Regularity of area minimizing currents mod $p$

We establish a first general partial regularity theorem for area minimizing currents $\mathrm{mod}(p)$, for every $p$, in any dimension and codimension. More precisely, we prove that the Hausdorff dimension of the interior singular set of an $m$-dimensional area minimizing current $\mathrm{mod}(p)$ cannot be larger than $m-1$. Additionally, we show that, when $p$ is odd, the interior singular set is $(m-1)$-rectifiable with locally finite $(m-1)$-dimensional measure.


Overview and main results.
In this paper we consider currents mod(p) (where p ≥ 2 is a fixed positive integer), for which we follow the definitions and the terminology of [15]. In particular, given an open subset Ω ⊂ R m+n , we will let R m (Ω) and F m (Ω) denote the spaces of m-dimensional integer rectifiable currents and m-dimensional integral flat chains in Ω, respectively. If C ⊂ R m+n is a closed set (or a relatively closed set in Ω), then R m (C) (resp. F m (C)) denotes the space of currents T ∈ R m (R m+n ) (resp. T ∈ F m (R m+n )) with compact support spt(T ) contained in C. Currents modulo p in C are defined introducing an appropriate family of pseudo-distances on F m (C): if S, T ∈ F m (C) and K ⊂ C is compact, then such that T − S = R + ∂Z + pP for some P ∈ F m (K) .
Two flat currents in C are then congruent modulo p if there is a compact set K ⊂ C such that F p K (T − S) = 0. The corresponding congruence class of a fixed flat chain T will be denoted by [T ], whereas if T and S are congruent we will write T = S mod(p) .
The symbols R p m (C) and F p m (C) will denote the quotient groups obtained from R m (C) and F m (C) via the above equivalence relation. The boundary operator ∂ has the obvious property that, if T = S mod(p), then ∂T = ∂S mod(p). This allows to define an appropriate notion of boundary mod(p) as ∂ p [T ] := [∂T ]. Correspondingly, we can define cycles and boundaries mod(p) in C: • a current T ∈ F m (C) is a cycle mod(p) if ∂T = 0 mod(p), namely if ∂ p [T ] = 0; • a current T ∈ F m (C) is a boundary mod(p) if ∃S ∈ F m+1 (C) such that T = ∂S mod(p), namely [T ] = ∂ p [S]. 1 As a first step to a better understanding of the singularities it is therefore desirable to give a bound on the Hausdorff dimension of the singular set. The present work achieves the best possible bound in the most general case, and in particular it answers a question of White, see [1,Problem 4.20]. Theorem 1.4. Assume that p ∈ N\{0, 1}, that Σ ⊂ R m+n is a C 3,a 0 submanifold of dimension m +n for some positive a 0 , that Ω ⊂ R m+n is open, and that T ∈ R m (Σ) is area minimizing mod(p) in Ω ∩ Σ. Then, H m−1+α (Sing(T )) = 0 for every α > 0.
Prior to the present paper, the state of the art in the literature on the size of the singular set for area minimizing currents mod(p) was as follows. We start with the results valid in any codimension.
(a) For m = 1 it is very elementary to see that Sing(T ) is discrete (and empty when p = 2); (b) Under the general assumptions of Theorem 1.4, Sing(T ) is a closed meager set in (spt p (T ) ∩ Ω) \ spt p (∂T ) by Allard's interior regularity theory for stationary varifolds, cf. [2] (in fact, in order to apply Allard's theorem it is sufficient to assume that Σ is of class C 2 ); (c) For p = 2, H m−2+α (Sing(T )) = 0 for every α > 0 by Federer's classical work [16]; moreover the same reference shows that Sing(T ) consists of isolated points when m = 2; for m > 2 the (m − 2)-rectifiability of Sing(T ) was first proved in [22] and the recent work [19] implies in addition that Sing(T ) has locally finite H m−2 measure, see below. We next look at the hypersurface case, namelyn = 1.
(d) When p = 2, H m−2 (Sing(T )) = 0 even in the case of minimizers of general uniformly elliptic integrands, see [20]; for the area functional, using [19], one can conclude additionally that Sing(T ) is (m − 3)-rectifiable and has locally finite H m−3 measure; (e) When p = 3 and m = 2, [24] gives a complete description of Sing(T ), which consists of C 1,α arcs where three regular sheets meet at equal angles; (f) When p is odd, [26] shows that H m (Sing(T )) = 0 for minimizers of a uniformly elliptic integrand, and that H m−1+α (Sing(T )) = 0 for every α > 0 for minimizers of the area functional; (g) When p = 4, [25] shows that minimizers of uniformly elliptic integrands are represented by immersed manifolds outside of a closed set of zero H m−2 measure.
In view of the examples known so far it is tempting to advance the following (c) p is odd and the codimensionn = 1. In all three cases, however, the conjecture follows from the much stronger fact that Sing f (T ) is empty: • the case (a) is an instructive exercise in geometric measure theory; • the case (b) follows from Allard's regularity theorem for stationary varifold; • the case (c) is a corollary of the main result in [26]. Note however that in all the other cases we cannot expect Sing f (T ) to be empty. Indeed the easiest case would be p = 4, m = 2 andn = 1. In this case it follows from the work [25] that, if S 1 and S 2 are integral currents representative mod(2) (see (3.1)) which are area minimizing mod (2), then T = S 1 + S 2 is a representative mod(4) which is area minimizing mod (4). Consider now two smooth minimal graphs in B 1 × R ⊂ R 3 , where u 1 , u 2 : B 1 → R are the corresponding functions. Endow the graphs with the natural orientation, and let S 1 and S 2 be the corresponding integral currents. Oriented minimal graphs in codimension 1 are then known to be area minimizing, both as integral currents and as currents mod (2). In particular T = S 1 + S 2 is then area minimizing mod (4). Observe therefore that, if u 1 and u 2 are distinct, then Sing(T ) is the intersection of the two graphs. It is easy to see that u 1 and u 2 might be chosen so that u 1 (0) = u 2 (0) = 0, ∇u 1 (0) = ∇u 2 (0) = 0 and u 1 and u 2 are anyway distinct. In particular 0 would be a singular point of T and the (unique) tangent cone to T at 0 is the (oriented) two dimensional horizontal plane π 0 = {x 3 = 0} with multiplicity 2. In such example we thus have 0 ∈ Sing f (T ).
In this paper we strengthen the result for p odd by showing that Conjecture 1.5 in fact holds in any codimension. Indeed we prove the following more general theorem. Theorem 1.6. Let T be as in Theorem 1.4 and Q < p 2 a positive integer. Consider the subset Sing Q (T ) of spt p (T ) \ spt p (∂T ) which consists of interior singular points of T where the density is Q (see Definition 7.1). Then H m−2+α (Sing Q (T )) = 0 for every α > 0.
The analysis of tangent cones (cf. Corollary 6.3) implies that if p is odd then Sing Q (T ) .
We thus get immediately Corollary 1.7. Conjecture 1.5 holds for every p odd in any dimension m and codimension n.
The fact above, combined with the techniques recently introduced in the remarkable work [19], allows us to conclude the following theorem. Theorem 1.8. Let T be as in Theorem 1.4 and assume p is odd. Then Sing(T ) is (m − 1)rectifiable, and for every compact K with K ∩ spt p (∂T ) = ∅ we have H m−1 (Sing(T ) ∩ K) < ∞.
In turn the above theorem implies the following structural result. in particular i Θ i (q) is an integer multiple of p for H m−1 -a.e. q ∈ Sing(T ) ∩ U .
It is tempting to advance the following conjecture. From the latter conjecture one can easily conclude an analogous structure theorem as in Corollary 1.9. Note that the conjecture is known to hold for p = 2 in every codimension (in which case, in fact, we know that Sing(T ) has dimension at most m − 2) and for p = 4 in codimension 1.
1.2. Plan of the paper. The paper is divided into five parts: the first four parts contain the arguments leading to the proof of Theorems 1.4 and 1.6, while the last part is concerned with the proof of the rectifiability Theorem 1.8 and of Corollary 1.9. Each part is further divided into sections. The proof of Theorems 1.4 and 1.6 is obtained by contradiction, and is inspired by F. Almgren's work on the partial regularity for area minimizing currents in any codimension as revisited by the first-named author and E. Spadaro in [8,10,11]. In particular, Part 1 contains the preliminary observations and reductions aimed at stating the contradiction assumption for Theorems 1.4 and 1.6, whereas Part 2, Part 3, and Part 4 are the counterpart of the papers [8], [10], and [11], respectively. An interesting feature of the regularity theory presented in this work is that Almgren's multiple valued functions minimizing the Dirichlet energy are not the right class of functions to consider when one wants to approximate a minimizing current mod(p) in a neighborhood of a flat interior singular point whenever the density of the point is precisely p 2 . Solving this issue requires (even in the codimensionn = 1 case) the introduction of a class of special multiple valued functions minimizing a suitably defined Dirichlet integral. The regularity theory for such maps (which we call linear theory) is the content of our paper [7]. Applications of multivalued functions to flat chains mod(p) were already envisioned by Almgren in [3], even though he considered somewhat different objects than those defined in [7]. Because of this profound interconnection between the two theories, the reading of [7] is meant to precede that of the present paper.
; η • f average of the (possibly special) multiple valued function f ; Gr(u) set-theoretical graph of a (possibly multi-valued) function u; T F integer rectifiable current associated (via push-forward) to the image of a (possibly special) multiple valued function; G u integer rectifiable current associated to the graph of a (possibly special) multiple valued function.

Preliminary reductions
We recall first that, as specified in [15, 4.2.26], for any S ∈ R m (Σ) we can find a representative mod(p), namely a T ∈ R m (Σ) congruent to S mod(p) such that for every Borel A ⊂ Σ. (3.1) In particular, such a representative has multiplicity function θ such that |θ| ≤ p/2 at T -a.e. point, and it satisfies M p ([T U ]) = T (U ) for every open set U and spt(T ) = spt p (T ) (observe in passing that the restriction to an open set U is defined for every current). It is evident that if T ∈ R m (Σ) is area minimizing mod(p) in Ω ∩ Σ then T is necessarily representative mod(p) in Ω ∩ Σ, in the sense that (3.1) holds true for every Borel A ⊂ Ω ∩ Σ. For this reason, we shall always assume that T is representative mod(p), and that the aforementioned properties concerning multiplicity, mass and support of T are satisfied. Note also that such T is area minimizing mod(p) in any smaller open set U ⊂ Ω. Moreover T is area minimizing mod (p) in Ω if and only if T Ω is area minimizing mod(p) in Ω. Also, for Ω sufficiently small the regularity of Σ guarantees that Σ ∩ Ω is a graph, and thus, if in addition Ω is a ball, Σ ∩ Ω is a Lipschitz deformation retract of R m+n . A current S ∈ R m (Σ ∩ Ω) is thus a cycle mod(p) if and only if it is a cycle mod(p) in R m+n . In these circumstances it does not matter what the shape of the ambient manifold Σ is outside Ω and thus, without loss of generality, we can assume that Σ is in fact an entire graph. By the same type of arguments we can also assume that ∂ p [T ] = 0 in Ω. We summarize these reductions in the following assumption (which will be taken as a hypothesis in most of our statements) and in a lemma (which will be used repeatedly).
Assumption 3.1. Σ is an entire C 3,a 0 (m +n)-dimensional graph in R m+n with 0 < a 0 ≤ 1, and Ω ⊂ R m+n is an open ball. T is an m-dimensional representative mod(p) in Σ that is area minimizing mod(p) in Σ ∩ Ω and such that (∂T ) Ω = 0 mod(p) in Ω.

Stationarity and compactness
Another important tool that will be used repeatedly in the sequel is the fact that the integral varifold v(T ) induced by an area minimizing representative mod(p) T is stationary in the open set Ω ∩ Σ \ spt p (∂T ).
Consider now an open ball B R = Ω ⊂ R m+n , a sequence of Riemannian manifolds Σ k and a sequence of currents T k such that each triple (Ω, Σ k , T k ) satisfies the Assumption 3.1. In addition assume that: (a) Σ k converges locally strongly in C 2 to a Riemannian submanifold Σ of R m+n which is also an entire graph; By the compactness theorem for integral currents mod(p) (cf. [15,Theorem (4.2.17) ν , p. 432]), we conclude the existence of a subsequence, not relabeled, of a current T ∈ R m (R m+n ) and of a compact set K ⊃ B R such that Let U δ be the closure of the δ-neighborhood of Σ and consider that, for a sufficiently small δ > 0, the compact set K ′ := B R ∩ U δ is a Lipschitz deformation retract of R m+n . For k sufficiently large, the currents T k B R are supported in K ′ and [15, Theorem (4.2.17) ν ] implies that spt(T ) ⊂ K ′ . Since δ can be chosen arbitrarily small, we conclude that spt(T ) ⊂ Σ and hence that T ∈ R m (Σ). At the same time, by Allard's compactness theorem for stationary integral varifolds, we can assume, up to extraction of a subsequence, that v(T k B R ) converges to some integral varifold V in the sense of varifolds.

Proposition 4.2.
Consider Ω, Σ k , T k , Σ, T and V as above. Then Proof. Let us simplify the notation by writing T k in place of T k B R . Recall that F p K (T k − T ) → 0 for some compact set K ⊃ B R . This means that there are sequences of rectifiable currents R k , S k and integral currents Q k 1 with support in K such that As above, denote by U δ the closure of the δ-neighborhood of the submanifold Σ. Observe next that, for every δ sufficiently small, K δ := U δ ∩ B R is a Lipschitz deformation retract. Moreover, for each k sufficiently large spt(T k ) ⊂ K δ . We can thus assume, without loss of generality, the existence of ak(δ) ∈ N such that Next, if we denote by U δ,k the closures of the δ-neighborhoods of Σ k , due to their C 2 regularity and C 2 convergence to Σ, for a δ > 0 sufficiently small (independent of k) the nearest point projections We now show that T is area minimizing mod(p) in B R ∩ Σ. Assume not: then there is a ρ < R and a currentT with spt(T −T ) ⊂ B ρ ∩ Σ such that and, for every where ε is independent of s because of the condition spt(T −T ) ⊂ B ρ . Denote by d : R m+n → R the map x → |x| and consider the slices S k , d, s . By Chebyshev's inequality, for each k we can select an s k ∈]ρ, R+ρ 2 [ such that Consider therefore the current: . Also, note that (4.5) implies that ∂S k has finite mass. Hence, by [21,Lemma 28.5(2)], In particular, combining the latter equality with (4.5), we get 1 Although the definition of flat convergence modulo p is given with Q k flat chains, a simple density argument shows that we can in fact take them integral.
where in the second line we have used that spt(T −T ) ⊂ B ρ ⊂ B s k . Since ∂(T −T ) = 0 mod(p) in Σ ⊂ R m+n , we conclude that ∂(T k − T k ) = 0 mod(p) in R m+n . However, considering (4.7), for k large enough the currentsT k , S k , R k , Q k , T andT are all supported in the domain of definition of the retraction p k . Since (p k ) ♯ T k = T k , we then have that ∂(T k − (p k ) ♯Tk ) = 0 mod(p) in Σ k . Consider also that, for each σ > 0 fixed, there is ak(σ) ∈ N such that all the currents above are indeed supported in U σ,k when k ≥k(σ). This implies in particular that, by (4.8), Up to extraction of a subsequence, we can assume that s k → s for some s ∈ [ρ, R+ρ 2 ]. Recalling the semicontinuity of the p-mass with respect to the flat convergence mod(p), we easily see that (since the T k 's and T are all representative mod(p)) Next, by the estimates (4.10) and (4.6) we immediately gain Finally, since the map p k is the identity on Σ k , again thanks to (4.8) and to the observation on the supports ofT k − T k , it turns out that spt( We thus have contradicted the minimality of T k .
Observe that, if in the argument above we replaceT with T itself, we easily achieve that, for every fixed ρ > 0, there is a sequence By this and by the semicontinuity of the p-mass under flat convergence, we easily conclude that lim The latter implies then that T k * ⇀ T in the sense of measures in B R . Consider now the rectifiable sets E k , E and the Borel functions Θ k : Let T q E k (resp. T q E) be the approximate tangent space to E k (resp. E) at H m -a.e. point q.

