Center Manifold: a case study

Following Almgren's construction of the"center manifold"in his Big regularity paper, we show the C^{3,\alpha} regularity of area-minimizing currents in the neighborhood of points of density one without using the nonparametric theory. This study is intended as a first step towards the understanding of Almgren's construction in its full generality.


Introduction
In this note we consider area-minimizing integral currents T of dimension m in R m+n . The following theorem is the cornerstone of the regularity theory. It was proved for the first time by De Giorgi [2] for n = 1 and then extended later by several authors (the constant ω m denotes, as usual, the Lebesgue measure of the m-dimensional unit ball).
Theorem 0.1. There exist constants ε, β > 0 such that, if T is an area-minimizing integral current and p is a point in its support such that θ(T, p) = 1, supp (∂T ) ∩ B r (p) = ∅ and T (B r (p)) ≤ (ω m + ε) r m , then supp (T ) ∩ B r/2 (p) is the graph of a C 1,β function f .
Once established this ε-regularity result, the regularity theory proceeds further by deriving the usual Euler-Lagrange equations for the function f . Indeed, it turns out that f solves a system of elliptic partial differential equations and the Schauder theory then implies that f is smooth (in fact analytic, using the classical result by Hopf [5]).
In his Big regularity paper [1], Almgren observes that an intermediate regularity result can be derived as a consequence of a more complicated construction without using the nonparametric PDE theory of minimal surfaces (i.e. without deriving the Euler-Lagrange equation for the graph of f ). Indeed, given a minimizing current T and a point p with θ(T, p) = Q ∈ N, under the hypothesis that the excess is sufficiently small, Almgren succeeds in constructing a C 3,α regular surface (called center manifold) which, roughly speaking, approximates the "average of the sheets of the current" (we refer to [1] for further details). In the introduction of [1] it is observed that, in the case Q = 1, the center manifold coincides with the current itself, thus implying directly the C 3,α regularity.
The aim of the present note is to give a simple direct proof of this remark, essentially following Almgren's strategy for the construction of the center manifold in the simplified setting Q = 1. At this point the following comment is in order: the excess decay leading to Theorem 0.1 remains anyway a fundamental step in the proof of this paper (see Proposition 1.2 below) and, as far as we understand, of Almgren's approach as well. One can take advantage of the information contained in Theorem 0.1 at several levels but we have decided to keep its use to the minimum.
1. Preliminaries 1.1. Some notation. From now on we assume, without loss of generality, that T is an area-minimizing integer rectifiable current in R m+n satisfying the following assumptions: with the small constant ε to be specified later.
In what follows, B m r (q), B n r (u) and B m+n r (p) denote the open balls contained, respectively, in the Euclidean spaces R m , R n and R m+n . Given a m-dimensional plane π, C π r (q) denotes the cylinder B m r (q) × π ⊥ ⊂ π × π ⊥ = R m+n and P π : π × π ⊥ → π the corresponding orthogonal projection. Central points, supscripts and subscripts will be often omitted when they are clear from the context.
We will consider different systems of cartesian coordinates in R m+n . A corollary of De Giorgi's excess decay theorem (a variant of which is precisely stated in Proposition 1.2 below) is that, when ε is sufficiently small, the current has a unique tangent plane at the origin (see Corollary 1.4). Thus, immediately after the statement of Corollary 1.4, the most important system of coordinates, denoted by x, will be fixed once and for all in such a way that π 0 = {x m+1 = . . . = x m+n = 0} is the tangent plane to T at 0. Other systems of coordinates will be denoted by x ′ , y or y ′ . We will always consider positively oriented systems x ′ , i.e. such that there is a unique element A ∈ SO(m + n) with x ′ (p) = A · x(p) for every point p. An important role in each system of coordinates will be played by the oriented m-dimensional plane π where the last n coordinates vanish (and by its orthogonal complement π ⊥ ). Obviously, given π there are several systems of coordinates y for which π = {y m+1 = . . . = y m+n = 0}. However, when we want to stress the relation between y and π we will use the notation y π .
