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Differentiability of the stable norm in codimension one

The real homology of a compact, n-dimensional Riemannian manifold M is naturally endowed with the stable norm. The stable norm of a homology class is the minimal Riemannian volume of its representatives. If M is orientable the stable norm on H_{n-1}(M,R) is a homogenized version of the Riemannian (n-1)-volume. We study the differentiability properties of the stable norm at points alpha in H_{n-1}(M,R). They depend on the position of alpha with respect to the integer lattice H_{n-1}(M,Z) in H_{n-1}(M,R). In particular, we show that the stable norm is differentiable at alpha if alpha is totally irrational.


Introduction
On every compact Riemannian manifold M the real homology vector spaces H m (M, R) are endowed with a natural norm, called stable or mass norm. The stable norm S(α) of α ∈ H m (M, R) is defined as the infimum of the Riemannian m-volumes of real singular cycles representing α. Equivalently, S(α) can be defined as the minimum of the masses of closed m-currents representing α. The term "stable norm" was coined by M. Gromov,cf. [Gr,Chapter 4]. The concept itself was introduced prior to this by H. Federer [Fe2,4.15]. If m = 1, or if M is n-dimensional and orientable and m = n − 1, then the stable norm is a homogenized version of the Riemannian length or (n−1)-volume functional, respectively. Here, homogenization is performed with respect to Z b acting as group of deck transformations on the Abelian covering of M, where b = b 1 (M) denotes the first Betti number.
We study differentiability properties of the stable norm S in the codimension one case, i.e. in the case m = n−1. At a point α ∈ H n−1 (M, R) the existence of two-sided directional derivatives of S at α depends on the position of α with respect to the integer lattice H n−1 (M, Z) in H n−1 (M, R).
1.1 Theorem. Let M be a compact, orientable, n-dimensional Riemannian manifold and S : H n−1 (M, R) → R ≥0 the associated stable norm on H n−1 (M, R). If α ∈ H n−1 (M, R), let V (α) denote the smallest linear subspace of H n−1 (M, R) that is spanned by integer classes and contains α. Then the restriction of S to V (α) is differentiable at α.
The extremal cases are that α is rationally independent, in which case V (α) = H n−1 (M, R) and S is differentiable at α, and the case that the direction of α = 0 is rational, in which case dim V (α) = 1 and the claim of Theorem 1.1 is obvious.
Due to the convexity and homogeneity of S the claim of Theorem 1.1 can be stated in the following alternative form.
The tangent cone of the unit ball B := β ∈ H n−1 (M, R) S(β) ≤ 1 at α ∈ ∂B splits as a product, with one factor a hyperplane in V (α).
There is strong evidence that this result is optimal, in the sense that for a large set of Riemannian metrics on an n-torus T n the stable norm on H n−1 (T n , R) ≃ R n is two-sided differentiable precisely in the directions covered by Theorem 1.1, cf. [Ba1], [Se2], [CdlL,10.4]. On the other hand, for flat metrics on T n the stable norm on H n−1 (T n , R) is induced by a scalar product. For this and other explicit examples, see [Fe2,4.15].
In the case of the 2-torus T 2 the theorem is proved in [Ba1], see also [Au] and [Ma]. For closed, orientable Riemannian surfaces F of genus s > 1 the boundary structure of the stable norm ball B ⊆ H 1 (F, R) ≃ R 2s is studied in [Ms1] and [Ms2]. In particular, in this case Theorem 1.1 follows from [Ms2,Corollary 3]. W. Senn [Se1], [Se2], [Se3] proved results analogous to Theorem 1.1 for Z n -periodic nonparametric variational problems. The case of the stable norm on H 1 (T n , R) for n > 2 is considerably more subtle, see [BIK].