Slicing formula mod(p)
In this section we prove a suitable version of the slicing formula for currents mod(p), which will be useful in several contexts. We let I p m (C) denote the group of integral currents mod(p), that is of classes Before coming to the proof of Lemma 5.1 we wish to point out two elementary consequences of the theory of currents mod(p) which are going to be rather useful in the sequel.
Proof. (i) is an obvious consequence of Federer's characterization in [15]: an integer rectifiable current T of dimension m is a representative mod(p) if and only if T (E) ≤ p 2 H m (E) for every Borel set E. By the coarea formula for rectifiable sets, this property is preserved for a.e. slice and thus (ii) is immediate. Moreover, again by Federer's characterization, if T is as in (iv), and if k(x) = arg min{|Θ(x) − kp| : k ∈ Z}, then setting Θ ′ (x) := Θ(x) − k(x) p we have that T ′ = Θ ′ K is a representative mod(p) of T , and thus, since |Θ ′ | = |Θ| p , (5.2) follows directly from M p (T E) = T ′ (E).
As for (iii), since T is a top-dimensional current, R m+1 (K) = {0}. We thus have and P ∈ F m (K)} . Observe however that, since K is m-dimensional, F m (K) consists of the integer rectifiable currents with support in K. A simple computation gives then and we can use (iv) to conclude.
Proof of Lemma 5.1. (ii) has been addressed already in Lemma 5.2, and (iii) is a simple consequence of Lemma 5.2 and of (i) with the choice Z = 0.
We now come to the proof of (i). By [17,Theorem 3.4], there exists a sequence {P k } ∞ k=1 of integral polyhedral chains and currents R k ∈ R m (Ω), S k ∈ R m+1 (Ω) and Q k ∈ I m (Ω), with the following properties for every k ≥ 1: Since P k is an integral current, by the classical slicing theory (cf. for instance [21,Lemma 28.5(2)]), the following formula holds for a.e. t ∈ R: The identity (5.3) implies that ∂S k has locally finite mass, and thus S k is an integral current. In particular, ∂ S k , f, t = − ∂S k , f, t . Furthermore, the slicing formula holds true for S k as well, that is for a.e. t ∈ R one has: Combining (5.3) and (5.9), we can therefore write: The identity (5.11) implies that ∂(R k +S k ) has locally finite mass, and thus in particular R k +S k is an integral current. Hence, for a.e. t ∈ R the slicing formula holds true for R k +S k , that is: From the identities (5.3) and (5.11), and using (5.7), (5.8), (5.12), and the slicing formula for Q k we easily conclude that the following holds for a.e. t ∈ R: Now,Q k {f < t} is an integral current and thus, setting K := Ω, we can estimate Since lim k M(R k ) + M(S k ) = 0, it remains to show that, for a.e. t, In order to see this, fix ε > 0. By [21,Lemma 28.5(1)], we have that there is a Borel set E k with measure |E k | ≤ ε 2 k such that In particular, if we set E := k E k , we have |E| ≤ 2ε, and using (5.10) we see that Hence lim k→∞ M( S k , f, t ) = 0 for all t ∈ E. Since ε is arbitrary, this concludes the proof.
Remark 5.3. We are actually able to give a much shorter proof of Lemma 5.1(i), provided one can prove that there exists an integral currentT such thatT = T mod(p). Indeed, in this case, sinceT is integral the classical slicing formula gives On the other hand, the conditionsT = T mod(p) and ∂T = ∂T = Z mod(p) imply that there are rectifiable currents R and Q such that T =T + pR and Z = ∂T + pQ, and thus we deduce as we wanted. The existence of an integral representative in any integral class mod(p) is in fact a very delicate question. If K is any given compact subset of R m+n then a class [T ] ∈ I p m (K) does not necessarily have a representative in I m (K) when m ≥ 2; see [17,Proposition 4.10]. Positive answers have been given, instead, when m = 1 in the class I m (K) for any given compact K in [17,Theorem 4.5], and in any dimension in the class K I m (K) in the remarkable work [28].

Monotonicity formula and tangent cones
From Lemma 4.1 and the classical monotonicity formula for stationary varifolds, cf. [2] and [21], we conclude directly the following corollary. Corollary 6.1. Let T, Σ and Ω = B R be as in Assumption 3.1. Then, if q ∈ spt(T ) ∩ Ω, the following monotonicity identity holds for every 0 < s < r < R − |q|: where Y ⊥ (x) denotes the component of the vector Y (x) orthogonal to the tangent plane of T at x (which is oriented by T (x)). In particular: ω m r m exists and is finite at every point q ∈ B R .
(iii) The map q → Θ T (q) is upper semicontinuous and it is a positive integer at H m -a.e. q ∈ spt(T ). In particular spt(T ) ∩ B R = {Θ T ≥ 1}.
Next, we introduce the usual blow-up procedure to analyze tangent cones at q ∈ spt(T ).
Definition 6.2. Fix a point q ∈ spt(T ) and define We denote by T q,r the currents Recalling Allard's theory of stationary varifolds, we then know that, for every sequence r k ↓ 0, a subsequence, not relabeled, of v(T q,r k ) converges locally to a varifold C which is a stationary cone in T q Σ (the tangent space to Σ at q). Combined with Proposition 4.2 we achieve the following corollary. Corollary 6.3. Let T, Σ and Ω = B R be as in Assumption 3.1, let q ∈ spt(T ) ∩ Ω, and let r k ↓ 0. Then there is a subsequence, not relabeled, and a current T 0 with the following properties: . Before coming to its proof, let us state an important lemma which will be used frequently during the rest of the paper. See [14,Theorem 7.6] for a proof. Lemma 6.4 (Constancy Lemma). Assume π ⊂ R m+n is an m-dimensional plane and let Ω ⊂ R m+n be an open set such that Ω ∩ π is connected.
. Proof of Corollary 6.3. Note that (v) is an obvious consequence of the constancy lemma and of (i). In order to prove the remaining statements, first extract a subsequence such that V k = v(T q,r k ) converges to a stationary cone C as above. Then observe that for every j ∈ N, using a classical Fubini argument and Lemma 5.1 we find a radius ρ(j) ∈ [j, j + 1] such that Thus we can find a subsequence to which we can apply the compactness Proposition 4.2. By a standard diagonal argument we can thus find a single subsequence r k with the following properties: Notice next that T j B ρ(i) = T i mod(p) for every i ≤ j. If we then define the current with ρ(−1) := 0, then the latter satisfies the conclusions (i), (ii) and (iv).
In the remaining part of the proof we wish to show (iii), after possibly changing T 0 to another representative mod(p) of the same class.
To this aim, consider that, by standard regularity theory for stationary varifolds, the closed set R = spt(C) is countably m-rectifiable, it is a cone with vertex at the origin and C = Θ C (x)H m R, where Θ C is the density of the varifold C. By the monotonicity formula and v(T ) = C we have If x is a point where the approximate tangent T x R exists, we then conclude easily that, up to subsequences, we can apply the same argument above and find that (T 0 ) x,r k with r k ↓ 0 converges locally mod(p) to a current S satisfying the corresponding conclusions: However, for S we would additionally know that it is supported in T x R, which is an mdimensional plane. We then could apply the Constancy Lemma and conclude that, if v 1 , . . . , v m is an orthonormal basis of T x R, then Θ C (x) ∈ N ∩ [1, p 2 ] and, for any ρ > 0, In particular we conclude that there is a Borel function ε : spt(C) = R → {−1, 1} such that where v(x) is an orienting Borel unit m-vector for T x R. Clearly, since R is a cone, we can choose v(x) with the additional property that v(x) = v(λx) for every positive λ. Also, since the varifold C is a cone, the density Θ C is 0-homogeneous as well. Moreover, at all points x where Θ C (x) = p 2 we can arbitrarily set ε(x) = 1, since this would neither change the class mod(p), nor the fact that T 0 is representative mod(p).
Fix now a radius s > 0 such that the conclusions of Lemma 5.1 hold with T = T 0 , f = |·|, and t = s, and consider the cone T ′ := T 0 , | · |, s × × {0}. Observe that ∂(T ′ − T 0 B s ) = 0 mod(p). We now make the following simple observation: if Z ∈ R m (R m+n ) with spt(Z) compact is such that ∂Z = 0 mod(p) in R m+n , then ∂(Z × × {0}) = Z mod(p). The proof is in fact a simple consequence of the definition, since ∂Z = 0 mod(p) implies the existence of integer rectifiable currents Q (1) k and Q (2) k and flat currents Q k such that We apply the above observation to Z = T ′ − T 0 B s . In that case we conclude however that the cone Z × × 0 is identically 0, because it is an (m+1)-dimensional rectifiable current supported in the countably m-rectifiable set R. We thus must necessarily have that T ′ − T 0 B s = 0 mod(p). Applying the argument of the previous paragraph, we of course again conclude that Consider now, as above, a point x ∈ B s where the approximate tangent plane to R exists.
However the two limits must be congruent mod(p) and, in case Θ C (x) < p 2 , this necessarily implies ε(x) = ε ′ (x). Fix now λ > 0. Since T ′ is a cone and s is arbitrary, we conclude that for H m a.e.
On the other hand we already have ε(x) = ε(λx) = 1 if Θ C (x) = p 2 . Hence we have concluded that ε(λx) = ε(x) for H m -a.e. x ∈ R. In particular (ι 0,λ ) ♯ T 0 = T 0 . The arbitrariness of λ implies now the desired conclusion (iii) and completes the proof of the corollary. Before proceeding, we need to recall the following definition. 3. An integral m-varifold V is called a k-symmetric cone (where 0 ≤ k ≤ m) if it can be written as the product of a k-dimensional plane passing through the origin times an (m − k)-dimensional cone. The largest plane passing through the origin such that the above holds is called the spine of V . If V is stationary, then the standard stratification of V is

Strata and blow-up sequence
As a consequence of Corollary 6.3 and of the classical Almgren's stratification theorem, we have now the following Then H m−1+α (Z) = 0 for every α > 0.
Proof. By Lemma 4.1, the varifold V = v(T ) is stationary in Σ ∩ Ω, thus we can consider the stratification of V as in (7.1) and (7.2). If q ∈ S m \ S m−1 then there is at least one tangent cone to V at q which is supported in a flat plane π 0 . Then there is a current T 0 as in Corollary 6.3, obtained as a limit T q,r k for an appropriate r k ↓ 0, which satisfies v(T 0 ) = V . Thus by the constancy lemma Θ T 0 (0) = Θ T (q) must belong to [1, p 2 ] ∩ N. This implies that Z ⊂ S m−1 . Our statement then follows immediately from the well known fact that dim H S k ≤ k for every 0 ≤ k ≤ m.
We shall also need the following elementary yet fundamental lemmas. Given v ∈ R m+n , we will adopt the notation τ v := ι v,1 , so that τ v (x) := x − v. Lemma 7.5. Assume T ∈ R m (R m+n ) is an m-dimensional integer rectifiable current such that ∂T = 0 mod(p) and the associated varifold v(T ) is a k-symmetric cone with spine R k × {0} ⊂ R m+n . Then