1.2. Lipschitz approximation of minimal currents. The following approximation theorem can be found in several accounts of the regularity theory for area-minimizing currents. It can also be seen as a special case of a much more general result due to Almgren (see the third chapter of [1]) and reproved in a simpler way in [3]. As it is customary the (rescaled) cylindrical excess is given by the formula (where π is the unit simple vector orienting π and the last equality in (1.1) holds when we assume P ♯ (T C π r ) = B r (p ).
Then, for s = r(1 − CE η ), there exists a Lipschitz function f : B s → R n and a closed set K ⊂ B s such that: This proposition is a key step in the derivation of Theorem 0.1. In the appendix we include a short proof in the spirit of [3]. Clearly, Theorem 0.1 can be thought as a much finer version of this approximation. However, an aspect which is crucial for further developments is that several important estimates can be derived directly from Proposition 1.1.

1.3.
De Giorgi's excess decay. The fundamental step in De Giorgi's proof of Theorem 0.1 is the decay of the quantity usually called "spherical excess" (where the minimum is taken over all oriented m-planes π): From now on we will consider the constant δ fixed. Its choice will be specified much later. Definition 1.3. For later reference, we say that a plane π is admissible in p at scale ρ (or simply that (p, ρ, π) is admissible) if for some fixed (possibly large) dimensional constant C m,n . Proposition 1.2 guarantees that, for every p and r as in the statement, there exists always an admissible plane π p,r . The following is a straightforward consequence of Proposition 1.2 which will be extensively used. Corollary 1.4. There are dimensional constants C, C ′ and C ′′ with the following property. For every δ, ε 0 > 0, there is ε > 0 such that, under the assumption (H): (a) if (p, ρ, π) and (p ′ , ρ ′ , π ′ ) are admissible (according to Definition 1.3), then (b) there exists a unique tangent plane π p to T at every p ∈ supp (T ) ∩ B 1/2 ; moreover, if (p, ρ, π) is admissible then |π − π p | ≤ C ′ ε 0 ρ 1−δ and, vice versa, if |π − π p | ≤ C ′′ ε 0 ρ 1−δ , then (p, ρ, π) is admissible; (c) for every q ∈ B m 1/4 , there exists a unique u ∈ R n such that (q, u) ∈ supp (T ) ∩ B 1/2 . Remark 1.5. An important point in the previous corollary is that the constant C ′′ can be chosen arbitrarily large, provided the constant C m,n in Definition 1.3 is chosen accordingly. This fact is an easy consequence of the proof given in the appendix.
Theorem 0.1 is clearly contained in the previous corollary (with the additional feature that the Hölder exponent β is equal to 1 − δ, i.e. is arbitrarily close to 1). In order to make the paper self-contained, we will include also a proof of Proposition 1.2 and Corollary 1.4 in the appendix.
1.4. Two technical lemmas. We conclude this section with the following two lemmas which will be needed in the sequel.
Consider two functions f : D ⊂ π 0 → π ⊥ 0 and f ′ : D ′ ⊂ π → π ⊥ , with associated systems of coordinates x and x ′ , respectively, and x ′ (p) = A · x(p) for every p ∈ R m+n . If for every q ′ ∈ D ′ there exists a unique q ∈ D such that (q ′ , f ′ (q ′ )) = A · (q, f (q)) and vice versa, then it follows that graph π 0 (f ) = graph π (f ′ ), where The following lemma compares norms of functions (and of differences of functions) having the same graphs in two nearby system of coordinates. Lemma 1.6. There are constants c 0 , C > 0 with the following properties. Assume that is given and f, g : B m 2r (q) → R n are Lipschitz functions such that Lip(f ), Lip(g) ≤ c 0 and |f (q) − u| + |g(q) − u| ≤ c 0 r.
Then, in the system of coordinates x ′ = A · x, for (q ′ , u ′ ) = A · (q, u), the following holds: (a) graph π 0 (f ) and graph π 0 (g) are the graphs of two Lipschitz functions f ′ and g ′ , whose domains of definition contain both B r (q ′ ); where Φ and Ψ are smooth functions.