Although the basic idea of our proof for Theorem 1.1 is simple, we meet some complications that are caused by the lack of regularity of the objects involved. Here, we give a rough sketch of the proof that does not attend to such subtleties. The proof is based on the duality between real homology and cohomology realized by flat cycles and cocycles from Geometric Measure Theory. Dual to the stable norm on H m (M, R) we have the comass norm on the de Rham cohomology vector space H m dR (M). Here the comass of a cohomology class l ∈ H m dR (M) is the infimum of the maximum norms of smooth closed m-forms representing l. An important point in the proof is the existence of a bounded, measurable, weakly closed m-form λ that represents l, realizes this infimum and that is defined by a process of differentiation from a flat cocycle representing l. This is due to J. H. Wolfe, cf. [Wh].
Differentiability of S|V (α) at α ∈ H n−1 (M, R) means that l 1 (β) = l 2 (β) whenever β ∈ V (α) and l 1 , l 2 ∈ H n−1 dR (M) are subderivatives of S at α. Hence, in order to prove Theorem 1.1, we will show that We let λ 1 , λ 2 denote the (n−1)-forms representing l 1 , l 2 mentioned above. We choose a smooth closed 1-form η that represents the Poincaré dual of β, so that Finally, there exists a closed (n−1)-current T representing α and of minimal mass M(T ) = S(α). Then the assumption that l 1 and l 2 are subderivatives of S at α can be used to prove that λ 1 and λ 2 coincide at Lebesgue almost all points of spt T . So, in order to prove (1) it suffices to show that According to [AB1] the current T can be represented as a measured lamination by minimizing hypersurfaces (possibly with singularities). The connected components of M spt T are called the gaps of T and it remains to prove that for every gap G of T . Using the fact that β ∈ V (α), one can see that η is exact on G, η = dg for some function g ∈ C ∞ (G, R). Hence, in the weak sense, we have on G. Now, one would like to integrate (3) over G using Stokes's theorem and to conclude that the boundary terms vanish since λ 1 = λ 2 on ∂G. This would prove (2). Actually, one has to be more careful at this point since on the one hand G is not a compact domain with smooth boundary, and on the other hand λ 1 and λ 2 need not be defined on ∂G.
In the course of the proof we obtain the following result which may be of independent interest. For the notions in this statement see the beginnings of Sections 3.1 and 4.
4.2 Theorem. Suppose that the flat cocycle L is a calibration and that the closed rectifiable current T ∈ R n−1 (M) is calibrated by L. Then the singular set of T is contained in the singular set of L.
In particular, the union of the singular sets of all closed T ∈ R n−1 (M) calibrated by L is a Lebesgue null set.
As the preceding statement shows we have to use notions and results from Geometric Measure Theory. In order to make the article reasonably comprehensive we give definitions for most of the concepts that are used in an essential way. These are of a functional analytic nature and easily comprehensible. In Section 2 we treat the existence of mass-minimizing currents in real homology classes. Section 3 develops the dual theory of flat cocycles, calibrations and their representation by weakly closed L ∞ -forms. In Section 4 we specialize to the codimension one case and prove Theorem 4.2 above. In Section 5 we summarize results on the structure of mass-minimizing currents in codimension one real homology classes. These are formulated for the lift of the current to the smallest (infinite) covering space on which this lift bounds. Finally, in Section 6, the proof of Theorem 1.1 is completed.
2 Currents and the Stable Norm