3)
and there exists an (m − k)-dimensional cone T ′ such that By the properties of M , modulo changing the sign of θ, we can also assume that the orienting unit m-vector field τ is a 0-homogeneous function such that Now, given two Lipschitz and proper maps f, g : R m+n → R m+n , and letting h : [0, 1] × R m+n → R m+n be the linear homotopy from f to g, namely the function defined by the homotopy formula (see [21,Equation 26.22]) states that Since ∂T = 0 mod(p), (7.5) yields Now, let v ∈ R k × {0}, and apply (7.6) with We can compute, for any ω ∈ D 1+m (R × R m+n ): Using that ω can be chosen arbitrarily, we conclude (7.3) from (7.6). Next, let p : R m+n → R m+n be the orthogonal projection operator onto R k × {0}. Using standard properties of the slicing of integer rectifiable currents (see e.g. [15, Theorem 4.3.2(7)]) and (7.3), we can conclude then that for every z, v ∈ R k × {0} such that the slices exist, or, equivalently, that T, p, z = (τ w−z ) ♯ T, p, w mod(p) (7.8) for every z, w ∈ R k × {0} such that the slices exist. Fix z such that T, p, z exists, and let T ′ ∈ R m−k (R m+n−k ) be such that T, p, z = (τ −z ) ♯ T ′ after identifying R m+n−k with {0} × R m+n−k . Then, the currentT := R k × T ′ satisfies Observe that we may write for a 0-homogeneous functionθ such thatθ( By the considerations above, the standard slicing theory of rectifiable currents (see e.g. [15,Theorem 4.3.8] for a Borel measurable unit (m − k)-vector field ζ = ζ z which is uniquely determined by τ and dφ. If z ∈ R k × {0} is such that both (7.9) and (7.11) hold, then By Fubini's theorem, the conclusion in (7.12) holds at H m -a.e. x ∈ M , so that (7.4) follows from (7.10) and the definition ofT .
If T is a representative mod(p), then T, p, z is a representative mod(p) for H k -a.e. z ∈ R k × {0}, and thus we can choose z such that the corresponding T ′ is a representative mod(p). With this choice,T is a representative mod(p) as well, and sinceθ(x) = θ(x) mod(p) for H m -a.e. x ∈ M we deduce that where ε(x) = 1 or |θ(x)| = p 2 . As a consequence, |θ| = |θ| H m M -a.e., which in turn implies that v(T ) = v(T ). The last conclusion of the lemma is elementary, and the details of the proof are omitted.
is an m-dimensional locally area minimizing current mod(p) without boundary mod(p) which is a cone (in the sense of Corollary 6.3 (iii)). Suppose, furthermore, that v(T 0 ) is (m − 1)-symmetric but not m-symmetric (namely not flat). Then, Since v(T 0 ) is (m − 1)-symmetric but not m-symmetric, by Lemma 7.5 T 0 = π × T ′ 0 mod(p), where π is the (m − 1)-dimensional spine of v(T 0 ), and T ′ 0 is a one-dimensional cone which has no boundary mod(p) and is locally area minimizing mod(p). Since Θ(T ′ 0 , 0) = Θ(T 0 , 0), we can reduce the proof of the lemma to the case when m = 1.
Thus we can assume that T 0 = i Q i ℓ i , where ℓ 1 , . . . , ℓ N are pairwise distinct oriented half lines in R 1+n with the origin as common endpoint and the Q i 's are integers. Without loss of generality we can assume that ∂ ℓ i = − 0 . Observe that and that i Q i = 0 mod(p) since T 0 has no boundary mod(p). If i Q i = 0, then T 0 would be an integral current without boundary, which in turn would have to be area minimizing. But since T 0 is by assumption not flat, this is not possible. Thus i Q i = kp for some nonzero integer k. This clearly implies which in turn yields Θ(T 0 , 0) ≥ p 2 . We are now ready to state the starting point of our proof of Theorem 3.3 and Theorem 7.2, which will be achieved by contradiction. Proposition 7.7 (Contradiction sequence). Assume Theorem 7.2 is false. Then there are integers m, n ≥ 1 and 2 ≤ Q < p 2 and reals α, η > 0 with the following property. There are (i) T, Σ and Ω as in Assumption 3.1 such that 0 ∈ Sing Q (T ); (ii) a sequence of radii r k ↓ 0 and an m-dimensional plane π 0 such that v(T 0,r k ) converges If Theorem 3.3 is false then either there is a sequence as above or, for Q = p 2 , there is a sequence as above where (iii) is replaced by Proof. Suppose first that Theorem 3.3 is false. Fix p ∈ N\{0, 1}, and let m ≥ 1 be the smallest integer for which the assertion of Theorem 3.3 is false. Observe that m > 1. Fix thus a T, Σ and Ω satisfying Assumption 3.1 for which there is an α > 0 with H m−1+α (Sing(T )) > 0. Then, by Proposition 7.4, there must be a Q ∈ N ∩ [1, p 2 ] such that H m−1+α (Sing Q (T )) > 0. By [21,Theorem 3.6], H m−1+α -a.e. point in Sing Q (T ) has positive H m−1+α ∞ -upper density: fix a point q with this property, and assume, without loss of generality, that q = 0 and that (∂T ) B 1 = 0 mod(p). Then, there exists a sequence of radii r k such that r k ↓ 0 as k → ∞ and such that Moreover, we can assume that the sequence of stationary varifolds v(T 0,r k ) converges to a stationary cone C ⊂ T 0 Σ. Consider the compact sets {Θ T 0,r k ≥ Q} ∩ B 1 and assume, without loss of generality, that they converge in the Hausdorff sense to a compact set K. As it is well known, by the monotonicity formula for stationary varifolds we must have Θ C (q) ≥ Q for every q ∈ K. On the other hand, this implies that every point q ∈ K belongs to the spine of the cone C; see [27]. In turn, by the upper semicontinuity of the H m−1+α ∞ measure with respect to Hausdorff convergence of compact sets, we have Recall that the spine of the cone C is however a linear subspace of R m+n , cf. again [27]. This implies in turn that C must be supported in a plane, which completes the proof under the assumption that Theorem 3.3 is false.
Now, let us suppose Theorem 7.2 is false. Then, we can find p, m, n and Q < p 2 , together with Ω, Σ, T as in Assumption 3.1, and α > 0 such that H m−2+α (Sing Q (T )) > 0. Arguing as above, we can then find a point q ∈ Sing Q (T ) with positive H m−2+α ∞ -upper density, and we can suppose, without loss of generality, that q = 0. Then, there is a sequence of radii r k with r k ↓ 0 as k → ∞ such that: • the blow-up sequence T 0,r k converges, in the sense of Corollary 6.3 (iv), to a current T 0 ∈ R m (T 0 Σ) satisfying properties (i), (ii), and (iii) of Corollary 6.3; • the sequence of varifolds v(T 0,r k ) converges to a stationary cone C in T 0 Σ; • C = v(T 0 ). • the spine of C is a linear subspace of T 0 Σ having dimension at least m − 1. Now, if the spine of C is (m − 1)-dimensional, then C is (m − 1)-symmetric but not flat, hence forcing Θ(T 0 , 0) ≥ p 2 by Lemma 7.6, which is a contradiction to the fact that 0 ∈ D Q (T ) with Q < p 2 . Thus, C is supported in an m-dimensional plane, and the proof is complete.

Part 2. Approximation with multiple valued graphs
Following the blueprint of Almgren's partial regularity theory for area minimizing currents, we now wish to show that any area minimizing current modulo p can be efficiently approximated, in a region where it is "sufficiently flat", with the graph of a multiple valued function which minimizes a suitably defined Dirichlet energy. Suppose that, in the region of interest, the current is a Q-fold cover of a given m-plane π, where Q ∈ 1, p 2 . The "classical" theory of Dir-minimizing Q-valued functions as in [12] is powerful enough to accomplish the task whenever Q < p 2 (which is always the case when p is odd). If p is even and Q = p 2 , on the other hand, Almgren's Q-valued functions are not anymore the appropriate maps, and we will need to work with the class of special multiple valued function defined in [7].

First Lipschitz approximation
From now on we denote by B r (x, π) the disk B r (x) ∩ (x + π), where π is some linear mdimensional plane. The symbol C r (x, π), instead, will always denote the cylinder B r (x, π) × π ⊥ . If we omit the plane π we then assume that π = π 0 := R m × {0}, and the point x will be omitted when it is the origin. Let e i be the unit vectors in the standard basis. We will regard π 0 as an oriented plane and we will denote by π 0 the m-vector e 1 ∧ . . . ∧ e m orienting it. We denote by p π and p ⊥ π the orthogonal projection operators onto, respectively, π and its orthogonal complement π ⊥ . If we omit the subscript we then assume again that π = π 0 .
We will make the following We next define the following relevant quantities.
Definition 8.2 (Excess measure). For a current T as in Assumption 8.1 we define the cylindrical excess E(T, C 4r (x)), the excess measure e T and its density d T : The subscript T will be omitted whenever it is clear from the context. We define the height function of T in the cylinder C 4r (x) by Note that, since T is a representative mod(p), we have T = T p , where T p denotes the Radon measure on R m+n defined by the mass mod(p). However, it is false in general that p ♯ T (A) = Q|A|, since p ♯ T is not necessarily a representative mod(p). The excess written above can thus be rewritten as , which is the standard cylindrical excess in the classical regularity theory for area minimizing currents. Of course, since p ♯ T p ≤ p ♯ T as measures, this "excess mod(p)" is, in general, larger than the classical excess. Definition 8.3. In general, given a measure µ on a domain Ω ⊂ R m we define its noncentered maximal function as ⊂ Ω . If f is a locally Lebesgue integrable non-negative function, we denote by mf the maximal function of the measure f L m .
The first Lipschitz approximation is given by the following proposition, according to which a representative mod(p) T as in Assumption 8.1 can be realized as the graph of a Lipschitz continuous multiple valued function in regions where the maximal function of its excess measure is suitably small. As already motivated, the approximating function needs to be a special multi-valued function whenever p is even and Q = p 2 . Concerning special multi-valued functions, we will adopt the notation introduced in [7]: in particular, the space of special Q-points . Given a function u : Ω → A Q (R n ) (possibly classical, namely with target A Q (R n )), we will let Gr(u) and G u denote the set-theoretic graph of u and the integer rectifiable current associated with it, respectively; see [7,Definition 4.1]. Also, we will let osc(u) denote the quantity Remark 8.4. The definition given in (8.3) for the quantity osc(u) is the special multi-valued counterpart of the definition provided in [8] for the A Q (R n )-valued case. In [10], on the other hand, the following comparable definition for the oscillation is used: More precisely one has To see the first inequality, let x, y ∈ Ω and v ∈ spt(u(x)), w ∈ spt(u(y)); then, for any q ∈ R n we have Taking the infimum over all q ∈ R n gives the claimed inequality. For the second inequality, fix any arbitrary y ∈ Ω and q ∈ spt(u(y)). Then, for any x ∈ Ω we have Taking the supremum over all x ∈ Ω and afterwards the infimum in q ∈ spt(u(y)) gives the desired bound. Set E := E(T, C 4s (x)), let 0 < δ < 1 be such that 16 m E < δ, and define

Then, there is a Lipschitz map u defined on B 3s (x) and taking either values in
for which the following facts hold.
We remark that in Proposition 8.5 we are not assuming that T is area minimizing modulo p. The proof of the proposition will require a suitable BV estimate for 0-dimensional slices mod(p), which is the content of the next section. This Jerrard-Soner type estimate is in fact a delicate point of the present paper, since the approach of [8] (which relies on testing the current with a suitable class of differential m-forms) is unavailable in our setting, since Assumption 8.1 only guarantees ∂T C 4s (x) = 0 mod(p) and not ∂T C 4s (x) = 0.

A BV estimate for slices modulo p
Recall that F k (C) denotes the group of k-dimensional integral flat chains supported in a closed set C. Definition 9.1. We define the groups On X we define the distance function Remark 9.2. Note that the following properties are satisfied: for some S ∈ X}, the non-trivial inclusion being a consequence of [17,Corollary 4.7]. Hence, the quotient groups X/mod(p) and X p /mod(p) coincide and they are characterized by X/mod is a complete metric space; the pseudo-metric d F p induces a complete metric space structure on the quotient X p , which we still denote d F p .
In the rest of the section we will use the theory of BV maps defined over Euclidean domains and taking values in metric spaces, as established in Ambrosio's foundational paper [4].

Lemma 9.3.
Assume T is a one-dimensional integer rectifiable current satisfying Assumption 8.1 in C 4 (that is, set m = 1, x = 0 and r = 1 in Assumption 8.1), and let T t be the slice Proof. Let us first observe that since (∂T ) C 4 = 0 mod(p) then by Lemma 5.1 for a.e. t ∈ J we have Arguing analogously for the t ∈ (−4, t 0 ) and integrating allows to concludê Next, we pass to the proof of (9.1). Without loss of generality, assume I = (a, b) to be an interval with a and b Lebesgue points for Φ. It is a consequence of [15,Theorem 4.5.9] (see also [13,Section 8.1]) that |DΦ|(I) equals the classical essential variation ess var(Φ) given by with t 0 , . . . , t N Lebesgue points for Φ .
Let t 0 , . . . , t N be as in (9.5), and let e denote the constant unit 1-vector orienting R × {0} ⊂ R 1+n . Then, one has where the first inequality has been deduced analogously to (9.3), and the last one follows from p ♯ T p ≤ p ♯ T as measures. This shows (9.1) and concludes the proof.

Comparison between distances
Another delicate point in the proof of Proposition 8.5 is that Lemma 9.3 is not powerful enough to guarantee the Lipschitz continuity of the approximating map u. To that aim, we shall need to combine the Jerrard-Soner type estimate (9.1) with the result of Theorem 10.1 below.
Let Q and p be positive integers with Q ≤ p 2 , and fix any A, B ∈ A Q (R n ). Observe that A, B ∈ F 0 (R n ). Furthermore, the flat chain A − B is an element of the subgroup X of Definition 9.1, so that we can compute F(A − B). Next, let us consider the flat chain A + B. In the case when Q = p 2 , we claim that A + B ∈X p , so that we can compute F p (A + B). Indeed, fix any z ∈ R n , and let h z : (0, 1) × R n → R n be the function defined by Then, the cone over A + B with vertex z, that is the 1-dimensional integral current R given by , which proves our claim. Furthermore, the above argument also shows that Having this in mind, we extend the norm F to A + B by setting We can now state the main result of this section.

Theorem 10.1. Let p and Q be positive integers with
In order to reach a proof of Theorem 10.1, we will need some preliminary results. First, for a given S ∈ R 1 (R n ), we say that S has the property (N C) (no cycles) if there exists no 0 = R ∈ R 1 (R n ) such that ∂R = 0 and

M(S) = M(R) + M(S − R).
We recall that I m (R m+n ) denotes the space of m-dimensional integral currents in R m+n .
Given S ∈ I 1 (R n ) satisfying the property (N C), we call a good decomposition of S a writing where θ j ∈ N, each S j is the integral current given by S j = γ j for γ j a simple Lipschitz curve of finite length, S j = S k if j = k and moreover The existence of a good decomposition for a current S ∈ I 1 (R n ) satisfying the property (N C) is a direct consequence of [15, 4.2.25]. We say that a good decomposition and that has the property (N T C).
Proof. Let S ∈ I 1 (R n ), and assume without loss of generality that S = 0. Among all currents S ′ ∈ I 1 (R n ) with the property (N C) and such that ∂S ′ = ∂S and M(S ′ ) ≤ M(S), and among all possible good decompositions of S ′ not satisfying the property (N T C) fix a current S ′ and a decomposition be a function such that (10.6) holds. Define: Now, consider the quantities and In any of the two cases, ∂S ′ ± = ∂S ′ = ∂S, and the obvious resulting decomposition of S ′ ± has at most N − 1 indexes. Hence, by minimality, the one of the two which does not increase the mass necessarily has the property (N T C). This concludes the proof.
(H2) S has the property (N C) and there exists a good decomposition Proof. Let S and Z be as above. Firstly, we claim that the set of indexes where the N ℓ 's (resp. the P ℓ 's) are not necessarily distinct, so that Consider any of the points P ℓ . By (10.5), the multiplicity of ∂S in P ℓ is at least p, and furthermore, since Next, assume by contradiction that for every j such that ∂S j is supported on spt(Z) one has θ j < p 2 . Fix, for instance, the point P 1 . Arguing as above, after possibly reordering the indexes (both in the family {S j } and {N ℓ }), we conclude that there exist N 1 and S 1 such that ∂S 1 = P 1 − N 1 . Moreover, by hypothesis, θ 1 < p 2 . This ensures that we can find P 2 and S 2 such that ∂S 2 = P 2 − N 1 , and again θ 2 < p 2 . The procedure can be iterated as long as the new points P ℓ+1 (resp. N ℓ+1 ) are distinct from the previous ones. Since the decomposition of S has the property (N T C) by hypothesis (H2), this would imply that the procedure can be iterated indefinitely, which gives the desired contradiction.
Proof of Theorem 10.1. Let us first consider case (a), with σ = 1.
It suffices to prove that

7) because the other inequality is obvious.
Suppose by contradiction that and let S ∈ I 1 (R n ) and 0 = Z ∈ R 0 (R n ) be such that We claim that there exist currents S 1 ∈ I 1 (R n ) and Z 1 ∈ R 0 (R n ) such that The conclusion trivially follows from the claim.
We proceed with the proof of (10.9). First observe that if S has a cycle R then the current . Therefore, we can assume without loss of generality that S has the property (N C). Next, applying Lemma 10.2 we can also assume that S has a good decomposition where the second inequality follows from θ j 0 ≥ p 2 . Let us now consider instead case (b), when σ = −1 and Q = p 2 . We know from (10.3) that (10.10) Observe that it cannot be Z = 0. Also, by Lemma 10.2 there is no loss of generality in assuming that S admits a good decomposition In that case, if we set R := z × × (A + B) then we have In order to complete the proof, it suffices to iterate this argument producing currents S k , Z k until M(Z k ) = 1.