Proof. Let P : R m×n → R m and Q : R m×n → R n be the usual orthogonal projections. Set π = A(π 0 ) and consider the maps F, G : B 2r (q) → π ⊥ and I, J : B 2r (q) → π given by ) and G(x) = Q(A((x, g(x))), ) and J(x) = P (A((x, g(x))). Obviously, if c 0 is sufficiently small, I and J are injective Lipschitz maps. Hence, graph π 0 (f ) and graph π 0 (g) coincide, in the new coordinates, with the graphs of the functions f ′ and g ′ defined respectively in D := I(B 2r (q)) andD := J(B 2r (q)) by f ′ = F •I −1 and g ′ = G•J −1 . If c 0 is chosen sufficiently small, then we can find a constant C such that and Clearly, (1.7) and (1.8) easily imply (a). Conclusion (c) is a simple consequence of the inverse function theorem. Finally we claim that, for small c 0 , (1.9) from which, using the change of variables formula for biLipschitz homeomorphisms and (1.7), (b) follows. In order to prove (1.9), consider any x ′ ∈ B r (q ′ ), set x := I −1 (x ′ ) and , and in this case (1.9) follows trivially. If this is not the case, the triangle with vertices p 1 , p 2 and p 3 is non-degenerate. Let θ i be the angle at p i . Note that, Therefore, if c 0 is small enough, we have 1 ≤ 2 sin θ 3 , so that, by the Sinus Theorem, thus concluding the claim.
The following is an elementary lemma on polynomials.
Lemma 1.7. For every n, m ∈ N, there exists a constant C(m, n) such that, for every polynomial R of degree at most n in R m and every positive r > 0, |R| for all k ≤ n and all q ∈ R m . (1.10) Proof. We rescale and translate the variables by setting S(x) = R(rx + q). The lemma is then reduced to show that for every polynomial S of degree at most n in R m , with C = C(n, m). Consider now the vector space V n,m of polynomials of degree at most n in m variables. V n,m is obviously finite dimensional. Moreover, on this space, the two quantities |S| are two norms. The inequality (1.11) is then a corollary of the equivalence of norms on finite-dimensional vector spaces.

The approximation scheme and the main theorem
The C 3,α regularity of the current T will be deduced from the limit of a suitable approximation scheme. In this section we describe the scheme and state the main theorem of the paper.
We start by fixing a nonnegative kernel ϕ ∈ C ∞ c (B m 1 ) which is radial and satisfies´ϕ = 1. As usual, for τ > 0, we set ϕ τ (w) := τ −m ϕ(w/τ ). Consider the area-minimizing current is the tangent plane to T at 0). From Corollary 1.4 (b) and (c), it is simple to deduce the following: if p = (q, u) ∈ π × π ⊥ , ρ ≤ 2 −6 and π form an admissible triple (p, 8 ρ, π) with p ∈ supp (T ) ∩ B 1/16 , then . From now on, we will assume that C m,n ε 2 where ε 1 is the constant of Proposition 1.1 and C m,n the constant of Proposition 1.2. This assumption guarantees the existence of the Lipschitz approximation of Proposition 1.1, which we restrict to B m 6ρ (q), f : B m 6ρ (q) ⊂ π → π ⊥ . Then, consider the following functions: In order to proceed further, we need to show the existence of g as in (I 3 ). We wish, therefore, to apply Lemma 1.6 to the functionf . First recall that |π − π 0 | ≤ Cε 0 |p| 1−δ ≤ Cε 0 by Corollary 1.4. Thus, assumption (i) in Lemma 1.6 is satisfied provided ε 0 is chosen sufficiently small. Next note that, by the interior estimates for the harmonic functions and (1.2a), one has Moreover, if we consider the ball B s (p) with s = ρE η/(2m) , by the monotonocity formula, Recalling that E ≤ Cε 2 0 ρ 2−2δ , we conclude that condition (ii) in Lemma 1.6 is satisfied when ε 0 is sufficiently small. Therefore Lemma 1.6(a) guarantees that the function g exists.

Remark 2.2.
It is obvious that in order to define the function g we could have used, in place of the f given by Proposition 1.1, the function whose graph gives the current T in B 6ρ (p). This would have simplified many of the computations below. However, as mentioned in the introduction, we hope that our choice helps in the understanding of the more general construction of Almgren.
