On some geometric properties of currents and Frobenius theorem

TL;DR

This paper explores the geometric structure of currents and their relation to Frobenius theorem, showing that certain currents cannot be tangent to non-involutive distributions and can be foliated when tangent to involutive ones.

Contribution

It establishes new connections between the theory of currents and Frobenius theorem, providing partial answers to longstanding questions in geometric measure theory.

Findings

Integral currents cannot be tangent to nowhere involutive distributions.

Normal currents tangent to involutive distributions can be locally foliated.

Partial resolution of a question by Frank Morgan.

Abstract

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].

Equations

$$v:=v_1\wedge \cdots \wedge v_k.$$

$$V(x):=\mathrm{span}(v(x)):=\mathrm{span}\big{\{}v_{1}(x),\dots,v_{k}(x)\big{\}}\quad\text{for every $x\in\Omega$.}$$

$$[v_i,v_j](x)\in V(x)\text{for every 1≤i,j≤k,}$$

$$[v_{i},v_{j}](x):=\big{\langle}\nabla v_{j}(x);v_{i}(x)\big{\rangle}-\big{\langle}\nabla v_{i}(x);v_{j}(x)\big{\rangle}=\frac{\partial v_{j}}{\partial v_{i}}(x)-\frac{\partial v_{i}}{\partial v_{j}}(x)\,,$$

$$\mathrm{Tan}(\Sigma ,x)=\mathrm{Tan}(S,x)\text{for Hk-a.e. x∈Σ∩S.}$$

$$\langle T;\omega\rangle:=\int_{\Omega}\big{\langle}\tau(x);\omega(x)\big{\rangle}\,d\mu(x)\,,$$

$$\langle T;\omega\rangle:=\int_{\Sigma}\big{\langle}\tau(x);\omega(x)\big{\rangle}\,\theta(x)\,d\mathscr{H}^{k}(x)\,.$$

$$0=⟨\partial T;\phi \alpha ⟩=⟨T;\mathrm{d}(\phi \alpha )⟩$$

$$=⟨T;\phi \mathrm{d}\alpha ⟩={\int }_{\Omega }⟨\tau ;\mathrm{d}\alpha ⟩\phi d\mu$$

$$\Sigma:=\big{\{}x\in S\,:\,\mathrm{Tan}(S,x)=V(x)\big{\}}$$

Full text

[version: October 20, 2017] Rend. Lincei Mat. Appl. 28 (2017), no. 4, 861-869

DOI 10.4171/RLM/788

**On some geometric properties of currents and Frobenius theorem

**

Giovanni Alberti, Annalisa Massaccesi

Abstract. 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].

Keywords: non-involutive distributions, Frobenius theorem, Sobolev surfaces, integral currents, normal currents, foliations, decomposition of normal currents.

MSC (2010): 58A30, 49Q15, 58A25, 53C17, 46E35.

1. Introduction

Consider a distribution of $k$-dimensional planes in $\mathbb{R}^{n}$, namely a map that associates to each point $x\in\mathbb{R}^{n}$ a $k$-dimensional subspace $V(x)$ of $\mathbb{R}^{n}$, and assume that $V$ is spanned by vectorfields $v_{1},\dots,v_{k}$ of class $C^{1}$. We say that $V$ is involutive at a point $x\in\mathbb{R}^{n}$ if the commutators of the vectorfields $v_{1},\dots,v_{k}$, evaluated at $x$, belong to $V(x)$ (see §2.1). Moreover, given a $k$-dimensional surface $S$ in $\mathbb{R}^{n}$, we say that $S$ is tangent to $V$ if the tangent space $\mathrm{Tan}(S,x)$ agrees with $V(x)$ for every $x\in S$.

In this context, the first part of Frobenius theorem states that, if $S$ is tangent to $V$, then $V$ must be involutive at every point of $S$. Or, in a slightly weaker form, that if $V$ is nowhere involutive then there exist no tangent surfaces (cf. [8], Theorem 14.5).

The classical version of this theorem requires that the surface $S$ is at least of class $C^{1}$, and it is then natural to ask if similar statements hold for weaker notions of surface. To this regard, we mention that a positive answer for Sobolev surfaces, that is, Sobolev images of open subsets of $\mathbb{R}^{2h}$, has been given in [9], Theorem 1.2, when $V$ is the distribution of $2h$-planes in $\mathbb{R}^{n=2h+1}$ corresponding to the horizontal distribution in the sub-Riemannian Heisenberg group $\mathbb{H}^{h}$.