Proof of Proposition 8.5
Since the statement is scaling and translation invariant, there is no loss of generality in assuming x = 0 and s = 1. Consider the slices T On the other hand, setting M( , it has to be σ(x)Q ≡ Q mod(p) as integers. We therefore have to distinguish between two cases: In this case, any measurable choice of σ : K → {−1, 1} would satisfy the condition σ(x)Q ≡ Q mod(p). On the other hand 11.1. Lipschitz estimate. Fix j ∈ {1, . . . , m}, and letp j : R m+n → R m−1 be the orthogonal projection onto the (m − 1)-plane given by span(e 1 , . . . , e j−1 , e j+1 , . . . , e m ). For almost every z ∈ R m−1 , consider the one-dimensional slice T j z := T,p j , z , and observe that Observe that T j z satisfies Assumption 8.1 with m = 1 for a.e. z. Let now p j be the orthogonal projection p j : R m+n → span(e j ), and for almost every Thus, from the definition of excess measure modulo p we deduce we can conclude that By [6, Lemma 7.3], one immediately obtains On the other hand, for a.e. x ∈ K we can regard Φ( In any case, Theorem 10.1 implies that in fact where P Q denotes the group of permutations of {1, . . . , Q}. If, on the other hand, σ(x) = σ(y), and to fix the ideas say that σ(x) = 1 and σ(y) = −1, then  Note that the points (i) and (iii) of the proposition are obvious by construction. Next observe that, since me T is lower semicontinuous, K is obviously closed. Let U := {me T > δ} be its complement. Fix r ≤ 3 and for every point Now, by the definition of the maximal function it follows clearly that B x ⊂ U ∩ B r+r 0 . In turn, by the 5r covering theorem we can select countably many pairwise disjoint B x i such that the corresponding concentric ballsB i with radii 5r(x i ) cover U ∩ B r . Then we get This shows claim (iv) of the proposition and completes the proof.

First harmonic approximation
Remark 12.1 (Good system of coordinates). Let T be as in Assumption 8.1 in the cylinder C 4r (x). If the excess E = E(T, C 4r (x)) is smaller than a geometric constant, then without loss of generality we can assume that the function Ψ : R m+n → R l parametrizing the manifold Σ satisfies Ψ(0) = 0, DΨ 0 ≤ C(E 1 /2 + rA) and D 2 Ψ 0 ≤ CA. This can be shown using a small variation of the argument outlined in [8,Remark 2.5]. First of all we introduce a suitable notion of nonoriented excess. Given the plane π 0 we consider the m-vector π 0 of mass 1 which gives the standard orientation to it. We then let where | · | is the norm associated to the standard inner product on the space Λ m (R m+n ) of m-vectors in R m+n , and define Consider next the orthogonal projection p : R m+n → π 0 and the corresponding slices T, p, y with y ∈ B 4r (x). For a.e. y, such a slice is an integral 0-dimensional current and we let M(y) ∈ N be its mass. Once again (cf. (11.2)), we observe that under the Assumption 8.
Thus, an elementary computation gives At this point we find clearly a point q ∈ spt(T ) ∩ C 4r (x) such that and we can proceed with the very same argument of [8,Remark 3.5].
If the coordinates are fixed as in Remark 12.1, then the Lipschitz approximation of T provided by Proposition 8.5 corresponding to the choice δ = E 2β will be called the E β -Lipschitz approximation of T in C 3s (x).
In the following theorem, we show that the minimality assumption on the current T and the smallness of the excess imply that the E β -Lipschitz approximation of T in C 3s (x) is close to a Dirichlet minimizer h, and we quantify the distance between u and h in terms of the excess. Theorem 12.3. For every η * > 0 and every β ∈ (0, 1 2m ) there exist constants ε * > 0 and C > 0 with the following property. Let T and Ψ be as in Assumption 8.1 in the cylinder C 4s (x), and assume that T is area minimizing mod(p) in there. Let u be the E β -Lipschitz approximation of T in B 3s (x), and let K be the set satisfying all the properties of Proposition and for which the following facts hold: Proof. Let us first observe that (12.3) implies (12.4): indeed, the estimate (8.4) implies: . Then, note that we can embed A Q (R n ) naturally and isometrically into A Q (R n ) using the map T ∈ A Q (R n ) → (T, 1). Hence, without loss of generality we may assume that u takes values in A Q (R n ). Furthermore, each Lipschitz approximation is of the form u(x) = (ū(x), Ψ(x,ū)) withū taking values in A Q (Rn).
Finally, since the statement is scale invariant we may assume x = 0 and s = 1.
We will now show the following.
Given any sequence of currents T k supported in manifolds Σ k = Gr(Ψ k ) and corresponding Lipschitz approximations u k satisfying all the assumptions in B 3 with then the following conclusions hold: (ii) One of the following holds true: either there is a single Dirichlet minimzing map or there are Dirichlet minimizing maps For sufficiently large k the conclusion of the Theorem therefore holds, since we can replace in . This can be seen as follows. Recall that by remark 12.1, we have Indeed, for any sequence of non-negative measurable functions a k , b k we havê where the measurable functions C j k (x), j = 1, 2, consist of a product of two first derivatives of Ψ k , and hence C j ) are uniformly bounded by (12.8), the last two integrals are o (1).
The remaining term can be estimated bŷ 12.1. Construction of the mapsh or h j . Let ι be the isometry defined in [7, Proposition 2.6], and define (v k ,w k , η •ū k ) = ι •ū k . As in [7, Definition 2.7], we set We distinguish if the limit Case b > 0 : After translating the currents T k vertically we may assume without loss of generality that ffl Since bothv k andw k vanish on sets of measure at least b > 0, we claim that there exists a constant C = C b such that Indeed, observe that the classical Poincaré inequality giveŝ which implies (12.9) again by (12.8).
Modulo passing to an appropriate subsequence, we therefore have that . When needed, we may identifyū + k with (ū + k , 1) taking values in A Q (Rn). We note that We used in the last line Poincaré's inequality forū − k that is vanishing on a set of uniformly positive measure. Now we can apply the concentration compactness lemma, [8,Proposition 4.3], to the sequence E , A Q j (Rn)) and points y j,k ∈ Rn such that the following properties are satisfied: ) . (12.12)

Lipschitz approximation of the competitors toh and h j .
We fix a radius s < 5 2 .
To be able to interpolate later betweenh (h k ) andū k and similarly between the currents T k and G u k , by using a Fubini type argument we may fix s < t < 5 2 such that for some C > 0 where, in (12.15), f is the function defined by f (y, z) := |y| for (y, z) ∈ π 0 × π ⊥ 0 . Also, in (12.14) we identified as beforeh k with the map (h k , 1) taking values in A Q (Rn) and used (12. For each k we fix now an interpolating map Observe that by our choice of the Lipschitz approximationh ε k we havê for large k (depending on ε) . (12.16) Moreover, observe that, by construction, lim sup k→∞ Lip(h ε k ) ≤ C * ε , where C * ε is a constant depending on ε but independent of k. Also, again for large values of k (depending on our fixed ε): (12.17) where the last inequality is a consequence of the fact that E In particular we can define competitors to E We have used (12.11), the closeness of the Dirichlet energies of c j and c ε j and (12.16). As we have seen in the calculations below point (ii) above, we can use the fact that DΨ k 0 + 12.4. Interpolating Currents. By our choice of t, (12.15), and the fact that the boundary operator commutes with slicing we have Using [15, (4.2.10) ν ], we can fix an isoperimetric filling S k , which can be assumed to be representative mod(p), such that 12.5. Dirichlet minimality. We can now finally define a competitor to T k by Observe that, by the hypotheses on T k , Lemma 5.1, and the choice of S k , we have Let us observe that by construction, and using once again the Taylor expansion of the mass of a special multi-valued graph [7, Equation (13.5)], we compute: where in the last equality we have used that By minimality of T k in C 3 we then have Hence dividing by E k and taking the lim sup as k → ∞ we deduce by (12.19) Since ε is arbitrary: (i) Choosing c j = h j , we see that lim sup k→∞ E −1 k e T k (B t \ K k ) = 0; (ii) By the arbitrariness of c j we conclude the Dirichlet minimality of h j . Afterwards by (12.11) we deduce that lim sup k→∞ E −1 k Dir(u k , B t ∩ K t ) − Dir(h k , B t ) = 0. In combination with (12.12) we obtain the second part of (ii), thus completing the proof. 13. Improved excess estimate and higher integrability So far, Proposition 8.5 and Theorem 12.3 have shown that if T is as in Assumption 8.1 then there is a Lipschitz continuous multiple valued function (possibly special, in case p is an even integer and Q = p 2 ) whose graph coincides with the current in a region where the excess measure is suitably small in a uniform sense; furthermore, if T is also area minimizing mod(p) then such an approximating Lipschitz multiple valued function is almost Dirichlet minimizing, and both the Dirichlet energy of the approximating function and the excess of the original current in the "bad region" decay faster than the excess. The goal of this section is to exploit the closeness of the Lipschitz approximation to a Dir-minimizer in order to deduce extra information concerning the behavior of the excess measure of T . We begin observing that the classical result on the higher integrability of the gradient of a harmonic function extends not only to classical multiple valued functions, as it is shown in [8, Theorem 6.1], but also to special multiple valued functions.
Proof. The proof is the very same presented in [8, Theorem 6.1]: one only has to replace the Almgren embedding ξ for A Q (R n ) used in there with the new version of the Almgren embedding ζ for A Q (R n ) introduced in [7, Theorem 5.1].
As a direct corollary of the first harmonic approximation and the higher integrability of the gradient we obtain the following result. Proof. By scaling and translating we may assume without loss of generality that x = 0 and s = 1. We fix β = 1 4m and η * > 0 to be determined below. Now let ε * = ε * (β, η * ) taken from Theorem 12.3. We distinguish the following two cases: In the latter case the inequality holds trivially with C = ε −2 * because In the first case, we can apply the first harmonic approximation, Theorem 12.3. Now let h(x) = (h(x), Ψ(x,h(x))), withh Dirichlet minimizing, the associated map as in (i). By (12.3) we directly conclude that where K is, as usual, the "good set" for the E β -Lipschitz approximation of T in C 3 as in Proposition 8.5. In order to estimate the e T measure of the portion of A inside K, we observe that The first addendum can be bounded by the Taylor expansion of mass by the second can be estimated using (12.5) and |Du| 2 − |Dh| 2 = (|Du| + |Dh|)(|Du| − |Dh|) by Using the higher integrability for Dirichlet minimizers we can estimate further Collecting all the estimates we get in conclusion Hence, the estimate in (13.1) follows also in this case after suitably choosing ε and η * depending on η.
For the following proof, we introduce the centered maximal function for a general radon measure µ on R m by setting Observe that one has the straightforward comparison between the centered and non-centered maximal functions Although the two quantities are therefore comparable, we decided to work for this proof with the centered version since in our opinion the geometric idea becomes more easily accessible. Furthermore we note that since the map x → µ(Bs(x)) ωm s m is lower semicontinuous, x → m c µ(x) is lower semicontinuous as it is the supremum of a family of lower semicontinuous functions. Theorem 13.3. There exist constants 0 < q < 1, C, ε > 0 with the following property. If T is area minimzing mod(p) in the cylinder C 4 and satisfies Assumption 8.1 with E ≤ ε then In particular this implies the following estimatê Remark 13.4. Observe that the excess measure e can be decomposed as where L m denotes the Lebesgue measure in R m , e sing ⊥ L m and d is the excess density as in Definition 8.2. Since d(x) ≤ m c e(x) for every x ∈ B 2 , we havê so that formula (13.3) in particular implies the following higher integrability of the excess density:ˆ{ Proof. Let us first observe that given any measure µ on R m we have that, for any fixed r > 0 and t > 0, if then for some constant C depending on m we have This can be seen as follows: we first note that for y ∈ B r (x) we have Hence, we deduce that if s ≥ 3r then µ(Bs(y)) ωms m ≤ t: in other words, if µ(Bs(y)) ωms m > t then we must have B s (y) ⊂ B 4r (x). This implies that so that (13.5) follows by a variation of the classical maximal function estimate applied to µ B 4r (x) . 2 Furthermore we recall that by classical differentiation theory of radon measures 3 one has as well µ (B r (x) ∩ {y : m c µ(y) ≤ t}) ≤ t|B r (x) ∩ {y : m c µ(y) ≤ t}|. (13.6) In what follows, for the sake of simplicity, we will work with the measure e = e B 4 , which is defined on the whole R m . Step

Proof of
Step 1: Let η > 0 be given, and let ε > 0 be given by Corollary 13.2 in correspondence with this choice of η. Also fix λ > 4 3 m . By the definition of r and the continuity of measures along increasing and decreasing sequence of sets, we easily see that ) ω m s m for all s > r. (13.9) Thus we can apply (13.5) with µ = e, thus deducing that then we can apply Corollary 13.2, which, together with (13.9), yields (13.10) Using (13.6) and (13.9), namely the identity t λ ω m r m = e(B r (x)) we have This implies that Using this estimate in (13.10) we deduce (13.8).
Step 2: For every η > 0 there exist positive constants λ, ε, C such that if We must have 0 < r x ≤ 1 4 , since m c e(x) > t ≥ t/λ, and since for each We apply the Besicovitch covering theorem to the family Since for each of these balls we have ω m r x m = λ t e(B rx (x)) ≤ λ t Eω m 4 m , we deduce B rx (x) ⊂ Br. Hence the result follows from where we used that by Step 1 e(∂B rx (x) = 0 for each of these balls, and then applying (13.8) of Step 1 to each.
Step 3: For every η > 0 there are constants C, λ, ε such that for every k ≥ 2 with (2λ) k E ≤ ε and r ≤ . Now (13.12) is a consequence of the following estimates (λ is sufficient large) In particular, the first estimates ensures that we may apply step 2 for each pair (t l , r l ).

Almgren's strong approximation theorem
We can finally state and prove the main Lipschitz approximation result for area minimizing currents mod(p), which contains improved estimates with respect to Proposition 8.5. Theorem 14.1 (Almgren's strong approximation). There exist constants ε, γ, C > 0 (depending on m,n, n, Q) with the following property. Let T be as in Assumption 8.1 in the cylinder C 4r (x), and assume it is area minimizing mod(p). Also assume that E = E(T, C 4r (x)) < ε. Then, there are u : , and a closed set K ⊂ B r (x) such that: Lip(u) ≤ C(E + A 2 r 2 ) γ and osc (u) ≤ Ch(T, C 4r (x), π 0 ) + Cr(E 1 /2 + rA) , The key improvement with respect to the conclusions of Proposition 8.5 lies in the superlinear power of the excess in (14.4) and (14.5). In turn, this gain is a consequence of the following improved excess estimate, analogous to [8, Theorem 7.1].