The function g is the main building block of the construction of this paper. It is called the (p, ρ, π)-interpolation of T or, if Ex(T, B 8ρ (p)) = Ex(T, B 8ρ (p), π), simply the (p, ρ)interpolation of T .
The main estimates of the paper are contained in the following proposition.
Proposition 2.3. There are constants α, C > 0 such that, if g, g ′ are respectively (p, ρ, π)and (p ′ , ρ, π ′ )-interpolations, then 2.1. Approximation scheme. Let 5 < n 0 < k 0 be natural numbers and consider the cube The corresponding closed cubes of the subdivision are then denoted by Q i and we consider below only those Q i 's which have nonempty intersection with Q. According to Corollary 1.4 and to the previous observations, for every c i there exists a unique u i such that Denote by A i the set of indices j such that Q j and Q i are adjacent. Note that the choice of ψ guarantees ψ i ψ j = 0 if j ∈ A i . Moreover the cardinality of A i is (bounded by) a dimensional constant independent of k and, if q ∈ Q i , then in a neighborhood of q we have We are now ready to state and prove the central theorem of this note.
Theorem 2.4. There are dimensional constants n 0 < k 0 with the following properties. Given an area-minimizing current T as in (H) and k ≥ k 0 , consider the functions h k : for some dimensional constants α > 0 and C (which, in particular, do not depend on k). Moreover, the graphs of h k converge, in the sense of currents, to T (Q × R n ) ∩ B 1/2 , thus implying that T is a C 3,α graph in a neighborhood of the origin.
Proof of Theorem 2.4. Given k, consider a cube Q i of the corresponding dyadic decomposition and a point q ∈ Q i . We already observed that, in a neighborhood of q, h k = j∈A i ψ j g j . Moreover, from the definition, we have that The C 0 estimate of h k follows trivially from (2.1a), since As for the C 1 estimate, we write the first derivative of h k as follows, from which, using (2.1a), (2.1b) and (2.4), we deduce With analogous computations, we obtain where C is a constant independent of k. Now, let q, q ′ ∈ B 1/2 and consider the cubes Q i and Q j such that q ∈ Q i and q ′ ∈ Q j . If the two cubes are adjacent, then we have |q − q ′ | ≤ C2 −k and, therefore, This concludes the proof of (2.3). We finally come to the convergence of the graphs of h k in the sense of currents. Obviously, by compactness we can assume that a subsequence of h k (not relabelled) converges in the C 3 (Q) norm to some limiting C 3,α function h. On the other hand, by Corollary 1.4 and Proposition 1.1, it follows easily that the support of T (Q × R n ) ∩ B 1/2 is contained in the graph of h. But then, by the Constancy Theorem, T (Q×R n )∩B 1/2 must coincide with an integer multiple of the graph of h. Our assumptions imply easily that the multiplicity is necessarily 1.

L 1 -estimate
The rest of the paper is devoted to the proof of Proposition 2.3. A fundamental point is an estimate for the L 1 distance between the harmonic functionf introduced in step (I 2 ) of the approximation scheme and the function f itself. A preliminary step is the following estimate on the Laplacian off , which is a simple consequence of the first variation formula for area-minimizing currents.
Proof. Let µ be the measure defined by µ(A) := T (A × π ⊥ ). We start showing that the approximation f given by Proposition 1.1 satisfies for every κ ∈ C 1 c (B 6ρ , R n ). Consider the vector field χ(x, y) = (0, κ(x)). From the minimality of the current T , we infer that the first variation of the mass in direction χ vanishes, δT (χ) = 0. We set T f = graph(f ). Since δT (χ) = 0, we get The first variation δT f (χ) is given by the formulâ

It follows then that
We next estimate the second term in the right hand side of (3.4): where we have used the Lipschitz bound on f to estimate the second integral in the right hand side of the first line. This concludes the proof of (3.3).
We now come to the proof of (3.1). From (3.3) and Proposition 1.1, it follows straightforwardly that Then, putting together the previous estimates, we conclude that Therefore, (3.1) follows choosing δ sufficiently small with respect to η. For the proof of (3.2), it is enough to notice that, from (3.3) and Proposition 1.1, we get Now we come to the L 1 -estimate for the harmonic approximationf .