In section 3 we give a positive answer for integral currents,111 The basic definitions and terminology concerning currents are recalled in Section 2. and more precisely we show that given an integral $k$-dimensional current $T$ which is tangent to $V$, then $V$ must be involutive on the support of $T$ (Theorem 3.6). Note that the assumption that $T$ is integral is crucial, and indeed the analogous statement for rectifiable sets does not hold, cf. Remark 3.7(a).

It turns out that Theorem 3.6 is an immediate consequence of the following geometric property of the boundary of integral currents, which is actually the heart of the matter: if $T$ is an integral $k$-dimensional current tangent to a continuous distribution of $k$-planes $V$, then $\partial T$ is tangent to $V$ as well (see §2.5 for the definition of tangency, and Theorems 3.1 and 3.2).

In Section 4 we turn to the other part of Frobenius theorem, which states that if $V$ is everywhere involutive, then $\mathbb{R}^{n}$ can be locally foliated with $k$-dimensional surfaces which are tangent to $V$. In Theorem 4.3 we prove the following generalization: if $V$ is everywhere involutive and $T$ is a $k$-dimensional normal current tangent to $V$, then $T$ can be locally foliated by a family $k$-dimensional integral currents tangent to $V$ (the definition of foliation, or mass decomposition, of a current is given in §4.1). Conversely, if $T$ can be foliated then $V$ must be involutive at every point in the support of $T$.

The first part of Theorem 4.3 gives a partial positive answer to a question raised by Frank Morgan in [1], namely if every normal current admits a foliation in terms of integral currents (other positive results were given in [6], [10], [12], see Remark 4.4). On the other hand, the second part shows that a normal current which is tangent to a nowhere involutive distribution of planes admits no foliation of a certain type: this result was first stated in [13], but in a form which is not correct (see Remark 4.4(c) for more details).

2. Notation

In this section we briefly recall some notation and basic definitions. For rectifiable sets and currents we essentially follow [7]. As usual, $\mathscr{H}^{k}$ stands for the $k$-dimensional Hausdorff measure and $\mathscr{L}^{n}$ for the Lebesgue measure on $\mathbb{R}^{n}$.

In the following we fix an open set $\Omega$ in $\mathbb{R}^{n}$.

2.1. The vectorfield $\boldsymbol{v}$

and the distribution of planes $\boldsymbol{V}$.

In the following we consider $v_{1},\dots,v_{k}$ continuous vectorfields on $\Omega$ with $0<k<n$, and the simple $k$-vectorfield

[TABLE]

Moreover we assume that $v$ is unitary, that is, $|v(x)|=1$ for every $x\in\Omega$, and we denote by $V$ the distribution of $k$-planes spanned by $v$, that is,

[TABLE]

With a slight abuse of language, we say that $V$ (or $v$) is of class $C^{1}$ to mean that $v_{1},\dots,v_{k}$ are of class $C^{1}$, and if this is the case we say that $V$ is involutive at a point $x\in\Omega$ if

[TABLE]

where $[v_{i},v_{j}]$ is the Lie bracket, or commutator, of $v_{i}$ and $v_{j}$.222 That is, the vectorfield defined by

[v_{i},v_{j}](x):=\big{\langle}\nabla v_{j}(x);v_{i}(x)\big{\rangle}-\big{\langle}\nabla v_{i}(x);v_{j}(x)\big{\rangle}=\frac{\partial v_{j}}{\partial v_{i}}(x)-\frac{\partial v_{i}}{\partial v_{j}}(x)\,,

where $\langle{~{};~{}}\rangle$ denotes the usual pairing of matrices and vectors.

2.2. Rectifiable sets, orientation.

A set $\Sigma$ in $\Omega$ is rectifiable of dimension $k$, or $k$-rectifiable, if it has finite $\mathscr{H}^{k}$ measure and can be covered, except for an $\mathscr{H}^{k}$-null subset, by countably many surfaces of dimension $k$ and class $C^{1}$. 333 Through this paper sets, maps and vectorfields are always (at least) Borel measurable.

Then at $\mathscr{H}^{k}$-a.e. $x\in\Sigma$ there exists an approximate tangent space $\mathrm{Tan}(\Sigma,x)$, which is characterized (for $\mathscr{H}^{k}$-a.e. $x\in\Sigma$) by the following property: for every $k$-surface $S$ of class $C^{1}$ there holds

[TABLE]

An orientation of $\Sigma$ is a simple $k$-vectorfield $\tau$ defined on $\Sigma$ such that $\tau(x)$ spans $\mathrm{Tan}(\Sigma,x)$ and has norm $1$ for $\mathscr{H}^{k}$-a.e. $x\in\Sigma$.