Theorem 14.2 (Almgren's strong excess estimate).
There exist constants ε * , γ * , C > 0 (depending on m,n, n, Q) with the following property. Assume T satisfies Assumption 8.1 and is area minimizing mod(p) in C 4 . If E := E(T, C 4 ) < ε * , then Finally, we take any 0 < σ < 1 and we estimate: We turn now to the proof of Theorem 14.2. We will use in an essential way the minimality mod(p) of T , and in order to do that we need to construct a suitable competitor. In this process, a key role will be played by the following result, analogous to [8, Proposition 7.3] Proposition 14.3. Let β ∈ 0, 1 2m , and assume that T satisfies Assumption 8.1 and is area minimizing mod(p) in C 4 . Let u be its E β -Lipschitz approximation. Then, there exist constants ε, γ, C > 0 and a subset of radii B ⊂ [ 9 /8, 2] with measure |B| > 1 /2 with the following property. If E(T, C 4 ) < ε, then for every σ ∈ B there exists a Q-valued map  (12.15) and the isoperimetric inequality mod(p), we deduce that there exists s ∈ B and an integer rectifiable current R which is representative mod(p) such that where u is the E β -Lipschitz approximation of T and ϕ(x) = |x|. Now, let g be the Lipschitz map given in Proposition 14.3 corresponding with the choice σ = s. Since g| ∂Bs = u| ∂Bs , it also holds G u − G g , ϕ, s = 0 mod(p). Furthermore, since (∂G g ) C s = 0 mod(p), and since g takes values in Σ, the current G g C s + R is a competitor for T in C s , and thus, using [7, Equation (4.1)], the minimality of T yields for some γ > 0: On the other hand, again by [7, Equation (4.1)] we also have: Combining (14.10) and (14.11) we conclude that e T (B s \ K) ≤ CE γ (E + A 2 ). Now, we are able to prove the estimate (14.6). Let A ⊂ B9 /8 be any Borel set. We get:  and we fix good Cartesian coordinates, then there exists a Dir-minimizingh :

Strong approximation with the nonoriented excess
In this section we show that it is possible to draw the same conclusions of the previous section replacing the cylindrical excess E(T, C 4r (x)) with the nonoriented E no (T, C 4r (x)) defined in (12.2). This will be vital, because in the remaining part of the paper we will in fact use mostly the nonoriented excess, which is structurally more suited to the arguments needed in the construction of the center manifold. Recall that in the classical regularity theory for integral currents the cylindrical excess already possesses the required structural features; see [8, Remark 2.5].
Theorem 15.1. There exist constants ε, γ, C > 0 (depending on m,n, n, Q) with the following property. Let T be as in Assumption 8.1 in the cylinder C 4r (x), and assume it is area minimizing mod(p). Also assume that E = E(T, C 4r (x)) < 1 2 and that E no := E no (T, C 4r (x)) ≤ ε. Then E(T, C 2r (x)) ≤ CE no (T, C 4r (x)) + CA 2 r 2 .

(15.1) and in particular all the conclusions of Theorem 14.1 (and of Theorem 14.4, provided r
Before coming to the proof we state a simple variant of Theorem 14.1, where the estimates are inferred in a radius which is just slightly smaller than the starting one. , and a closed set K ⊂ B 4ρ (x) such that: Lip(u) ≤ C(E + r 2 A 2 ) γ/2 and osc (u) ≤ Ch(T, C 4r (x), π 0 ) + Cr(E 1 /2 + rA) , (15.3) Proof. For every point y ∈ B 4r(1−(E+r 2 A 2 ) ω ) (x) and a corresponding cylinder C y := C 4r(E+r 2 A 2 ) ω (y), note that Thus, by choosingε suitably small compared to ε in Theorem 14.1 we fall under its assumptions. In particular, we find a function u y defined on the ball B y := B r(E+r 2 A 2 ) ω (y) taking values into either A Q (R n ) or A Q (R n ) (depending on whether Q < p 2 or Q = p 2 ) and a set K y for which the following conclusions hold: . . ,N − 1 for a geometric constantc(m) > 0. We now consider for each B i = B y i the corresponding setsK i := K y i and functions u i := u y i . We next define the sets We then set K := i K i and observe that, by (o2), (o3) and (15.10), we must have Next, we find a globally defined function g on K by setting g| K i := u i K i . This function certainly enjoys the estimate Lip( If ℓ := |z − w| ≥ c(m)r(E + r 2 A 2 ) ω , we use the chain of balls B i of (o5) and remark that, thanks to the estimate on |B i \ K i |, we can guarantee the existence of intermediate points This proves that g has the global Lipschitz bound C(E + r 2 A 2 ) γ/2 on K. Furthermore, since the graph G g is mod(p) equivalent to the current T in the cylinder K × R n , we have osc(g) ≤ C h(T, C 4r (x), π 0 ), see Remark 8.4. Now we can proceed as in Proposition 8.5 or Theorem 14.1. More precisely, we write g = i (h, Ψ(·, h)) , with h : The map h satisfies Lip(h) ≤ C(E + r 2 A 2 ) γ/2 and osc(h) ≤ C h(T, C 4r (x), π 0 ). Hence, taking advantage of [  In order to show (15.1) we start observing that, by scaling and translating, we can assume x = 0 and r = 1. We then argue in several steps.
Step 1. First of all we claim that, for every δ > 0 there is ε sufficiently small such that E(T, C 3 ) < δ. Otherwise, by contradiction, there would be a sequence {T k } ∞ k=1 of area minimizing currents mod(p) satisfying the hypotheses in Assumption 8.1 in C 4 together with E(T k , C 4 ) < 1 2 for which E no (T k , C 4 ) → 0 and M p (T k C 3 ) ≥ (Q + δ)ω m 3 m . In particular, because of the uniform bound on the excess, we can assume that T k converge, up to subsequences, to a T which is an area minimizing current mod(p) and satisfies Assumption 8.1. By convergence of the M p in the interior, we also know that On the other hand, since we can assume by Proposition 4.2 that v(T k C 4 ) → v(T C 4 ) as varifolds, and since the nonoriented excess is continuous in the varifold convergence, we must have E no (T, C 4 ) = 0. Moreover, since T is a representative mod(p) we must have T (C 4 ) ≤ ω m (Q + 1 2 )4 m by the hypothesis that E(T k , C 4 ) < 1 2 for every k. The first condition implies that T is supported in a finite number of planes parallel to π 0 . By the constancy Lemma 6.4 we can assume that T is a sum of integer multiples of m-dimensional disks of radius 4 parallel to B 4 (0, π 0 ). We thus have that the sum of the moduli of such integers must be at most Q. This contradicts (15.12).
Step 2. First of all, if E := E(T, C 3 ) ≤ A 2 , then there is nothing to prove. Hence, without loss of generality assume that E ≥ A 2 .
Now apply Proposition 15.2 to obtain a Lipschitz map u : 14) Now we set r 1 := 3 − CE ω , E 1 := E(T, C r 1 ) and we consider the following three alternatives: In the first case, assuming ε sufficiently small, since C 2 ⊂ C r 1 , we have concluded our desired estimate (15.1). In the second case observe first that from the estimates above we easily conclude Consider now that, using T K × R n = G u K × R n and standard computations, we have We thus can combine these two estimates and claim In particular we easily get and again we have proved (15.1). Finally, if we are in case (c) we iterate the step above and get a Lipschitz approximation in the cylinder C r 2 where r 2 = 3 − CE ω − CE ω 1 and the new excess is E 2 := E(T, C r 2 ). We keep iterating this procedure which we stop at a certain radius Observe that as long as the procedure does not end we have the recursive property E i ≤ E i−1 2 . We can thus estimate Since ω is a fixed exponent, provided δ > E is sufficiently small (which from the first step can be achieved by choosing ε sufficiently small), we have r k ≥ 2. Thus, if the procedure stops we have proved (15.1). If the procedure does not stop, since E k → 0 we conclude easily that: (i) A = 0; (ii) If we set r ∞ := lim k→∞ r k , then 2 ≤ r ∞ and E(T, C r∞ ) = 0.
This implies that T (C r∞ ) = Qω m r m ∞ . Given that p ♯ T C r∞ = Q B r∞ (0, π 0 ) mod(p), this is only possible if the current T in C r∞ consists of a finite number of disks parallel to B r∞ (0, π 0 ) counted with integer multiplicities θ i so that i |θ i | = Q. In particular, since 2 ≤ r ∞ , obviously E(T, C 2 ) = 0 ≤ E no (T, C 4 ), which shows the validity of (15.1) even in this case.

Part 3. Center manifold and approximation on its normal bundle
This part of the paper deals with the construction of the center manifold. As it is the case with the proof of the partial regularity result for area minimizing currents in codimension higher than one, one might now attempt a proof of Theorems 3.3 and 7.2 carrying on the following program: (1) Apply Almgren's strong approximation Theorem 14.1 to construct a sequence of Lipschitz maps u k approximating T 0,r k : here, r k is the contradiction sequence of radii appearing in Proposition 7.7, and the maps u k take values in A Q (π ⊥ 0 ) or in A Q (π ⊥ 0 ) depending on whether Q < p 2 or Q = p 2 , respectively; (2) Apply Theorem 14.4 to show that, after suitable normalization, a subsequence of the u k converges to a multiple valued map u ∞ minimizing the Dirichlet energy (as in [12] if Q < p 2 or as in [7] if Q = p 2 ); (3) Use (iii) (resp. (iii)s) in Proposition 7.7 to infer that u ∞ has a singular set of positive H m−2+α measure (resp. of positive H m−1+α measure), thus contradicting the linear theory in [12] if Q < p 2 or in [7] if Q = p 2 , respectively. The obstacle towards the success of this program is making point (3) work, namely, showing that the "large" singular set of the currents persists in the limit as the approximating functions u k converge to u ∞ . As it was just stated, this is false: at this stage, nothing forces u ∞ to actually exhibit any singularities. The center manifold construction is needed precisely to address this issue: when we approximate the current from the center manifold, we "subtract the regular part" of the Dir-minimizer in the limit, which in turn allows us to close the contradiction argument.
In the first section of this part we will outline the arguments and present the statements of the main results. The subsequent sections will then be devoted to the proofs.

Preliminaries for the construction of the center manifold.
Notation 16.1 (Distance and nonoriented distance between m-planes). Throughout this part, π 0 continues to denote the plane R m × {0}, with the standard orientation given by π 0 = e 1 ∧ . . . ∧ e m . Given a k-dimensional plane π in R m+n , we will in fact always identify π with a simple unit k-vector π = v 1 ∧ . . . ∧ v k orienting it (thereby making a distinction when the same plane is given opposite orientations). By a slight abuse of notation, given two k-planes π 1 and π 2 , we will sometimes write |π 1 − π 2 | in place of | π 1 − π 2 |, where the norm is induced by the standard inner product in Λ k (R m+n ). Furthermore, for a given integer rectifiable current T , we recall the definition of | T (y) − π 0 | no from (12.1). More in general, if π 1 and π 2 are two k-planes, we can define |π 1 − π 2 | no by It is understood that |π 1 − π 2 | no does not depend on the choice of the orientations π 1 and π 2 .

Definition 16.2 (Excess and height). Given an integer rectifiable m-dimensional current
T which is a representative mod(p) in R m+n with finite mass and compact support and an m-plane π, we define the nonoriented excess of T in the ball B r (x) with respect to the plane π as The height function in a set A ⊂ R m+n with respect to π is Observe that in general the plane optimizing the nonoriented excess is not necessarily unique and h(T, B r (x), π) might depend on the optimizer π. Since for notational purposes it is convenient to define a unique "height" function h(T, B r (x)), we call a plane π as in (16.2) and ( In fact the point of (16.3) is to ensure that the planes π "optimizing the nonoriented excess" always satisfy |π − π 0 | = |π − π 0 | no .
We are now ready to formulate the main assumptions of the statements in this section.
Assumption 16.5. ε 0 ∈]0, 1] is a fixed constant and Σ ⊂ B 7 √ m ⊂ R m+n is a C 3,ε 0 (m +n)dimensional submanifold with no boundary in B 7 √ m . We moreover assume that, for each q ∈ Σ, Σ is the graph of a C 3,ε 0 map Ψ q : T q Σ ∩ B 7 √ m → T q Σ ⊥ . We denote by c(Σ) the number sup q∈Σ DΨ q C 2,ε 0 . T 0 is an m-dimensional integer rectifiable current of R m+n which is a representative mod(p) and with support in Σ ∩B 6 √ m . T 0 is area-minimizing mod(p) in Σ and moreover Θ(T 0 , 0) = Q and ∂T 0 B 6 √ m = 0 mod(p), (16.5) Here, Q is a positive integer with 2 ≤ Q ≤ ⌊ p 2 ⌋, and ε 2 is a positive number whose choice will be specified in each subsequent statement.
Constants depending only upon m, n,n and Q will be called geometric and usually denoted by C 0 . Remark 16.6. Note that (16.8) where A Σ denotes, as usual, the second fundamental form of Σ and C 0 is a geometric constant. Observe further that for q ∈ Σ the oscillation of Ψ q is controlled in T q Σ ∩ B 6 √ m by C 0 m 1 /2 0 . In what follows we set l := n−n. To avoid discussing domains of definitions it is convenient to extend Σ so that it is an entire graph over all T q Σ. Moreover we will often need to parametrize Σ as the graph of a map Ψ : R m+n → R l . However we do not assume that R m+n × {0} is tangent to Σ at any q and thus we need the following lemma. Lemma 16.7. There are positive constants C 0 (m,n, n) and c 0 (m,n, n) such that, provided ε 2 < c 0 , the following holds. If Σ is as in Assumption 16.5, then we can (modify it outside B 6 √ m and) extend it to a complete submanifold of R m+n which, for every q ∈ Σ, is the T 0 is still areaminimizing mod(p) in the extended manifold and in addition we can apply a global affine isometry which leaves R m × {0} fixed and maps Σ onto Σ ′ so that and Σ ′ is the graph of a C 3,ε 0 map Ψ : R m+n → R l with Ψ(0) = 0 and DΨ C 2,ε 0 ≤ C 0 m 1 /2 0 . From now on we assume w.l.o.g. that Σ ′ = Σ. The next lemma is a standard consequence of the theory of area-minimizing currents (we include the proofs of Lemma 16.7 and Lemma 16.8 in Section 17 for the reader's convenience).