The stable norm
Throughout the paper, M will denote an oriented, n-dimensional Riemannian manifold and m an integer, 0 ≤ m ≤ n. In this subsection we assume in addition that M is compact.
The stable (or mass) norm S(α) of a real homology class α ∈ H m (M, R) is defined as the infimum of the Riemannian volumes of all real Lipschitz cycles c = r i σ i representing α, see [Gr,4.C]. Here the volume vol m (c) of c is |r i | vol m (σ i ), where vol m (σ i ) denotes the m-dimensional Riemannian volume of the Lipschitz simplex σ i : ∆ m → M. To see that S(α) > 0 if α = 0, note that there exists a de Rham cohomology class β ∈ H m dR (M) such that whenever c represents α and the closed m-form ω represents β.
In general, there will not exist a real Lipschitz cycle c in a given homology class α ∈ H m (M, R) such that the volume of c equals the stable norm S(α) of α. Geometric Measure Theory provides an appropriate notion of weak solution to this problem -the normal currents.

Normal and locally normal currents
In the following we do not assume that M is compact. We consider the chain complex where ω x denotes the (pointwise) comass norm of ω x ∈ Λ m T M x , ω x = max ω x (e 1 , . . . , e m ) e i ∈ T M x and |e i | ≤ 1 for 1 ≤ i ≤ m .
In this case T is representable by integration, i.e., there exist a positive Radon measure µ T and a µ T -measurable unit m-vector field T such that 2.1 Definition. A locally normal current T on M is an element of Ω m 0 M * such that both T and ∂T have locally finite mass. We let N loc m (M) denote the set of locally normal currents on M. A normal current is a locally normal current T whose support spt T (= spt µ T ) is compact. We let N m (M) denote the set of normal currents.
Note that the mass M is a norm on the R-vector space N m (M). Here are the most important classes of examples of (locally) normal currents. Contrary to what is stated in [Gr,4.17], inequality can occur. If parts of the singular simplices cover each other with different orientations, these parts add to vol m (c), while they cancel in [[c]]. If m < n this situation does not occur for generic chains.
Then one has A similar equality is also true for 1 < m < n − 1 if one uses an appropriate norm on (n−m)-covectors (namely the norm dual to the comass norm on m-vectors with respect to the duality induced by the wedge product and the volume form). Obviously, one has T η ∈ N m (M) if and only if η ∈ Ω n−m 0 M.

Flat norm
With respect to the mass norm neither the subspace of smooth currents with compact support nor the subspace of Lipschitz chains is dense in N m (M). However, this is true with respect to the flat norm F, which is weaker than the mass norm M.
One can show that

Homologically mass-minimizing currents
whenever U ⊆ M is open and relatively compact and S ∈ N m+1 (M) has support in U.
For the rest of this subsection we additionally assume that M is compact. We approach the question of existence of a mass-minimizing current in every real homology class. Using de Rham's Theorem, the Hahn-Banach Theorem and the fact that real Lipschitz m-chains are F-dense in N m (M), cf. Section 2.3, one can conclude that the homology H m (M, R) of the chain com- Remark. In the framework of Geometric Measure Theory it is natural to take the right hand side of the preceding equation as definition of the stable norm of α. We chose the definition of S(α) as infimum of the volumes of Lipschitz cycles representing α, since it is geometrically intuitive.
Proof. By [Fe2,3.9] the minimum on the right hand side is attained. Denote, for the moment, the quantity on the right hand side by S ′ (α). By Example 1 above, clearly For the converse, note that since S and S ′ are both norms on H m (M, R), it suffices to show that they coincide on the dense subset H m (M, Q). Let α ∈ H m (M, Q), T ∈ N m (M) with ∂T = 0 and [T ] = α, and let ε > 0. We suppose that M is isometrically embedded into some R N . Choose a tubular neighborhood U of M in R N so small that the nearest point projection p : U → M satisfies Lip(p) m ≤ min 1 + ε/M(T ), 2 . By [Fe2,5.8] there is a closed rectifiable current R ∈ R m (M) (in fact an integral Lipschitz chain) and k ∈ N such that 1 k R is homologous to T and M 1 k R ≤ M(T ) + ε. (For the notion of rectifiable current see the beginning of Section 4.) By [Fe1,Lemma 4.2.19] there exists a closed integral polyhedral chain P ∈ P m (U) (i.e. a linear combination of affine simplices with integer coefficients) and where the ∆ i are affine simplices belonging to a simplicial complex.
Therefore and since f is a diffeomorphism, when calculating the mass there is no cancelation. Puttingc := Projectingc to M, we get the desired Lipschitz cycle c representing α and satisfying Note that a closed current T ∈ N m (M) with [T ] = α is homologically massminimizing if and only if M(T ) = S(α). Such a current is called a (mass-) minimizing current in α. For m = 1 and m = n − 1 the structure of homologically mass-minimizing closed currents is well understood, see [Ba2] for the case m = 1 and [AB1], [AB2] and Section 5 for the case m = n − 1.
The norm on H m dR (M) induced by the comass norm on Ω m M is equally called comass norm, and will be denoted by see [Fe2,4.10] or [Gr,4.35]. Since we need the arguments from the proof of this duality we will reprove it in Section 3.4, Theorem 3.8.