Proposition 3.2. Let (p, 8ρ, π) be admissible andf be as in (I 2 ). Then, there exists α > 0 such that Proof. First we estimate the L 1 distance betweenf andf . Using the Poincaré inequality and a simple integration by parts, we infer that ≤ C ρ m+3+λ . In order to prove (3.6), then it is enough to prove the following inequality, For every z ∈ B 4ρ , from the definition off we havê (3.8) To simplify the notation assume z = 0 and rewrite (3.8) aŝ dw.
More generally, for every z ∈ B 4ρ , we havef (z) − f (z) =´∇f (w) · Φ(w − z) dw and Since ϕ is radial, the function Φ is a gradient. Indeed, it can be easily checked that, for any ψ, the vector field ψ(|w|) w is curl-free. Moreover, supp (Φ) is compactly contained in B ρ . Hence, we can apply (3.2) and get By a simple computation, The last integral is a constant which depends only on ϕ. Thus, (3.7) follows from (3.9).
A simple consequence of the L 1 -estimate is a comparison between harmonic approximations at different scales. Corollary 3.3. Assume (p, 16 r, π) is an admissible triple and letf 1 andf 2 be as in (I 2 ), with ρ = r and ρ = 2 r respectively. Then, if p = (q, u) ∈ π × π ⊥ , Proof. It is enough to show that because then the conclusion of the lemma follows easily from the classical mean-value property of harmonic functions. Clearly, from the admissibility of (p, 16 r, π) and Corollary 1.4, it follows that |π − π p | ≤ C r 2−2δ . Hence, always by the same corollary E 2 := Ex(T, B 16r (p), π) ≤ Cr 2−2δ . Then, in view of Proposition 3.2, in order to show (3.11), it suffices to prove Note first that f 1 and f 2 coincide on a set K with |B 2r \ K| ≤ CE 1+η 2 r m . Moreover, since the Lipschitz constants of f 1 and f 2 are bounded by a universal constant C, we have |f 1 (z) − f 2 (z)| ≤ Cr for every z ∈ B 2r . Therefore, we conclude (3.12) from

Proof of Proposition 2.3
The proof of (2.1a) in Proposition 2.3 is given by a simple iteration of Corollary 3.3 on dyadic balls. Lemma 4.1. Let g 1 , g 2 be respectively the (p, ρ, π)and the (p, 2 N ρ, π)-interpolation (under the assumption of admissibility (1.4)). Then, for p = (q ′ , u ′ ) ∈ π 0 × π ⊥ 0 , it holds Proof. Recalling Lemma 1.6, it suffices to show (4.1) for the functionf 1 . Let n 0 be the biggest integer such that 2 n 0 +3 ρ ≤ 1 2 and for every k ≤ n 0 − 1 set r k = 2 k ρ. If π k is such that Ex(T, B 8r k , π k ) = Ex(T, B 8r k ), then, by Corollary 1.4 (b), |π−π k | ≤ C r 1−δ k . Hence, we conclude that the admissibility condition (1.4) holds with r = r k , so that we can consider the approximationf k as in (I 2 ) for r k . From Corollary 3.3, we get Note that the series i 2 −i(3+α−ℓ) is summable for ℓ ≤ 3. Therefore, f 1 C 3 ≤ C + f n 0 C 3 . On the other hand, since r n 0 > 1/32, it is easy to see that f n 0 C 3 ≤ C for some universal constant C, so that f 1 C 3 ≤ C. In the same way we have D 4f 1 C 0 ≤ C ρ α−1 . Then, (4.1) follows from Lemma 1.6 (c) .
Finally, Corollary 3.3 obviously implies that Hence, using again Lemma 1.6, we concludê (4.5) Let P k and P k+1 be the third order Taylor polynomials at q ′ of g k and g k+1 . From the estimate D 4 g k , D 4 g k+1 ≤ Cr α−1 k and (4.5), we easily infer Hence, applying Lemma 1.7, we then get Arguing as above, the estimate (4.2) follows from (4.6) and a simple iteration.