Construction of the center manifold.
From now we will always work with the current T of Lemma 16.8. We specify next some notation which will be recurrent in the paper when dealing with cubes of π 0 . For each j ∈ N, C j denotes the family of closed cubes L of π 0 of the form where 2 ℓ = 2 1−j =: 2 ℓ(L) is the side-length of the cube, a i ∈ 2 1−j Z ∀i and we require in addition −4 ≤ a i ≤ a i + 2ℓ ≤ 4. To avoid cumbersome notation, we will usually drop the factor {0} in (16.13) and treat each cube, its subsets and its points as subsets and elements of R m . Thus, for the center x L of L we will use the notation x L = (a 1 + ℓ, . . . , a m + ℓ), although the precise one is (a 1 + ℓ, . . . , a m + ℓ, 0, . . . , 0). (w2) the interiors of any pair of distinct cubes L 1 , L 2 ∈ W are disjoint; (w3) if L 1 , L 2 ∈ W have nonempty intersection, then 1 2 ℓ(L 1 ) ≤ ℓ(L 2 ) ≤ 2 ℓ(L 1 ). Observe that (w1) -(w3) imply sep (Γ, L) := inf{|x − y| : x ∈ L, y ∈ Γ} ≥ 2ℓ(L) for every L ∈ W . (16.14) However, we do not require any inequality of the form sep (Γ, L) ≤ Cℓ(L), although this would be customary for what is commonly called a Whitney decomposition in the literature. The algorithm for the construction of the center manifold involves several parameters which depend in a complicated way upon several quantities and estimates. We introduce these parameters and specify some relations among them in the following As we can see, β 2 and δ 2 are fixed. The other parameters are not fixed but are subject to further restrictions in the various statements, respecting the following "hierarchy". As already mentioned, "geometric constants" are assumed to depend only upon m, n,n and Q. The dependence of other constants upon the various parameters p i will be highlighted using the notation C = C(p 1 , p 2 , . . .). (e) ε 2 is smaller than c(β 2 , δ 2 , M 0 , N 0 , C e , C h ) (which will always be positive).
The functions C and c will vary in the various statements: the hierarchy above guarantees however that there is a choice of the parameters for which all the restrictions required in the statements of the next propositions are simultaneously satisfied. To simplify our exposition, for smallness conditions on ε 2 as in (e) we will use the sentence "ε 2 is sufficiently small".
Thanks to Lemma 16.8, for every L ∈ C , we may choose y L ∈ π ⊥ 0 so that p L := (x L , y L ) ∈ spt(T ) (recall that x L is the center of L). y L is in general not unique and we fix an arbitrary choice. A more correct notation for p L would be x L + y L . This would however become rather cumbersome later, when we deal with various decompositions of the ambient space in triples of orthogonal planes. We thus abuse the notation slightly in using (x, y) instead of x + y and, consistently, π 0 × π ⊥ 0 instead of π 0 ⊕ π ⊥ 0 . Definition 16.12 (Refining procedure). For L ∈ C we set r L := M 0 √ m ℓ(L) and B L := B 64r L (p L ). We next define the families of cubes S ⊂ C and W = W e ∪ W h ∪ W n ⊂ C with the convention that S j = S ∩ C j , W j = W ∩ C j and W j = W ∩ C j for = h, n, e. We define W i = S i = ∅ for i < N 0 . We proceed with j ≥ N 0 inductively: if no ancestor of L ∈ C j is in W , then h but it intersects an element of W j−1 ; if none of the above occurs, then L ∈ S j . We finally set (16.17) Observe that, if j > N 0 and L ∈ S j ∪ W j , then necessarily its father belongs to S j−1 .
We will prove Proposition 16.13 in Section 18. Next, we fix two important functions ϑ, ̺ : R m → R. 17 16 ] m , [0, 1] is identically 1 on [−1, 1] m . ̺ will be used as convolution kernel for smoothing maps z defined on m-dimensional planes π of R m+n . In particular, having fixed an isometry A of R m onto π, the smoothing will be given by Observe that since ̺ is radial, our map does not depend on the choice of the isometry and we will therefore use the shorthand notation z * ̺ λ . Definition 16.15 (π-approximations). Let L ∈ S ∪ W and π be an m-dimensional plane. If T C 32r L (p L , π) fulfills the assumptions of Theorem 15.1 in the cylinder C 32r L (p L , π), then the resulting map u given by the theorem, which is defined on B 8r L (p L , π) and takes values is called a π-approximation of T in C 8r L (p L , π). The mapĥ : B 7r L (p L , π) → π ⊥ given byĥ := (η • u) * ̺ ℓ(L) will be called the smoothed average of the π-approximation. Definition 16.16 (Reference plane π L ). For each L ∈ S ∪ W we letπ L be an optimal plane in B L (cf. Definition 16.3) and choose an m-plane π L ⊂ T p L Σ which minimizes |π L − π L |.
The following lemma, which will be proved in Section 18, deals with graphs of multivalued functions f in several systems of coordinates.
For the sake of simplicity, in the future we will sometimes regard g L as a map g L : B 4r L (x L , π 0 ) → π ⊥ 0 rather than as a map g L : B 4r L (p L , π 0 ) → π ⊥ 0 . In particular, we will sometimes consider g L (x) with x ∈ B 4r L (x L , π 0 ) even though the correct writing is the more cumbersome g L ((x, y L )). Definition 16.18 (Interpolating functions). The maps h L and g L in Lemma 16.17 will be called, respectively, the tilted L-interpolating function and the L-interpolating function. For each j let P j := S j ∪ j i=N 0 W i and for L ∈ P j define ϑ L (y) := ϑ( y−x L ℓ(L) ). Set (16.21) letφ j (y) be the firstn components ofφ j (y) and define ϕ j (y) := φ j (y), Ψ(y,φ j (y)) , where Ψ is the map of Lemma 16.7. ϕ j will be called the glued interpolation at the step j.  The following is then a corollary of Theorem 16.19 and the construction algorithm; see Section 20 for the proof. (iii) T, p, q = Q q for every q ∈ Φ(Γ).
The next main goal is to couple the center manifold of Theorem 16.19 with a good approximating map defined on it. is the normal part of F . In the definition above it is not required that the map F approximates efficiently the current outside the set Φ Γ ∩ [− 7 2 , 7 2 ] m . However, all the maps constructed will approximate T with a high degree of accuracy in each Whitney region: such estimates are detailed in the next theorem, the proof of which will be tackled in Section 20. Moreover, for any a > 0 and any Borel V ⊂ L, we have (for

Separation and domains of influence of large excess cubes.
We now analyze more in detail the consequences of the various stopping conditions for the cubes in W . We first deal with L ∈ W h .

Proposition 16.26 (Separation).
There is a constant C ♯ (M 0 ) > 0 with the following property. Assume the hypotheses of Theorem 16.24 and in addition C 2m h ≥ C ♯ C e . If ε 2 is sufficiently small, then the following conclusions hold for every L ∈ W h : A simple corollary of the previous proposition is the following. (a) L 0 ∈ W e and L i ∈ W n for all i > 0; We use this last corollary to partition W n .
Definition 16.28 (Domains of influence). We first fix an ordering of the cubes in W e as {J i } i∈N so that their sidelengths do not increase. Then H ∈ W n belongs to W n (J 0 ) (the domain of influence of J 0 ) if there is a chain as in Corollary 16.27 with L 0 = J 0 . Inductively, W n (J r ) is the set of cubes H ∈ W n \ ∪ i<r W n (J i ) for which there is a chain as in Corollary 16.27 with L 0 = J r .
16.5. Splitting before tilting. The following proposition contains a "typical" splittingbefore-tilting phenomenon: the key assumption of the theorem (i.e. L ∈ W e ) is that the excess does not decay at some given scale ("tilting") and the main conclusion (16.30) implies a certain amount of separation between the sheets of the current ("splitting"); see Section 21 for the proof.
, if the hypotheses of Theorem 16.24 hold and if ε 2 is chosen sufficiently small, then the following holds. If L ∈ W e , q ∈ π 0 with dist(L, q) ≤ 4 √ m ℓ(L) and Ω = Φ(B ℓ(L)/4 (q, π 0 )), then (with C, C 3 = C(β 2 , δ 2 , M 0 , N 0 , C e , C h )): 16.6. Persistence of multiplicity Q points. We next state two important properties triggered by the existence of q ∈ spt(T ) with Θ(T, q) = Q, both related to the splitting before tilting. Their proofs will be discussed in Section 22.  (ii) for the center manifold M ′ of T ′ relative to π and the M ′ -normal approximation N ′ as in Theorem 16.24, we havê

Height bound and first technical lemmas
We can now discuss the proofs of the main results outlined in the previous section. We begin with a mod(p) version of the sheeting lemma appearing in [10, Theorem A.1].
Remark 17.2. The proof that we are going to present is substantially different from the one in [10, Theorem A.1], and it could be easily adapted to the case of area minimizing integral currents as well. The statement above is sufficient for our purposes; nonetheless, the proof is actually going to give us more. In particular, in dimension m ≥ 3 the result holds with a better estimate on the bandwidth of the various S i , namely with σ 2m . In dimension m = 2, the proof below also produces the height bound with the optimal estimate featuring σ = O(E 1 /2 ), but only in the cylinder C r 2 (x 0 ). Proof. In the rest of the proof we denote by p the orthogonal projection onto π 0 = R m × {0}. The last part of the statement, where E is replaced with E no follows from Theorem 15.1. Moreover, we assume x 0 = 0 and r = 1 after appropriate translation and rescaling. We also observe, as in the proof of [10, Theorem A.1] that (iii) is a corollary of the interior monotonicity formula (the only ingredients of the argument in there are the stationarity of the varifold induced by T i := T (C ρ ∩ S i ) and the inequality M(T i ) ≤ ω m ρ m (|Q i | + E)).
We therefore focus on (i) and (ii) and since the case Q < p 2 is entirely analogous, for the sake of simplicity we assume Q = p 2 . We first prove (i). We start by considering an approximation as in Proposition 15.2. We thus find an exponent ω > 0 (which depends only on Q, m and n), a Lipschitz map u : B 1−(E+A 2 ) ω → A Q (R n ) and a K ⊂ B 1−(E+A 2 ) ω with the following properties: . We consider first the case m > 2. Recall the Poincaré inequality and find a point 1 m }, whereC is a constant which will be later chosen sufficiently large. Using (17.1) and Chebyshev's inequality, we easily conclude In particular, for any fixedη, ifC is chosen large enough, we reach the estimate Consider now the setK := K ∩ K * and observe that, by choosing ε sufficiently small, we reach To fix ideas assume now that T 0 = ( J j=1 k j p j , 1), where the p j 's are pairwise distinct and all k j are positive. Let spt(T 0 ) = {p 1 , . . . p J }. From (ii) and the definition ofK, it follows easily that dist(spt(T 0 ), p ⊥ (spt ( T, p, x)) ≤C(E + A 2 ) 1 m for x ∈K. Define thus the sets , and by the monotonicity formula T (B where c 0 is a geometric constant. This is however incompatible with (17.5) as soon as 2η is chosen smaller than c 0 , thus showing that spt(T ) ∩ C 1−2 (E+A 2 ) ω ⊂ U ′ . We can now subdivide U ′ in a finite number of disjoint stripes S i of widthC(E + A 2 ) 1 m , whereC is larger thanC by a factor which depends only on Q. This shows therefore the claim (i) of the theorem when m > 2.
The case m = 2 is slightly more subtle. Observe first that |Du| 2 ≤ min{m c e, 1} and hence we can use the same argument as in the proof of Theorem 13.3 to achievê The subtlety is in losing at most (E + A 2 ) ω in the radius of the ball; as usual, the price to pay is a slightly worse estimate, cf. (17.6) with (13.3). Since |B 1−(E+A 2 ) ω \ K| ≤ E 1+ω , if we choose q small enough we easily reach the estimate In particular, if we set in this case K * := {x ∈ B 1−(E+A 2 ) ω : G s (u(x), T 0 ) ≤C(E + A 2 ) 1 4 } then from Morrey's embedding follows that K * = B 1−(E+A 2 ) ω , providedC is chosen large enough. (17.3) is thus trivially true and the rest of the argument remains unchanged.
We now come to claim (ii). By the constancy theorem, it is easy to see that for some integer Q i ∈ {−(Q − 1), . . . , −1, 0, 1, . . . , Q}. However, recall that for x ∈K: • the support S of the current Z i (x) := T, p, x C 1−2(E+A 2 ) ω ∩ S i consists of at most Q points; • either all points in S have positive integer multiplicity, or they all have negative integer multiplicity; We thus conclude that each Q i is nonzero and that |Q i | = M(Z i (x)). Now, since M( T, p, x ) = Q, we must have |Q i | = Q. On the other hand Hence i Q i = Q mod(p). Hence we conclude that either all Q i 's are positive or they are all negative.
The proposition is a simple geometric observation, and its proof is left to the reader.
Now, both T 0 (q) and − T 0 (q) orient a plane contained in T q Σ. We can thus apply Proposition 17.3 provided m 0 is sufficiently small. From it we conclude that p TqΣ (π 0 ) is an m-dimensional plane with |p TqΣ (π 0 )−π 0 | ≤ C 0 m 1 /2 0 . From this inequality we then conclude following literally the final arguments of [10, Proof of Lemma 1.5].
Proof of Lemma 16.8. We follow the proof of [10,Lemma 1.6] given in [10,Section 4]. First of all we notice that, once (16.10) and (16.11) are established, (16.12) follows from Theorem 17.1, since we clearly have that E no (T, C 11 √ m/2 , π 0 ) ≤ CE no (T 0 , B 6 √ m , π 0 ). Moreover, recall that there is a set of full measure A ⊂ B 5 √ m such that T, p π 0 , x is an integer rectifiable current for every x ∈ A. For any such x we have thus T, is a finite collection of points and each k i (x) is an integer. In particular we must have i k i (x) = Q mod(p) and since 1 ≤ Q ≤ p 2 , at least one k i (x) must be nonzero, which means in turn that spt( T, p π 0 , x ) = ∅. Hence we conclude that spt(T )∩p −1 π 0 (x) = ∅ for every x ∈ A, and by the density of A we conclude that spt(T ) ∩ p −1 π 0 (x) = ∅ for every x ∈ B 5 √ m . We next come to (16.10) and (16.11). As in the proof of [10, Lemma 1.6], we argue by contradiction and assume that one among (16.10) and (16.11) fails for a sequence T 0 k of currents which satisfy Assumption 16.5 with ε 2 = ε 2 (k) ↓ 0. The compactness property given by Proposition 4.2 ensures the existence of a subsequence, not relabeled, converging to a current T 0 ∞ in the F p K norm for every compact K ⊂ B 6 √ m : in fact Proposition 4.2 ensures also that T 0 ∞ is area minimizing mod(p) in a suitable (m +n)-dimensional plane (the limit of the Riemannian manifolds Σ k ) and that the varifolds induced by T 0 k converge to the varifold induced by T 0 ∞ . In particular, ∂T 0 ∞ = 0 mod(p) in B 6 √ m and the tangent plane to T 0 ∞ is parallel to π 0 T 0 ∞ -almost everywhere. Observe that by upper semicontinuity of the density, (16.5) implies that 0 is a point of density Q for T 0 ∞ . On the other hand (16.6) Hence, by the monotonicity formula, T 0 ∞ must be a cone. Observe that if q ∈ spt(T 0 ∞ ) is a point where the approximate tangent space π q exists, since T 0 ∞ is a cone, we must have that q ∈ π q . Thus q ∈ π 0 for T 0 ∞ -a.e. q, which in turn implies that spt(T 0 ∞ ) ⊂ π 0 . In conclusion √ m mod(p), and moreover the varifold convergence holds in the whole R m+n . Again by the monotonicity formula, spt(T 0 k ) is converging locally in the sense of Hausdorff to spt(T 0 ∞ ). In particular if we set T k := T 0 k B 23 √ m/4 , for k large T k will have no boundary mod(p) in C 11 √ m/2 . Hence it must be (16.11) which fails for an infinite number of k's. On the other hand we certainly have (p π 0 ) ♯ T k C 11 √ m/2 = Q k B 11 √ m/2 mod(p). Notice that by the varifold convergence we have T 0 k (C 11 √ m/2 \B 23 √ m/4 ) → 0 as k → ∞. In particular the limit of the currents (p π 0 ) ♯ T k C 11 √ m/2 is the same as the limit of the currents (p π 0 ) ♯ T 0 and thus it must be Q k = Q mod(p) for k large enough.
18. Tilting of planes and proof of Proposition 16.13 Following [10], the first important technical step in the proof of the existence of the center manifold is to gain a control on the tilting of the optimal planes as the cubes get refined. The following proposition corresponds to [ In particular, the conclusions of Proposition 16.13 hold.
Proof. First of all we observe that, if we replace (ii), (iii) and (iv) with (iv)no |π H − π 0 | no ≤Cm 1 /2 0 , then the arguments given in the [10, Proof of Proposition 4.1] can be followed literally with minor adjustments. Indeed those arguments depend only on: • the monotonicity formula; • the triangle inequality |α − γ| ≤ |α − β| + |β − γ|; • the elementary geometric observation that, for every subset E and every pair of mplanes α and β, we have the inequality where C is a geometric constant.
However, it can be easily verified that all such properties remain true if we replace | · | with | · | no . We next come to (ii), (iii) and (iv). First observe that both π H and the (oriented) mplane with the same support and opposite orientation belong to T p H Σ. For this reason, the definition of π H implies that |π H −π H | no = |π H −π H |, thus allowing us to infer (ii) from (ii)no.
Next, recall that we have |π H − π 0 | = |π H − π 0 | no , cf. Remark 16.4. Hence (iv)no implies (iv). Now, combining (iv) for two planes H and L as in statement (iii) of the proposition, we conclude that |π H − π L | ≤ |π H − π 0 | + |π L − π 0 | ≤ Cm 1 /2 0 . Hence, again assuming that ε 2 is sufficiently small, we can apply Proposition 17.3, in particular conclusion (Eq): Arguing as in [10,Section 4.3] we get the following existence theorem with very minor modifications (the only adjustment that needs to be taken into consideration is that the identities [10, (4.9)], [10, (4.10)] and the subsequent analogous ones must be replaced with the same equalities mod(p)): . Then, (i) for π = π H ,π H , (p π ) ♯ T C 32r L (p L , π) = Q B 32r L (p L , π)) mod(p) and T satisfies the assumptions of 15.1 in the cylinder C 32r L (p L , π); (ii) Let f HL be the π H -approximation of T in C 8r L (p L , π H ) and h HL := (η • f HL ) * ̺ ℓ(L) be its smoothed average. Set κ H := π ⊥ H ∩ T p H Σ and consider the maps Then there is a smooth g HL :