Subderivatives of the Stable Norm and Calibrations 3.1 Calibrations
A flat m-cochain is a linear functional on N m (M) that is continuous with respect to the flat norm F. It is called a flat cocycle if it vanishes on the space of boundaries Note that for a flat cocycle L its flat norm coincides with its comass norm Every flat m-cochain can naturally be extended to the F-closure F m (M) of N m (M), the space of m-dimensional flat chains. These flat chains appear in Lemma 3.2, where they are needed in the course of the proof, even if one restricts the statements to the case of normal currents. They will be avoided in the rest of the paper. Proof. Since the mass is lower semicontinuous with respect to flat convergence we have This proves (a). Statement (b) follows directly from the definitions. This proves also (c) for the case of step functions. Using (a) and approximation by step functions one obtains (c). (Note that the currents T g n , where the g n are step functions approximating g, need not belong to N m (M) even if T g does.) 3.3 Definition. A locally normal m-current T ∈ N loc m (M) is calibrated by the calibration L if T A is calibrated by L for every compact set A such that T A ∈ N m (M).

Subderivatives of the stable norm
In this subsection we assume that M is compact. A subderivative of the stable norm S : H m (M, R) → R at α ∈ H m (M, R) is a linear form l ∈ H m (M, R) * such that l(α) = S(α) and l(β) ≤ S(β) for all β ∈ H m (M, R). i.e., L calibrates T .

The canonical representative of a flat cochain
According to their definition flat m-cochains are objects purely from functional analysis. They are elements of N m (M) * that are continuous with respect to the flat norm. But it is well known that flat m-cochains can be represented by bounded Lebesgue measurable m-forms in the following sense, cf. [Wh,IX,Theorem 5A] or [Fe1,4.1.19].
If L is a flat m-cochain then, by [Wh,IX,Theorem 5A] or [Fe1,4.1.19] there exists a bounded Lebesgue-measurable m-form λ such that for every smooth m-current T η , η ∈ Ω n−m 0 M, we have We say that λ is a representative of L or that λ represents L.
For the proof of Theorem 4.2 it will be important that we can find a canonical representative of L, denoted by D L , by a process of differentiation. For the following discussion, based on Whitney's book [Wh], we work in standard Euclidean space R n . Using charts we can apply the results to manifolds (see below).
Given an oriented m-dimensional affine simplex σ, we denote by P (σ) the oriented m-dimensional affine subspace of R n that contains σ, by ξ(σ) ∈ Λ m R n the unit m-vector orienting P (σ), and by vol(σ) = M [[σ]] the mdimensional Euclidean volume of σ.
The thickness (or fullness) Θ(σ) of σ is defined by If p ∈ R n , then the p-thickness Θ p (σ) of σ is defined by The following definition is implicit in [Wh,IX,§ 4].
3.5 Definition. Let L be a flat cochain of degree m on R n and p ∈ R n . We say that p is a regular point for L if there exists an m-covector ϕ ∈ Λ m R n such that for every m-vector ξ ∈ Λ m R n and all ε, η > 0 there exists δ > 0 with the property that every m-simplex σ ⊆ R n with p ∈ σ, ξ(σ) = ξ, diam(σ) < δ, and Θ(σ) ≥ η satisfies Clearly in this case the covector ϕ is unique. We denote it by D L (p).
The set of all regular points for L is denoted by reg L, its complement, the set of all singular points, by sing L.
By [Wh,IX,Theorem 5A], sing L is a Lebesgue null set and the function D L : R n sing L → Λ m R n is measurable. Moreover, D L (p) ≤ F(L) and the bounded m-form λ = D L represents L in the sense of (5) By [Wh,X,Theorem 9A], regularity is invariant under local diffeomorphisms and D L (p) behaves like an m-covector. Hence, using local charts, reg L and D L are also defined for flat cochains L on a Riemannian manifold.
Proposition 3.6 below proves that D L (p) can be obtained by a local blow-up of L at p.
For p ∈ R n and r ≥ 0 we denote by µ r,p : R n → R n the homothety with center p and factor r, i.e., µ r,p (p + x) = p + rx.
For L ∈ N m (R n ) * a flat m-cochain, p ∈ R n , and r > 0, we set L r,p := 1 r m (µ r,p ) # L, i.e., for every T ∈ N m (R n ) we have L r,p (T ) = 1 r m L (µ r,p ) # T . It is easy to see that if L is represented by the m-form λ then L r,p is represented by λ r,p where λ r,p (p + x) = λ(p + rx).
If p ∈ reg L, let L 0,p denote the flat m-cochain represented by the constant m-form λ 0,p , where λ 0,p (x) := D L (p).
3.6 Proposition. Let L be a flat m-cochain on R n .
(a) If p ∈ R n and 0 < r ≤ 1 then F(L r,p ) ≤ F(L).
(b) If p ∈ reg L then, for r → 0, L r,p converges to L 0,p in the weak- * -topology on N m (R n ) * , i.e., lim r→0 L r,p (T ) = L 0,p (T ) for every T ∈ N m (R n ).
Proof. We may assume that p = 0. In the following we omit the subscript p. By [Fe1, 4.1.14], for 0 ≤ r ≤ 1 and every T ∈ N m (R n ) we have Therefore, The fact that polyhedral currents are F-dense in N m (R n ) and (a) imply that lim r→0 L r (T ) = L 0 (T ) for every T ∈ N m (R n ).