2.3. Currents, boundary, mass, normal currents.

A $k$-dimensional current, or $k$-current, in $\Omega$ is a (continuous) linear functional on the space of smooth $k$-forms with compact support on $\Omega$. The boundary of a $k$-current $T$ is the $(k-1)$-current $\partial T$ defined by $\langle\partial T;\omega\rangle:=\langle T;\mathrm{d}\omega\rangle$, where $\mathrm{d}\omega$ is the exterior differential of the form $\omega$.

The mass of $T$, denoted by $\mathbb{M}(T)$, is the supremum of $\langle T;\omega\rangle$ over all forms $\omega$ such that $|\omega(x)|\leq 1$ for every $x$. A current $T$ with finite mass can be represented as a vector measure, that is, there exist a positive finite measure $\mu$ on $\Omega$ and a map $\tau$ from $\Omega$ to the set of $k$-vectors with norm $1$, called orientation, such that

[TABLE]

where $\langle{~{};~{}}\rangle$ is the usual pairing of $k$-vectors and $k$-covectors. In this case we simply write $T=\tau\mu$. Note that the mass of $T$ is $\mathbb{M}(T)=\mu(\Omega)=\|\mu\|$.

A current $T$ is called normal if both $T$ and $\partial T$ have finite mass.

2.4. Rectifiable and integral currents.

A $k$-current $T$ is called rectifiable (with integral multiplicity) if there exist a $k$-rectifiable set $\Sigma$, an orientation $\tau$ of $\Sigma$, and a positive, integer-valued multiplicity $\theta\in L^{1}(\Sigma,\mathscr{H}^{k})$ such that

[TABLE]

In this case we write $T=\llbracket\Sigma,\tau,\theta\rrbracket$. Note that the mass of $T$ agrees with the $k$-dimensional measure of $\Sigma$ counted with multiplicity, that is, $\mathbb{M}(T)=\int_{\Sigma}\theta\,d\mathscr{H}^{k}$; in particular $\mathbb{M}(T)$ is finite.

A current $T$ is called integral if both $T$ and $\partial T$ are rectifiable; in particular every integral current is normal.

2.5. Notions of tangency.

Take $v$ and $V$ as in §2.1 We say that an $h$-rectifiable set $\Sigma$ with $h\leq k$ is tangent to $V$ if the tangent space $\mathrm{Tan}(\Sigma,x)$ is contained in $V(x)$ for $\mathscr{H}^{h}$-a.e. $x\in\Sigma$.

Accordingly, a rectifiable $h$-current $T=\llbracket\Sigma,\tau,\theta\rrbracket$ is tangent to $V$ if the supporting rectifiable set $\Sigma$ is so. More generally, an $h$-current with finite mass $T=\tau\mu$ is tangent to $V$ if the span of the $h$-vector $\tau(x)$ is contained in $V(x)$ for $\mu$-a.e. $x$.444 The span of a $h$-vector $w$ in $\mathbb{R}^{n}$ is defined as the smallest subspace $W$ of $\mathbb{R}^{n}$ such that $w$ is also a $h$-vector in $W$. If $w$ is a simple vector we recover the usual definition.

Moreover we say that a rectifiable $k$-current $T=\llbracket\Sigma,\tau,\theta\rrbracket$ is oriented by $v$ if $\tau(x)=v(x)$ for $\mathscr{H}^{k}$-a.e. $x\in\Sigma$, and more generally, a $k$-current with finite mass $T=\tau\mu$ is oriented by $v$ if $\tau(x)=v(x)$ for $\mu$-a.e. $x$.

Remark 2.6.

If $T=\tau\mu$ is a $k$-current with finite mass, then $T$ is tangent to $V$ if and only if $\tau(x)=\pm v(x)$ for $\mu$-a.e. $x$ (recall that $v$ is unitary). In particular if $T$ is oriented by $v$ then it is also tangent to $V$, but clearly the converse does not hold.

3. Geometric structure of the boundary

Through this section, $v$ and $V$ are taken as in §2.1.

The next two statements are the main results in this section, and establish a natural (and apparently obvious) relation between the tangent space of a current $T$ and the tangent space of the boundary $\partial T$, namely that, under suitable assumptions, the former contains the latter.