The final step in the proof of Proposition 2.3 consists in comparing two different interpolating functions defined at the same scale but for nearby balls and varying planes π. We do this in the following two separate lemmas.
Lemma 4.2. Let g 1 and g 2 be the (p, ρ, π)and (p, ρ, π ′ )-interpolating functions where as usual (p, 8ρ, π) and (p, 8ρ, π ′ ) are admissible. Then, Proof. As before, we first show that Denote by f 1 , f 2 the Lipschitz approximations given by Proposition 1.1 in the coordinates associated to π, π ′ and let h 1 , h 2 : B ρ (q) → π ⊥ 0 be the Lipschitz functions whose graphs coincide with the graphs of f 1 and f 2 respectively. From Lemma 1.6 and Proposition 3.2, we have where (q 1 , u 1 ), (q 2 , u 2 ) and (q, u) are the coordinates of p in π × π ⊥ , π ′ × π ′⊥ and π 0 × π ⊥ 0 respectively. Therefore, for (4.8) it is enough to show To see this, consider the set A = {h 1 = h 2 }. From Proposition 1.1 if follows that Then, if x ∈ A and y ∈ B 3ρ/2 \ A, since h 1 (y) = h 2 (y) and Lip(h i ) ≤ C, we have From (4.8) we are ready to conclude. Let x ∈ B ρ (q) and P i be the third order Taylor expansions of g i at x. Arguing as in Lemma 4.1, we conclude Using Lemma 1.7 we then conclude On the other hand, since D k P i (x) = D k g i (x), (4.9) implies the desired estimates.
Lemma 4.3. Let g 1 and g 2 be, respectively, the (p, ρ, π)and (p ′ , ρ, π)-interpolating functions, where (p, ρ, π) and (p ′ , ρ, π) are admissible. Assume that p = (q, u), The proof of this lemma exploits only a portion of the same computations used for Lemma 4.2 and is left to the reader.
The proof of (2.1b) follows straightforwardly from Lemma 4.2 and Lemma 4.3; while the proof of (2.1c) is given below.

Appendix A. De Giorgi's regularity result
In this section we provide a proof of De Giorgi's regularity Theorem 0.1 in its more refined version of Corollary 1.4. The overall strategy proposed here is essentially De Giorgi's celebrated original one [2]; however, in many points we get advantage from some new observations contained in our recent work [3].
In the following we keep the conventions of the rest of the paper, but we use the various Greek letters α, β, . . . for other parameters and other functions. Moreover, given a current T in R m+n , a Borel set A ⊂ R m+n and a simple m-vector τ , we define the following excess measures: A.1. Lipschitz approximation. In this section we prove Proposition 1.1. To this aim, we assume without loss of generality that π = π 0 and consider an area-minimizing integer rectifiable m-dimensional current T in C r such that where P : R m+n → R m is the orthogonal projection. The proof is in the spirit of the approximation result in [3] and is made in three steps.
A.1.1. BV estimate. Consider the push-forwards into the vertical direction of the 0-slices T, P, x through the projection P ⊥ : R m+n → R n : x . These integer 0-currents (i.e. sums of Dirac deltas with integer coefficients) are characterized by the following identity (see [6, Section 28]): Hence, from the characterization of the slices T x , it follows that where in the last equality we used the hypothesis ∂T = ∅ on B m r × R n . Now, observe that the m-form dψ ∧ α has no dx component, since Write T = ( T · e m ) e m + S (see [6,Section 25] for the scalar product on m-vectors). We then conclude that T, dψ ∧ α = S · T , dψ ∧ α . If ϕ ∞ ≤ 1, then |dψ ∧ α| ≤ Dψ ∞ ϕ ∞ ≤ 1. Hence, by Cauchy-Schwartz inequalitŷ Taking the supremum over all ϕ with L ∞ -norm less or equal 1, we conclude (A.1).
A.1.2. Maximal Function truncation. Here, we show how we determine the Lipschitz approximation. Given α > 0, we set Then, there exists h : B r ′ → R n such that: Proof. Note that x / ∈ K if and only if there exists 0 < r x < r E 1−2α m such that Hence, recalling the standard Maximal Function estimate (see, for example, [7]), we deduce easily (A.6).