Smoothing flat cochains
In the proof of Theorem 1.1 we have to pass from the merely bounded and measurable form λ representing the flat cochain L to a smooth approximation, obtained by convolution. The following lemma is formulated and proved for the case of M = R n and the usual convolution in R n . In the general case, one can embed M into some R N and perform the convolution using a tubular neighborhood of the submanifold M. See also [Fe2,4.7].
3.7 Lemma. Suppose L ∈ N m (R n ) * is a flat cochain, represented by the bounded, measurable m-form λ. For ε > 0, let λ ε originate from λ by convolution with the kernel ϕ ε .
Then we have lim Remark. Note that the statement is standard if λ is a smooth form or if T is given by a smooth form.
Hence, it suffices to show that T ε (λ) = T (λ ε ) holds also for the merely bounded and Lebesgue-measurable form λ.
Since the support of T is compact, we may assume that also the support of λ is compact and hence that λ is integrable w.r.t. Lebesgue measure. So we can find a sequence of smooth forms λ i converging to λ in L 1 . Let λ i ε originate from λ i by convolution with ϕ ε . Then Therefore, λ i ε converges, for i → ∞, uniformly to λ ε . So for the current On the other hand, since λ i is smooth, we have T (λ i ε ) = T ε (λ i ), and since T ε is smooth, Using Lemma 3.4 and Lemma 3.7 we can easily prove (4) Remark. This statement follows from [Fe2,4.10]. The proof given here elaborates the one sketched in [Gr,4.35].
It remains to show that By convolution we mollify a measurable m-form λ representing L to obtain closed forms λ ε ∈ Ω m M such that lim ε→0 λ ε ∞ = ess sup x∈M λ(x) = 1 .

Calibrations and Minimizing Currents in Codimension One
In this section we will show that the singular set of a codimension one calibration L contains the singular sets of all closed integer multiplicity rectifiable currents calibrated by L. For the definition of the set R loc n−1 (M) of integer multiplicity rectifiable currents, for the definition of the regular part reg T of a homologically minimizing T ∈ R loc n−1 (M) and for the regularity theory for such currents we refer to [Si,§27 and §37]. Here, we note that if T ∈ R loc n−1 (M) and if ∂T has locally finite mass, then T ∈ N loc n−1 (M). Moreover, if T ∈ R loc n−1 (M) is closed and homologically minimizing then reg T is a smooth hypersurface in M oriented by a smooth unit (n−1)-vector field T and T (ω) = reg T ω for all ω ∈ Ω n−1 0 M.