The key construction estimates
Having at disposal the Existence Proposition 18.2 we can now come to the main estimates on the building blocks of the center manifold, which in fact correspond precisely to [10,Proposition 4.4] and are thus restated here only for the reader's convenience.  The only adjustement needed is in the argument for claim (iii). Following the one of [10, Section 6.1] we conclude that at every q ∈ Φ(Γ), if we denote by π the oriented tangent plane to M at q, then the current Q π is the unique tangent mod(p) of T at q, in the sense of Corollary 6.3. We then can use Proposition 4.2 to conclude that Θ(T, q) = Q.
For Theorem 16.24 we can repeat the arguments of [10, Section 6.2] in order to prove the existence of the M-normal approximation and the validity of (16.22) and (16.23). As for (16.25) we can repeat the arguments of [10,Section 6.3], whereas in order to get (16.24) we make the following adjustments to the first part of [10,Section 6.3]. The paragraphs leading to [10,Eq. (6.11)] are obviously valid in our setting. However [10,Eq. (6.11)] must be replaced with the following analogous estimatê We next come to the proof of Proposition 16.29. A first important ingredient is the Unique continuation property of [10, Lemma 7.1], which we will now prove it is valid for A Q (R n ) minimizers as well. Proof. We follow partially the argument of [10, Section 7.2] for [10, Lemma 7.1]. In particular, the second part of the argument, which reduces the statement to the following claim, can be applied with no alterations: (UC) if Ω is a connected open set and w ∈ W 1,2 (Ω, A Q (R n )) is Dir-minimizing in any every bounded Ω ′ ⊂⊂ Ω, then either w is constant or´J |Dw| 2 > 0 for every nontrivial open J ⊂ Ω. However, the proof given in [10, Section 7.1] of (UC) when w ∈ W 1,2 (Ω, A Q (R n )) cannot be repeated in our case, since it uses heavily the fact that the singular sets of A Q (R n )-valued Dir-minimizers cannot disconnect the domain, a property which is not enjoyed by A Q (R n )valued Dir-minimizers. We thus have to modify the proof somewhat, although the tools used are essentially the same.
Assume by contradiction that there are a connected open set Ω ⊂ R m , a map w ∈ W 1,2 loc (Ω, A Q (R n )) and a nontrivial open subset J ⊂ Ω such that (a) w is Dir-minimizing on every open Ω ′ ⊂⊂ Ω; (b) w is not constant, and thus´Ω ′ |Dw| 2 > 0 for some Ω ′ ⊂⊂ Ω; (c)´J |Dw| 2 = 0. Observe first that, from the classical unique continuation of harmonic functions, either η • w is constant, or it has positive Dirichlet energy on any nontrivial open subset of Ω. Since however the Dirichlet energy of η • w is controlled from above by that of w, (c) excludes the second posssibility. Thus η • w is constant and hence, without loss of generality, we can assume η • w ≡ 0.
Next assume, without loss of generality, that J is connected. Clearly, w is constantly equal to some P ∈ A Q (R n ) on J. Since, without loss of generality, we could "flip the signs of the Dirac masses" which constitute the values of u, we can always assume that P = ( i P i , 1). We then distinguish two cases.
First Case. The diameter of spt(P ) is positive, namely |P i − P j | > 0 for some i = j. In this case consider the interior U of the set {w = P }. We want to argue that U = Ω, which contradicts (b).
Since Ω is open and connected, it suffices to show that ∂U ∩ Ω = ∅. In order to show this, consider a point x ∈ ∂U . If x ∈ Ω, using the continuity of the map w, we know that in a sufficiently small ball B ρ (x) there is an A Q (R n )-valued map z such that w(y) = (z(y), 1) for all y ∈ B ρ (x). As such, z must be a Dir-minimizer to which we can apply [10, Section 7.2]: since´J ′ |Dz| 2 = 0 for some nontrivial open J ′ ⊂ B ρ (x), we must have that z is constant on B ρ (x). But then we would have B ρ (x) ⊂ U , thus contradicting the assumption that x ∈ ∂U .
Second Case. The remaining possibility is that P = Q η • w(x) = Q 0 (which equals both (Q 0 , 1) and (Q 0 , −1), since the latter points are identified in A Q (R n )). Define therefore and notice thatK ⊂ K since w is continuous. The setK is necessarily nonempty. If it were empty, we could first apply the classical characterization of Federer of sets of finite perimeter, cf. [15,Theorem 4.5.11], to infer that K is a set of finite perimeter, and subsequently we could then apply the classical structure theorem of De Giorgi to conclude that, since the reduced boundary of K would be empty, D1 K = 0. The latter would imply that 1 K is constant on the connected set Ω, namely that Ω \ K has zero Lebesgue measure, which in turn would contradict (b). Fix a point x ∈K. Clearly it must be´B ρ (x) |Dw| 2 > 0 for every ρ > 0, otherwise w would be constant in a neighborhood of x and thus x would be an interior point of K. Denoting I x,w (·) the frequency function of w at x as in [7, Define then the maps y → w r (y), whose positive and negative parts are given by and observe that a subsequence of {w r k } k∈N , not relabeled, is converging to a nontrivial w 0 ∈ W 1,2 loc (R m , A Q (R n )) which minimizes the Dirichlet energy on every Ω ′ ⊂⊂ R m and is I 0 -homogeneous.
Next define the sets K r k := r −1 k (K − x), where the maps w r k vanish identically, and observe that, by (21.1), lim inf k |K r k ∩ B 1 | > 0. Since the sets K r k ∩ B 1 are compact we can, without loss of generality, assume that they convergence in the sense of Hausdorff to some set K 0 . The limiting map w 0 vanishes on such set because the w r k are converging locally uniformly to w 0 . On the other hand it is elementary to see that the Lebesgue measure is upper semicontinuous under Hausdorff convergence and we thus conclude |K 0 | > 0.
We can now repeat the procedure above on some point y = 0 where the Lebesgue density of K 0 does not exist or it is neither zero nor one. We find thus a corresponding tangent function w 1 that has all the properties of w 0 , namely • it is nontrivial, • it vanishes identically on a set of positive measure, • it is I 1 -homogeneous for some positive constant I 1 , • and it minimizes the Dirichlet energy on any bounded open set. In addition w 1 is invariant under translations along the direction y |y| . Assuming, after rotations, that such vector is e m = (0, 0, . . . , 0, 1), the function w 1 depends therefore only on the variables x 1 , . . . , x m−1 and can thus be treated as a function defined over R m−1 . Iterating m − 2 more times such procedure we achieve finally a function w m−1 : R → A Q (R n ) with the following properties: (A) w m−1 is identically Q 0 on some set of positive measure; Observe that, because of (B), at least one of the c i 's is nonzero. Therefore ε cannot be equal to 1, otherwise w m−1 would give an A Q (R n )-valued Dir-minimizer on the real line with a singularity, which is not possible. However, since (Q 0 , 1) = (Q 0 , −1), if ε equals −1 we reach precisely the same contradiction. This completes the proof.
We keep following the strategy of [10, Section 7.2] towards a proof of Proposition 16.29. First of all, we introduce some useful notation. Definition 21.2. Let w : E → A Q (R n ), let E + , E − and E 0 be the canonical decomposition of E induced by w and let w + , w − and η • w the corresponding maps, as in [7,Definition 2.7]. For any f : We next show that if the energy of an A Q (R n )-valued Dir-minimizer w does not decay appropriately, then the map must "split", in other wordsw cannot be too small compared to η • w. As in [10, Section 7.2], we fix λ > 0 such that Then, if we letw be as in Definition 21.2, the following holds: The proof of [10, Proposition 7.2] can be literally followed for our case using the Unique continuation Lemma 21.1 in combination with the next simple algebraic computation (which is the counterpart of [10, Lemma 7.3]).

Lemma 21.4.
Let B ⊂ R m be a ball centered at 0, w ∈ W 1,2 (B, A Q (R n )) Dir-minimizing andw as in Definition 21. 2 We then have The detail of the necessary modifications to the argument in [10, Proof of Proposition 7.2] towards proving Proposition 21.3 are left to the reader; we will instead show how to prove the lemma above.
Proof. Let u := η • w and observe that it is harmonic. Thus, using the mean value property of harmonic functions and a straightforward computation we get On the other hand, using again the mean value property of harmonic functions, it is easy to see that In particular, we get (0) Next, a simple algebraic computations showŝ  1 2 [ such that, for every s <s, there existsε(s, C * ,δ) > 0 with the following property. If T is as in Theorem 15.1, E no := E no (T, C 4 r (x)) <ε, r 2 A 2 ≤ C ⋆ E no and Θ(T, (p, q)) = Q at some (p, q) ∈ C r/2 (x), then the approximation f of Theorem 14.1 satisfieŝ it is easy to see that the proof is in fact valid for stationary varifolds and as such can be applied to mod(p) area-minimizing currents. We formulate the precise theorem here for the reader's convenience.

Lemma 22.2.
There is a constant C depending only on m, n andn with the following property. If Σ ⊂ R m+n is a C 2 (m +n)-dimensional submanifold with A Σ ∞ ≤ A, U is an open set in R m+n and V an m-dimensional integral varifold supported in Σ which is stationary in Σ ∩ U , then for every ξ ∈ Σ ∩ U the function ρ → exp(CA 2 ρ 2 )ρ −m V (B ρ (ξ)) is monotone on the interval ]0,ρ[, whereρ := min{dist(x, ∂U ), (CA) −1 }. Remark 22.3. The proof of Theorem 22.1 can also be given following the alternative argument of Spolaor in [23], which uses the Hardt-Simon inequality and the classical version by Allard of Moser's iteration for subharmonic functions on varifolds. While Spolaor's argument is more flexible and indeed works for integral currents which are not minimizing but sufficiently close to minimizing ones in a suitably quantified way, we prefer to adhere to the strategy of [8] because it is more homogeneous to our notation and terminology.

Proof of Proposition 16.32
The proof follows the one of [10,Proposition 3.7] given in [10, Section 9] with minor modifications. The necessary tools used there, namely the splitting before tilting Propositions, the height bound and the reparametrization theorem are all available from the previous sections.