**Theorem 3.1. **(See [2].)

If $T$ is an integral $k$-current oriented by $v$, then the boundary $\partial T$ is tangent to $V$.

**Theorem 3.2. **(See [3].)

If $V$ is of class $C^{1}$ and $T$ is an integral $k$-current tangent to $V$, then $\partial T$ is tangent to $V$.

Remark 3.3.

(a) Theorem 3.2 can be viewed as the “non-oriented version” of Theorem 3.1, and under the assumption that $V$ is of class $C^{1}$ it is actually a stronger statement (cf. Remark 2.6).

(b) Theorem 3.1 can be proved in a slightly stronger form, and under slightly weaker assumptions on the current $T$ (see [2] for more details); the key step of the proof consists in taking the blow-up of $T$ at “almost every point of the boundary”, and here the assumption that $T$ is integral (or slightly less) plays an essential role.

(c) Theorem 3.2 can be proved under much weaker assumptions on the current $T$, including the case where $T$ is normal and $\partial T$ is singular with respect to $T$.555 Here both $T$ and $\partial T$ are viewed as (vector-valued) measures. The proof is completely different from that of Theorem 3.1, and relies heavily on the fact that $V$ is of class $C^{1}$ (see [3]). Note that this regularity assumption on $V$ can perhaps be weakened, but cannot be entirely dropped: indeed in [2] we construct a continuous distribution $V$ of $2$-planes in $\mathbb{R}^{3}$ and an integral $2$-current $T$ such that $T$ is tangent to $V$ but $\partial T$ is not.666 This current is actually (supported on) the graph of a continuous Sobolev function.

(d) In [3] we also show that if $V$ is everywhere involutive then Theorem 3.2 holds for every normal $k$-current $T$. This is no longer true if $V$ is not everywhere involutive, the counterexample being any current on $\Omega$ of the form $T:=v\mu$ where $\mu:=\rho\mathscr{L}^{n}$ and $\rho$ is a function of class $C^{1}$ whose support is compact and contained in the (open) set of all points where $v$ is not involutive.

The relation between the geometric property of the boundary of $T$ proved in Theorem 3.2 and Frobenius theorem is made clear in the following statement.

Proposition 3.4.

Assume that $V$ is of class $C^{1}$, and let $T$ be a normal $k$-current tangent to $V$ such that $\partial T$ is also tangent to $V$. Then $V$ is involutive at every point of the support of $T$.777 By support of a current with finite mass $T=\tau\mu$ we mean the support of the measure $\mu$, that is, the smallest closed set $F$ such that $\mu(\mathbb{R}^{n}\setminus F)=0$. If $T$ is rectifiable, that is $T=\llbracket\Sigma,\tau,\theta\rrbracket$, then the support of $T$ turns out to be the closure of the set of all points $x\in\Sigma$ where the $k$-dimensional density of $\Sigma$ is $1$.

This result is an immediate consequence of the following lemma:

**Lemma 3.5. **(See [2].)

If $V$ is of class $C^{1}$ and is not involutive at a point $x_{0}\in\Omega$, then there exists a $(k-1)$-form $\alpha$ of class $C^{1}$ on $\Omega$ such that

• (i)

for every $x\in\mathbb{R}^{n}$ the restriction of $\alpha(x)$ to $V(x)$ is zero; 888 That is, $\langle w;\alpha(x)\rangle=0$ for every ($k-1$)-vector $w$ whose span is contained in $V(x)$.

• (ii)

$\langle v(x_{0});\mathrm{d}\alpha(x_{0})\rangle\neq 0$.

• Proof of Proposition 3.4.

We write $T=\tau\mu$, and we assume by contradiction that there exists a point $x_{0}$ in the support of $\mu$ where $v$ is not involutive.

We take $\alpha$ as in Lemma 3.5. Then for every smooth function $\varphi$ with compact support on $\Omega$ there holds

[TABLE]

(the first equality follows from the fact that $\partial T$ is tangent to $V$ and property (i) in Lemma 3.5; the third one from the identity $\mathrm{d}(\varphi\alpha)=\mathrm{d}\varphi\wedge\alpha+\varphi\,\mathrm{d}\alpha$; the fourth one by the fact that $T$ is tangent to $V$ and the restriction of the $k$-form $\mathrm{d}\varphi\wedge\alpha$ to $V$ is null, again by property (i) in Lemma 3.5).

