5: µ ′ x E i ≪ H d − for i = 1 , . . . , k . We prove this claim by applyingProposition 2.24 to the open set Ω d , the current T = vµ = τ | µ | and the vectorfield v i .To this end we notice that • | µ | x Ω d ≪ H d by statement (ii) and the definition of Ω d ; • v i ∧ v = 0 everywhere in Ω and then also in Ω d ; • v i ∧ τ ′ = 0 on E i by the definition of E i . Step 6: µ ′ x Ω d ≪ H d − , which implies statement (iii) . To prove the first partof the claim we use that Ω d = F ∪ E ∪ · · · ∪ E k and Steps 4 and 5. To provestatement (iii) use that N ( V, d ) ⊂ Ω d . (cid:3) Proof of Corollary 1.4. Since the set N ( V ) is open, proving that the support of µ does not intersect N ( V ) is equivalent to showing that µ x N ( V ) = 0.If condition (a) holds, then µ is singular with respect to I dt for every d ≥ k + 1,and this fact and Theorem 1.3(ii) imply µ x N ( V, d ) = 0, and since N ( V ) is theunion of all N ( V, d ) with d ≥ k + 1, we obtain µ x N ( V ) = 0, as desired.If condition (b) holds then Theorem 1.3(i) yields µ x N ( V ) = 0. eometric structure of currents 25 To conclude the proof we notice that condition (c) implies condition (a), andcondition (d) implies condition (b). (cid:3) Proof of Theorem 1.1. Apply Corollary 1.4 with condition (c). (cid:3) Proof of Theorem 1.7. We begin with the implication (i) ⇒ (ii). The geomet-ric property of the boundary for T , namely inclusion (1.1), together with Theo-rem 3.3(ii) imply that V = b V µ ′ a -a.e., where µ ′ a is the absolutely continuous partof µ ′ with respect to µ . Since V ( x ) = b V ( x ) at every x ∈ N ( V ) we infer that µ ′ a x N ( V ) = 0 and using Theorem 1.3(i) we deduce that µ x N ( V ) = 0. Since N ( V )is open, this means that the support of µ does not intersect N ( V ).We now prove (ii) ⇒ (i). By assumption we have that b V ( x ) = V ( x ) for | µ | -a.e. x ;this fact and Theorem 3.3(ii) imply that span( τ ′ ( x )) ⊂ V ( x ) for µ ′ a -a.e. x . On theother hand this inclusion holds also for µ ′ s -a.e. x by Theorem 3.3(i), and therefore T has the geometric property of the boundary. (cid:3) Sketch of proof of Theorem 1.12. Assume by contradiction that there exists z ∈ A such that u ( z ) ∈ N ( V ). Since N ( V ) is open and u is continuous and ofclass W ,p loc we can find a ball U centered at z such that • u ( U ) is contained in N ( V ); • the restriction of u to ∂U belongs to W ,p ( ∂U ).Then the graph of the restriction of u to U , denoted by Γ, is a k -dimensional recti-fiable set with H k (Γ) < + ∞ . 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Massaccesi , Currents with coefficients in groups, applications and other problems in Geo-metric Measure Theory , PhD thesis, Scuola Normale Superiore, Pisa, 2014.[18] L. Simon , Lectures on geometric measure theory , vol. 3 of Proceedings of the Centre for Math-ematical Analysis, Australian National University, Canberra, 1983.[19] F. W. Warner , Foundations of differentiable manifolds and Lie groups , vol. 94 of GraduateTexts in Mathematics, Springer-Verlag, New York-Berlin, 1983. Corrected reprint of the 1971edition.[20] M. Zworski , Decomposition of normal currents , Proc. Amer. Math. Soc., 102 (1988), pp. 831–839.G.A.Dipartimento di Matematica, Universit`a di Pisalargo Pontecorvo 5, 56127 Pisa, Italye-mail: [email protected] A.M.Dipartimento di Tecnica e Gestione dei Sistemi Industriali (DTG), Universit`a di Padovastradella S. Nicola 3, 36100 Vicenza, Italye-mail: [email protected]