Lemma.
Suppose that the flat cocycle L ∈ N n−1 (M) * is a calibration and that T ∈ R loc n−1 (M) is closed and calibrated by L. Then, for every p ∈ reg T ∩ reg L, we have D L (p) = T (p) ♭ .
Proof. Let p ∈ reg T ∩ reg L. Since the statement is local we may work in a local chart, i.e. in R n equipped with a Riemannian metric g. Mass and volume will be defined w.r.t. g. To distinguish metric terms which refer to g from the Euclidean ones, we will mark them by a subscript or superscript g. We may assume that p is the origin, that the metric g p at the origin coincides with the standard Euclidean scalar product and that reg T is a hyperplane P through the origin, oriented by the unit (n−1)-vector ξ := T (p).
4.2 Theorem. Suppose that the flat cocycle L ∈ N n−1 (M) * is a calibration and that T ∈ R loc n−1 (M) is closed and calibrated by L. Then sing T ⊆ sing L. In particular, the union of the singular sets of all closed T ∈ R loc n−1 (M) calibrated by L is a Lebesgue null set.
Proof. Note that T is homologically mass-minimizing. Assume that p ∈ spt T is a regular point of L. We have to show that p ∈ reg T . As above, since the statement is of local nature, we may work in local coordinates. By the Decomposition Theorem for codimension one rectifiable currents, cf. [Si,27.6], we may assume that T is of multiplicity one. We identify some neighborhood of p in M with R n , equipped with a Riemannian metric g, the point p with the origin 0, such that g p = g 0 corresponds to the standard scalar product of R n . Mass w.r.t. g is denoted by M g while all other metric terms refer to the Euclidean metric.
The regularity theory for mass-minimizing currents implies that there exists a tangent cone C of T at p, i.e., there exists a sequence r i > 0 converging to zero such that the sequence T i := µ 1/r i # T converges weakly to a closed multiplicity one current C ∈ R loc n−1 (R n ) (cf. [Si,Theorem 37.4]). This current C is a cone, i.e., µ r# C = C for every r > 0. The tangent cone C is massminimizing with respect to the metric g 0 . Hence C is given by the smooth hypersurface reg C (which is a cone, too) with a possible singular set of dimension ≤ n − 8.
We will show that C is calibrated by the cochain L 0 represented by the constant (n−1)-form λ 0 (x) := D L (p). From this we will conclude that spt C is a hyperplane, and then regularity theory shows that p is a regular point of T .
Let W ⊆ R n be an open set with compact closure that contains the origin such thatC := C W ∈ N n−1 (R n ). By the arguments used in the proofs of [Si,Theorem 37.2] and [Fe1,5.4.2], there exists a sequence of compact sets K i ⊆ R n such that the currents S i : If we set L i := 1 r n−1 i (µ r i ) # L, Proposition 3.6 (b) gives us lim i→∞ L i (S) = L 0 (S) for every S ∈ N n−1 (R n ). By Proposition 3.6 (a) the sequence of cochains L i is uniformly bounded w.r.t. the flat norm. Hence where L 0 is given by the constant (n−1)-form λ 0 (x) = D L (p). Since L 0 is closed and F(L 0 ) = D L (p) ≤ 1, L 0 is a calibration.
Using the fact that L calibrates T , we get In particular, C(x) does not depend on x ∈ reg C. Since ∂C = 0, we can conclude that spt C is a hyperplane. Since C is of multiplicity one this implies that the density of C at p, and hence the density of T at p, is one, and it follows from [Fe1,5.4.6] together with [Fe1,5.4.5 (2)] that p ∈ reg T . This proves the first statement.
The second one follows from the fact that, by [Wh,IX,Theorem 5A], sing L is a Lebesgue null set.