Since $\varphi$ is arbitrary we infer that $\langle\tau;\mathrm{d}\alpha\rangle=0$ $\mu$-a.e., and since $\tau=\pm v$ (because $T$ is tangent to $V$, cf. Remark 2.6) we obtain that $\langle v;\mathrm{d}\alpha\rangle=0$ $\mu$-a.e.

On the other hand, property (ii) in Lemma 3.5 implies that $\langle v;\mathrm{d}\alpha\rangle\neq 0$ in a neighbourhood of $x_{0}$. Since $x_{0}$ is in the support of $\mu$, this neighbourhood has positive $\mu$ measure, and we have a contradiction. ∎

Using Theorem 3.2 and Proposition 3.4 we immediately obtain the following:

Theorem 3.6.

Assume that $V$ is of class $C^{1}$ and that $T$ is an integral $k$-current tangent to $V$. Then $V$ is involutive at every point in the support of $T$.

Remark 3.7.

(a) The analogue of Theorem 3.6 for rectifiable sets does not hold. Indeed in [2] we show that for every distribution $V$, even a nowhere involutive one, it is possible to find a $k$-dimensional surface $S$ of class $C^{1}$ whose tangency set

[TABLE]

has positive $\mathscr{H}^{k}$-measure; in particular $\Sigma$ is a non-trivial $k$-rectifiable set tangent to $V$. (This result was first proved in a slightly less general form in [4], Theorem 1.4.)

(b) Using Theorem 3.6 we can partly recover (and even extend) the Frobenius theorem for Sobolev surfaces proved in [9], Theorem 1.2. To be precise, by Sobolev surface we mean a $k$-rectifiable set $\Sigma$ of the form $\Sigma=f(U)$ where $U$ is an open set in $\mathbb{R}^{k}$ and $f:U\to\Omega$ is a continuous map of class $W^{1,p}$ with $p>k$, and we can show the following (see [2]): if $V$ is of class $C^{1}$ and $\Sigma$ is a Sobolev surface tangent to $V$, then $V$ is involutive at $\mathscr{H}^{k}$-a.e. point of $\Sigma$.

4. Foliations of normal currents

We begin this section by giving the definition or foliation of a current, and then we show that for a normal current which is tangent to a distribution of planes $V$ of class $C^{1}$ the existence of a foliation is strictly related to the involutivity of $V$ (Theorem 4.3).

4.1. Foliations of currents.

Let $T=\tau\mu$ be a $k$-current with finite mass in $\Omega$, and let $\{R_{t}\}$ be a family of rectifiable $k$-currents in $\Omega$, where $t$ varies in some index space $I$ endowed with a measure $dt$.999 We also assume that the function $t\mapsto\mathbb{M}(R_{t})$ and $t\mapsto\langle R_{t};\omega\rangle$ are Borel measurable for every $k$-form $\omega$ on $\Omega$ of class $C^{\infty}_{c}$ (or, equivalently, of class $C_{0}$). We say that $\{R_{t}\}$ is a mass decomposition, or foliation, of $T$ if

• (i)

$\langle T;\omega\rangle=\int_{I}\langle R_{t};\omega\rangle\,dt$ for every $k$-form $\omega$ on $\Omega$ of class $C^{\infty}_{c}$;

• (ii)

$\mathbb{M}(T)=\int_{I}\mathbb{M}(R_{t})\,dt$.

If $T$ is a normal current, we may also consider the following additional conditions:

• (iii)

$\int_{I}\mathbb{M}(\partial R_{t})\,dt<+\infty$;

• (iv)

$\mathbb{M}(\partial T)=\int_{I}\mathbb{M}(\partial R_{t})\,dt$.

Remark 4.2.

(a) Condition (i) is often written in compact form: $T=\int_{I}R_{t}\,dt$.

(b) If $T$ is oriented by a continuous $k$-vectorfield $v$, then condition (ii) implies that $R_{t}$ is oriented by $v$ for a.e. $t\in I$.101010 Conversely, if $\int_{I}\mathbb{M}(R_{t})\,dt$ is finite, (i) holds, and $R_{t}$ is oriented by $v$ for a.e. $t$, then (ii) holds. This explains the term “foliation”.

(c) Conditions (i) and (iii) imply that $\partial T=\int_{I}\partial R_{t}\,dt$. Condition (iv) is stronger than (iii), and implies that the family $\{\partial R_{t}\}$ is a foliation of $\partial T$.