Intervals of flattening
Our argument for Theorem 3.3 is by contradiction, and we start therefore fixing a current T , a submanifold Σ, an open set Ω, an integer 2 ≤ Q ≤ p 2 , positive reals α and η and a sequence r k ↓ 0 of radii as in Proposition 7.7. In this section we proceed as in [11,Section 2] and define appropriate intervals of flattening ]s j , t j ], which are intervals over which we will construct appropriate center manifolds. These intervals, which will be ordered so that t j+1 ≤ s j will satisfy several properties, among which we anticipate the following fundamental one: aside from finitely many exceptions, each radius r k belongs to one of the intervals. In particular, if they are finitely many, then 0 is the left endpoint of the last one, whereas if they are infinitely many, then t j ↓ 0. The definition of these intervals is taken literally from [11, Section 2.1], the only difference being that we take advantage of Theorem 16.19 in place of [10,Theorem 1.17]. However we repeat the details for the reader's convenience.
Without loss of generality we assume that B 6 √ m (0) ⊂ Ω, and we fix a small parameter ε 3 ∈]0, ε 2 [, where ε 2 is the constant appearing in (16.8) of Assumption 16.5. Then, we take advantage of Proposition 7.7 and of a simple rescaling argument to assume further that: Observe that {0} ∪ R is a closed set and that, since E no (T, B 6 √ mr k ) → 0 as k ↑ ∞, r k ∈ R for k large enough.
The intervals of flattening will form a covering of R. We first define t 0 as the maximum of R. We then define inductively s 0 , . . . , t j , s j in the following way.
Let us first assume that we have defined t j and we wish to define s j (in particular this part is applied also with j = 0 to define s 0 ). We first consider the rescaled current T j := ((ι 0,t j ) ♯ T ) B 6 √ m , Σ j := ι 0,t j (Σ)∩B 7 √ m ; moreover, consider for each j an orthonormal system of coordinates so that, if we denote by π 0 the m-plane R m × {0}, then E no (T j , B 6 √ m , π 0 ) = E no (T j , B 6 √ m ) (alternatively we can keep the system of coordinates fixed and rotate the currents T j ). Definition 24.1. We let M j be the corresponding center manifold constructed in Theorem 16.19 applied to T j and Σ j with respect to the m-plane π 0 . The manifold M j is then the graph of a map ϕ j : π 0 ⊃ [−4, 4] m → π ⊥ 0 , and we set Φ j (x) := (x, ϕ j (x)) ∈ π 0 × π ⊥ 0 . We then let W (j) be the Whitney decomposition of [−4, 4] m ⊂ π 0 as in Definition 16.9, applied to T j . We denote by p j the orthogonal projection on the center manifold M j , which, given the C 3,κ estimate on ϕ j , is well defined in a "slab" U j of thickness 1 as defined in point (U) of Assumption 16.21.
Next we distinguish two cases: (Go) For every L ∈ W (j) , ℓ(L) < c s dist(0, L) , (24.5) For every interval of flattening I j =]s j , t j ] ∈ F, we let N j be the normal approximation of T j on the center manifold M j of Thereom 16.24. As in [11,Section 3] we introduce the corresponding frequency functions and state the main analytical estimate, which allows us to exclude infinite order of contact of the normal approximations with the center manifolds M j .
Definition 25.1 (Frequency functions). For every r ∈]0, 3] we define: where d j (q) is the geodesic distance on M j between q and Φ j (0), and dq is short for dH m (q).
If H j (r) > 0, we define the frequency function I j (r) := (25.1) To simplify the notation, in this section we drop the index j and omit the measure H m in the integrals over regions of M. The proof exploits four identities collected in Proposition 25.4, which is the analog of [11,Proposition 3.5] and whose proof will be discussed in the next sections. Following [11,Section 3] we introduce further auxiliary functions in order to express derivatives and estimates on the functions D, H and I. We also remind the reader that in principle we must distinguish two situations: • If Q < p 2 , then the normal approximations are A Q (R m+n )-valued maps and thus all the quantities considered here coincide literally with the ones defined in [11, Section 3]; • If Q = p 2 , then the normal approximations take values in A Q (R m+n ); in this case we use the notational conventions of [7, Subsection 7.1] and thus, although at the formal level the definitions of the various objects are identical, the notation is underlying the fact that all integrals involved in the computations must be split into three domains to be reduced to integrals of expressions involving the A Q (R m+n )-valued maps N + , N − and Q η • N . Definition 25.3. We let ∂r denote the derivative with respect to arclength along geodesics starting at Φ(0). We set As in [11,Section 3] we observe that the estimate Theorem 25.2 follows from the latter four estimates and from (25.4) through the computations given in [11,Section 3]. The proofs of the estimates (25.5) and (25.8) given in [11,Section 3] are valid in our case as well, since they do not exploit the connection between the approximation and the currents, but they are in fact valid for any map N satisfying I ≥ 1. We therefore focus on (25.6) and (25.7) which are instead obtained from first variation arguments applied to the area minimizing current T j . In our case the current is area minimizing mod(p), however a close inspection of the proofs in [11] shows that the computations in there can be transferred to our case because the varifold induced by T j is stationary (and the required estimates relating the varifold induced by the graph of N j in the normal bundle of M j and the current T j have been proved in the previous section).
In the rest of the section we omit the subscript j from T j , Σ j , M j and N j .

First variations.
We recall the vector field used in [11]. We will consider: Note that X i is the infinitesimal generator of a one parameter family of bilipschitz homeomorphisms Φ ε defined as Φ ε (q) := Ψ ε (p(q)) + q − p(q), where Ψ ε is the one-parameter family of bilipschitz homeomorphisms of M generated by Y . Consider now the map F (q) := i q + N i (q) and the current T F associated to its image: in particular we are using the conventions of [9] in the case Q < p 2 (i.e. when N takes values in A Q (R m+n )) and the conventions introduced in [7, Definition 11.2] in the case Q = p 2 (i.e. when N takes values in A Q (R m+n )). As in [11,Section 3.3] we observe that, although the vector fields X = X o and X = X i are not compactly supported, it is easy to see that δT (X) = δT (X T ) + δT (X ⊥ ) = δT (X ⊥ ), where X = X T + X ⊥ is the decomposition of X in the tangent and normal components to T Σ.
Then, we have . (25.9) In order to simplify the notation we set ϕ r ( where H M is the mean curvature vector of M. In particular we conclude Err o j , (25.14) where Err o 4 and Err o 5 denote the terms Err 4 and Err 5 of (25.9) when X = X o . We follow the same arguments with X = X i , applying this time [9,  and where Err i 4 and Err i 5 denote the terms Err 4 and Err 5 of (25.9) when X = X i .

Error estimates.
We next proceed as in [11,Section 4]. First of all, since the structure and estimates on the size of the cubes of the Whitney decomposition are exactly the same, we can define the regions of [ First of all we split the latter error into the terms I 1 and I 2 of [11, Page 596]. The term I 1 is estimated in the same way. Fo r the term I 2 we can use the same argument when Q < p 2 and hence F is A Q -valued. However, we need a small modification in the case Q = p 2 , when F is A Q -valued.
If q ∈ Im 0 (F ) ∪ Im + (F ), as in [11, Page 597] we set h j q := h j q ( T F (q)) and h q = l j=1 h j q ν j (q) .
We proceed however differently for q ∈ Im − (F ): in this case we set h j q := h j q (− T F (q)) and h q = l j=1 h j q ν j (q) .
Combining the latter inequality with [7, Theorem 13.1] we can bound After the extraction of a further subsequence, we can assume the existence of r such that Br\B 3r 4 |N j | 2 → 0, (26.5) and the existence of a mod(p) area-minimizing cone S such that (ι 0,t j ) ♯ T → S. Recall that S is a representative mod(p). By (26.4), the cone S cannot be an integer multiple of an m-dimensional plane. We argue as in [11, and conclude that, if M is the limit of a subsequence (not relabeled) of the M j , then there are two radii 0 < s < t such that spt(S)∩B t (0)\B s (0) ⊂ M. In particular, by the Constancy Theorem mod(p) we conclude that S B t (0) \ B s (0) = Q 0 M ∩ B t (0) \ B s (0) mod(p) for an integer Q 0 with |Q 0 | ≤ p 2 . Since S is a cone and a representative mod(p) we can in fact infer that S B t (0) = Q 0 0 × × M ∩ ∂B t (0) mod(p) (in fact it can be easily inferred from the argument in [11, Pages 601-602] that Q 0 = Q, although this is not needed in our argument). Since 0 × × M ∩ ∂B t (0) induces a stationary varifold and M is the graph of a function with small C 3,ε 0 norm, we can applied Allard's Theorem to conclude that in fact 0 × × M ∩ ∂B t (0) is smooth. This implies that the latter is in fact π ∩ B t (0) for some m-dimensional plane π, contradicting the fact that S is not a flat cone.
Finally, Theorem 26.1 can be used as in [11,Section 5]

Final contradiction argument
In this section we complete the proof of Theorem 1.4 showing that, by Proposition 7.7, under the assumption that the theorem is false, we get a contradiction. In particular fix T, Σ, Ω and r k as in Proposition 7.7. We have already remarked that for each k there is an interval of flattening I j(k) =]s j(k) , t j(k) ] containing r k . We proceed as in [11,Section 6] and introduce the following new objects: • We first apply Corollary 26.3 to r = r k t j(k) and sets k := t j(k) σ k , so thats k t j(k) • We rescale our geometric objects, namely (U1) The currentsT k , the manifoldsΣ k and the center manifoldsM k are given respectively byT k = (ι 0,r k ) ♯ T j(k) = ((ι 0,r k t j(k) ) ♯ T ) B 6 √ m/r k (27.1) Σ k = ι 0,r k (Σ j(k) ) = ι 0,r k t j(k) (Σ) (27.2) M k = ι 0,r k (M j(k) ) . (27.3) (U2) In order to define the rescaled mapsN k onM k we need to distinguish two cases. When Q < p 2 , the mapN k takes values in A Q (R m+n ) and is defined bȳ In the case Q = p 2 , the mapN k takes values in A Q (R m+n ) and is defined analogously. The reader might either use the decomposition of M j(k) into M + j(k) , M − j (k) and (M j(k) ) 0 or, using the original notation in [7,Definition 2.2], and ε(·) ∈ {−1, 1}. Without loss of generality we can assume that T 0 Σ = R m+n ×{0}, thus the ambient manifolds Σ k converge to R m+n × {0} locally in C 3,ε 0 . Observe in addition that 1 2 < r k r k t j(k) < 1 and hence it follows from Proposition 7.7(ii) that Indeed Proposition 7.7(ii) implies thatT k converge to Q π 0 both in the sense of varifolds and in the sense of currents mod(p). Finally, we recall that, by where α is a positive number and C 0 a geometric constant. As in [11,Section 6] we claim the counterpart of [11, Lemma 6.1], namely Lemma 27.1, which implies thatM k converge locally to the flat m-plane π 0 . We also introduce the exponential maps e k : B 3 ⊂ R m ≃ Tq kM k →M k denotes the exponential map atq k = Φ j(k) (0)/r k ( here and in what follows we assume, w.l.o.g., to have applied a suitable rotation to eachT k so that the tangent plane Tq kM k coincides with R m × {0}). We are finally ready to define the blow-up maps N b k : B 3 ⊂ R m → A Q (R m+n ), when Q < p 2 and N b k : where h k := N k L 2 (B 3 2 ) .
Lemma 27.1 (Vanishing lemma). LetT k ,r k ,M k andΣ k be as above. We then have: (i) min{m j(k) 0 ,r k } → 0; (ii) the rescaled center manifoldsM k converge (up to subsequences) to π 0 = R m × {0} in C 3,κ/2 (B 4 ) and the maps e k converge in C 2,κ/2 to the identity map id : B 3 → B 3 ; (iii) there exists a constant C > 0, depending only on T , such that, for every k, Proof. The argument for (i) can be taken from [11, Proof of Lemma 6.1]. As for part (ii) the argument given in [11,Section 6] for the convergence of the center manifolds can be shortened considerably observing that it is a direct consequence of Proposition 24.3(v) and the convergence of the currentsT k . The C 2,κ/2 convergence of the exponential maps follow then immediately from [11,Proposition A.4]. Finally, (iii) is an obvious consequence of Corollary 26.3.
Having defined the blow-up maps, the final contradiction comes from the following statements. The two theorems would contradict [12, Theorem 0.11] in case Q < p 2 since, arguing as in [11,Section 6] we easily conclude that Υ is a subset of the singularities of N b ∞ . In the case Q = p 2 we infer instead from [7,Proposition 10.3 on the whole B 3/2 , which in turn would imply N ∞ b = Q 0 . This however contradicts N ∞ b L 2 (B 3/2 ) = 1.

Proof of Theorem 27.2.
Without loss of generality we may assume thatq k := r −1 k Φ j(k) (0) coincide all with the origin. We then define a new mapF k on the geodesic ball B 3/2 ⊂M k distinguishing, as usual, the two cases Q < p 2 and Q = p 2 . In the first case we follow the definition of [11, Section 7.1], namely we set In the case Q = p 2 the mapF k takes values in A Q (R m+n ) and it is induced byN k in the sense explained at point (N) of [7, Assumption 11.1]. The argument given in [11, Section 7.1] works Next, following [11, Section 6.2, Step 2], for every q ∈ Λ k we definez k (q) = p π k (q) (where π k is the reference plane for the center manifold related to T j(k) ) and x k (q) := (z k (q),r −1 k ϕ j(k) (r kzk (q))) .
Observe thatx k (q) ∈M k . We next claim the existence of a suitably chosen geometric constant 1 > c 0 > 0 (in particular, independent of σ) such that, when k is large enough, for each q ∈ Λ k there is a radius ̺ q ≤ 2σ with the following properties: B ̺q (x k (q)) ⊂ B 4̺q (q) . (27.23) The argument given in [11, Section 6.2, Step 2] can be routinously modified in our case. In particular we define the points q k :=r k q, z k :=r kzk (q) and x k =r kxk (q) = (z k , ϕ j(k) (z k )) and discuss the three different possibilities depending on whether z k belongs to a cube L ∈ W j(k) or to the contact set Γ j(k) .
The first case, z k ∈ L ∈ W j(k) h can be excluded with the same argument given in [11, Section 6.2, Step 2], where we replace [10, Proposition 3.1] with Proposition 16.26, because q k is a multiplicity Q point for the current T j(k) .
Following the argument in [11, Section 6.2, Step 2], when z k ∈ W j(k) n ∪ W j(k) e we find a t(q) ≤ σ with the property that In the case z k ∈ Γ j(k) we find a t(q) < σ such that We choose ϕ k j so that 0 ≤ ϕ k j ≤ 1 and dϕ k j 0 ≤ C(r k j ) −1 , where C is a geometric constant. We then define ϕ k := j ϕ k j .
Recall that M((∂T i ) U ) = sup{∂T i (ω) : ω c ≤ 1 , ω ∈ D k (U )} . We therefore fix a smooth (m − 1)-form ω with compact support in U and we are interested in bounding ∂T i (ω) = T i (dω). Observe that ϕ k ↑ 1 T i -a.e. on U . Hence we can write On the other hand, since ϕ k ω is supported in an open set V ⊂⊂ U \ K we conclude Hence we can estimate Letting k ↑ ∞ we thus conclude This shows that (∂T i ) U has finite mass. Point (a) follows therefore from the Federer-Fleming boundary rectifiability theorem. In order to show (c), consider the set K ′ of points q ∈ K where • K has an approximate tangent plane T q K; • q is a Lebesgue point for all Θ i 's with Θ i (q) ∈ Z. By a standard blow-up argument, it follows that, for every fixed q ∈ K ′ , any limit S of the currents (ι q,r ) ♯ (T i ) as r ↓ 0 is an area-minimizing current on R m+n with boundary either −Θ i (q) T q K or +Θ i (q) T q K . By the boundary monotonicity formula, We therefore conclude that lim inf Fix any natural number N . We then conclude from (29.2) that In particular we conclude that This shows that This completes the proof of (c) and of the structure theorem.