The case of boundaries
It is not difficult to prove that N loc n (M) is precisely the set of currents of the form T f = [ [M]] f , where f ∈ BV loc (M) is of locally bounded variation, cf. [Fe1,4.5.7], and Hence a codimension one boundary T ∈ N loc n−1 (M) is given as T = ∂T f for some f ∈ BV loc (M). Additionally, we will assume that T is homologically minimizing, cf. Definition 2.3. We will give an overview over results on the structure of such T , which are proved in [Fe1,4.5.9] and [AB1], [AB2].
The BV-function f can be chosen to be (upper or lower) semicontinuous. For We have: 1. T s+ = T s− for all but countably many s ∈ R.
3. Every T s+ , T s− is homologically minimizing, so The sets G s are called the gaps of T .
Then we have For the boundary of G s in the sense of currents, The set theoretic boundary ∂G of G s consists of connected components of spt T s− ∪ spt T s+ . Outside the singular sets sing T s− ∪ sing T s+ this boundary is smooth and consists of connected components of the hypersurfaces reg T s− and reg T s+ . We denote the smooth part of ∂G by reg ∂G.
We call reg T := s∈R reg T s+ ∪ reg T s− the regular set of T and sing T := s∈R sing T s+ ∪ sing T s− = spt T reg T the singular set of T . Although the regularity theory for mass-minimizing rectifiable (n−1)-currents implies that the singular set of each T s± is of Hausdorff dimension ≤ n − 8, a priori it is not clear whether their union sing T is small. However, the following lemma implies that it is at least a Lebesgue null set, cf. Corollary 5.3.

Lemma. Suppose T ∈ N loc
n−1 (M) is a boundary calibrated by the flat cocycle L. Then: (a) For every p ∈ reg T ∩ reg L, the (n−1)-covector D L (p) is uniquely determined by T , D L (p) = T s± (p) ♭ if p ∈ reg T s± .
(b) sing T ⊆ sing L. In particular, sing T is a Lebesgue null set.
Proof. Since T is calibrated by L, it is homologically mass-minimizing. So we can apply the list of statements above. Points 6 and 8 imply that L calibrates each T s+ and T s− , cf. Lemma 3.2.
Then, by Lemma 4.1, we have D L (p) = T s± (p) ♭ for every p ∈ reg T s± ∩ reg L. This proves (a). Statement (b) follows immediately from Theorem 4.2.
5.3 Corollary. Suppose T ∈ N loc n−1 (M) is a locally mass-minimizing boundary. Then sing T (as defined above) is a Lebesgue null set.
Proof. Since the statement is of local nature we may assume that T has compact support. Using the Hahn-Banach Theorem like in Section 3.2 we get a calibration that calibrates T (cf. [Fe2,4.10]). Then the statement follows from Lemma 5.2.
Recall that in the introduction we defined V (α) to be the smallest linear subspace of H n−1 (M, R) that is spanned by integer classes and contains α. From the preceding discussion one can conclude that V (α) is the orthogonal complement of K(α) with respect to I, V (α) = β ∈ H n−1 (M, R) I(k, β) = 0 for all k ∈ K(α) .
Consider the covering p : M α → M associated to α and letT := p # T be the lift of T to M α . Then M spt T = p(M α sptT ), where M α sptT = s∈J G s is the union of at most countably many gaps, cf. Section 5. By Corollary 5.6, for every gap G, the restriction p|G : G → M of p to G is injective. Hence it suffices to show that G p * η ∧ (λ 1 − λ 2 ) = 0 for every gap G.
Therefore we can find r > r 0 such that ∂B r is a smooth hypersurface which meets the regular part reg ∂G of ∂G transversely and such that vol n (G B r ) < δ 8C and vol n−1 (G ∩ ∂B r ) < δ 24Cr .
Since the (n−1)-formsλ 1 ,λ 2 are not smooth, we have to pass to smooth forms in order to apply Stokes's Theorem. We do this by mollifying the λ i by means of convolution. Thus we find smooth approximations λ ε i of theλ i which satisfy sup|λ ε i | ≤ 3 2 and G∩Br dg ∧ (ω ε − ω) < δ 4 , where ω ε = λ ε 1 − λ ε 2 . Since, for i = 1, 2, L i is a cocycle,λ i is weakly closed and hence dλ ε i = 0. Now, the boundary currents T + := T s+ ∂G s and T − := T s− ∂G s of G = G s are calibrated by the liftL i = p # L i , represented byλ i , cf. Lemma 5.5. By Lemma 3.2 (c) we havē L i (T ± g) B r = reg T ± ∩Br g d vol n−1 .