(d) A current of finite mass $T=\tau\mu$ may admit no foliation. For example this happens if $\mu$ is a Dirac mass, or more generally a measure supported on a set $E$ which is purely $k$-unrectifiable.111111 That is, $\mathscr{H}^{k}(E\cap\Sigma)=0$ for every $k$-rectifiable set $\Sigma$. Or if $\mu$ is the restriction of $\mathscr{H}^{k}$ to a $k$-surface $S$ but $\tau$ does not span the tangent bundle of $S$.121212 The point is that for currents with finite mass the measure $\mu$ can be chosen independently of the orientation $\tau$. This is not the case with normal currents, and indeed none of these examples is a normal current.

While the question of the existence of foliations for currents with finite mass is not particularly interesting, the same question for normal currents is quite relevant, and was first formulated by Frank Morgan (see [1], Problem 3.8). The next result answers this question for normal currents which are tangent to a distribution of planes of class $C^{1}$. If no regularity assumption is made on the tangent bundle of the currents there are a few partial results (see Remark 4.4) and the question is not completely settled.

**Theorem 4.3. **(See [2].)

Let $V$ be a distribution of $k$-planes of class $C^{1}$ on $\Omega$.

(i)* If $V$ is everywhere involutive, then every point of $\Omega$ admits a neighbourhood $U$ such that every normal $k$-current in $U$ tangent to $V$ admits a foliation $\{R_{t}\}$ satisfying conditions (i), (ii), (iv) in §4.1.*

(ii)* Conversely, if $T$ is a normal $k$-current in $\Omega$ which is tangent to $V$ and admits a foliation $\{R_{t}\}$ satisfying conditions (i), (ii) in §4.1 and such that the currents $R_{t}$ are integral, then $V$ is involutive at every point in the support of $T$.*

Remark 4.4.

(a) Statement (ii) is an immediate consequence of Theorem 3.6.

(b) Statement (ii) shows that the answer to Morgan’s question is negative whenever $1<k<n$, at least if we require that the currents in the foliation are integral (and not just rectifiable); the example is given by any normal current $T$ tangent to distribution of $k$-planes which is nowhere involutive, for example the normal $k$-current $T$ given in Remark 3.3(d).

(c) A negative answer to Morgan’s question was first given by M. Zworski, who proposed the following variant of statement (ii) above (see [13], Theorem 2): if $v$ is a nowhere involutive $k$-vectorfield and $T$ is a current of the form $T=v\mathscr{L}^{n}$, then $T$ admits no foliation. However this statement is not correct, because it does not require that the currents in the foliation are integral, and for $k=n-1$ it contradicts the fact that every normal current admits a foliation, see remark (e) below.

(d) Every normal $1$-current in $\Omega$ admits a foliation satisfying conditions (i), (ii), (iii) in §4.1; this is essentially a consequence of the decomposition result by S. Smirnov [12] (see also [10]), even though it is not explicitly stated there. This result does not hold if we require that condition (iv) holds.

(e) Consider a normal $(n-1)$-current $T$ in $\Omega$. A consequence of the coarea formula for $BV$ functions is that if $T$ is a boundary then it admits a foliation satisfying conditions (i), (ii), (iv) in §4.1 (see [5], Theorem 4.5.9(13)). Such a foliation exists also if the boundary of $T$ is rectifiable, as proved in [13], Theorem 1, using an idea from [6]. By modifying the argument in [6] we prove in [2] that that every normal $(n-1)$-current admits a foliation.131313 In this case the currents in the foliation are no better than rectifiable.

(f) The existence of foliations for normal currents of dimension $d=1$ or $d=n-1$ mentioned in items (d) and (e) above has no counterpart for $2\leq d\leq n-2$. Indeed, for any $n\geq 4$, Andrea Schioppa constructed in [11] a normal current $T$ of codimension $2$ in $\mathbb{R}^{n}$ whose support is purely $2$-unrectifiable. Clearly such $T$ admits no foliation, and more precisely it cannot even be decomposed as $T=\int_{I}R_{t}\,dt$ with the only assumption that $\int_{I}\mathbb{M}(R_{t})\,dt$ is finite.

Acknowledgements

Part of this research was carried out while the second author was visiting the Mathematics Department in Pisa, supported by the University of Pisa through the 2015 PRA Grant “Variational methods for geometric problems”. The research of the first author has been partially supported by the Italian Ministry of Education, University and Research (MIUR) through the 2011 PRIN Grant “Calculus of variations”, and by the European Research Council (ERC) through the 2011 Advanced Grant “Local structure of sets, measures and currents”. The research of the second author has been partially supported by the European Research Council through the 2012 Starting Grant “Regularity theory for area minimizing currents”.