In order to define the approximation h, recall that´A T x ≤ T (A × R n ) for every open set A (cp. to [6,Lemma 28.5]). Therefore, Hence, since E < 1 and P ♯ T = B r , we have that 1 ≤ T x < 2 for almost every point in K. Thus, T x = δ g(x) for some measurable function g.
Therefore, by a standard argument (see, for instance, [4, 6.6.2]), this implies the existence of a constant C > 0 such that, for every x, y ∈ K Lebesgue points of Φ ψ , Taking the supremum over a dense, countable set of ψ ∈ C ∞ c (R n ) with Dψ ∞ ≤ 1, we deduce that We can hence extend g to all B r ′ , obtaining a Lipschitz function h with Lipschitz bound CE α . Clearly, since h| K = g| K and T x = δ g(x) , we conclude graph(h| K ) = T (K × R n ).
Remark A.3. Note that from Lemma A.2 it follows that for some dimensional constant C > 0 A.1.3. Proof of Proposition 1.1. We start fixing positive constants α, σ, θ, γ such that Consider the Lipschitz approximation h given by Lemma A.2 corresponding to the exponent α (we keep the same notation as above). By a slicing argument, we find s ∈ [r(1−E σ ), r(1− E θ )] such thatˆB s+rE θ \B s−2rE θ (With a slight abuse of notation, we write (T − graph(h)) ∂B s for T − graph(h), ϕ, s , where ϕ(x) = |x|.) Moreover, setting for a standard kernel ϕ it is simple to verify that Lip(g) ≤ CE α and, furthermore, for some δ > 0, where L is as in Lemma A.2. Indeed, we can estimate the energy of g in two steps as follows. First in the annulus B s \ B s−rE θ : Hence, in B s−rE θ : where the first term is estimated in turn aŝ and the second one as follows, Hence, by the choice of the constants in (A.10), (A.13) follows. Next, we observe that, from ∂ T − graph(h) ∂B r = 0, by the isoperimetric inequality and (A.12), there is an integer rectifiable current R such that Moreover, being g| ∂Bs = h| ∂Bs , we can use graph(g) + R as competitor for the current T . In this way we obtain, for a suitable τ > 0, On the other hand, again using the Taylor expansion for the area functional, Hence, from (A.14) and (A.15), we deduce We are now in the position to conclude the proof of Proposition 1.1. Let β < α be such that 2β < τ and let f be the Lipschitz approximation given by Lemma A.2 corresponding to β. Clearly, (1.2a) follows once we take η ≤ β. Moreover, since {M T > E β /2 m } ⊂ L, from (A.6) and (A.16) we get (1.2b) if η is accordingly chosen. Finally, for (1.2c), we use again the Taylor expansion of the area functional to conclude: A.2. Convergence to harmonic functions. Let (T l ) l∈N be a sequence of minimizing m-currents in B 1 ⊂ R m+n such that It is immediate to see that, up to subsequences, the T l converge in the sense of currents to a flat m-dimensional disk centered at the origin. By the monotonicity formula, there is also Hausdorff convergence of the supports of T l to the flat disk in every compact set In particular, there exist radii r l → 1 such that ∂ T l C r l = 0 in C r l and P # T l C r l = B m r l . In the following proposition we prove the convergence to a harmonic function for the rescaled Lipschitz approximations. the averages, converge in W 1,2 loc to a harmonic function u. Proof. Note that, by (A.8) it follows that sup l´B r l |Du l | 2 < ∞. Hence, since ffl u l = 0, by the Sobolev embedding and the Poincaré inequality, there exists a function u : B 1 → R n such that, for every s < 1, u l → u in L 2 (B s ) and Du l ⇀ Du in L 2 (B s ).
In particular, Du is not identically 0. Since from Remark A.5´B 1 |Du| 2 ≤ 2ω m , from (A.25) and Lemma A.6, we get where A l = ffl B 1/2 Df l and τ l is as in (A.20). Rescaling by E l and passing to the limit in l, for (Du) s = ffl Bs Du, we get against the decay property of harmonic functions. This gives the contradiction and concludes the proof.