Quantitative regularity for $p$-minimizing maps through a Reifenberg Theorem

In this article we extend to generic $p$-energy minimizing maps between Riemannian manifolds a regularity result which is known to hold in the case $p=2$. We first show that the set of singular points of such a map can be quantitatively stratified: we classify singular points based on the number of almost-symmetries of the map around them, as done by Cheeger and Naber in 2013. Then, adapting the work of Naber and Valtorta, we apply a Reifenberg-type Theorem to each quantitative stratum; through this, we achieve an upper bound on the Minkowski content of the singular set, and we prove it is $k$-rectifiable for a $k$ which only depends on $p$ and the dimension of the domain.


Introduction
The aim of this article is to study the singular set S(u) of maps that take values in a smooth Riemannian manifold and minimize the p-energy functional: for p ∈ (1, ∞); we assume that u : B ⊂ M → N , where B is a geodesic ball in the Riemannian smooth manifold M, and N is a smooth closed Riemannian manifold. Here by singular set we mean the set of points where the map is not continuous. The information we obtain is twofold: 1. First we obtain a Minkowski-type estimate on the singular set: if u : B ⊂ M → N , and the total p-energy is bounded by Λ, then where B ′ is in general a smaller geodesic ball. In particular, the Minkowski dimension of S(u) is at most dim M − ⌊p⌋ − 1, and the upper Minkowski content (in B ′ ) is bounded by a universal constant C(M, N , Λ, p).
In what follows, we set m . = dim M. Our results complete the following scheme: • p = 2: in 1982, Schoen and Uhlenbeck proved through a dimension reduction argument that any 2-energy minimizing map is C 0,α outside of a set of Hausdorff dimension at most m − 3 (see [SU82]). Furthermore, by standard elliptic regularity, the regularity outside the singular set can be improved to C ∞ .
• p ∈ (1, ∞): in 1987, Schoen and Uhlenbeck's result was extended to p-harmonic maps by Hardt and Lin [HL87]. The best regularity one can achieve outside the singular set is C 1,α ; so any p-energy minimizing map is C 1,α outside of a set of Hausdorff dimension at most m − ⌊p⌋ − 1. Notice that the case m ≤ p was already completely solved here: in this case, there are no singular points, and the map is everywhere C 1,α . The only case worth studying is m > p.
• p = 2: in 2013, Cheeger and Naber [CN13] proved that the singular set of a 2minimizing map with energy bounded by Λ satisfies the following estimate: for any ε > 0; it is assumed that the dimension of the domain is at least 3. This implies that the Minkowski dimension of S(u) is at most m−3, but gives no bound on the Minkowski content. Here a notion of "quantitative stratification" of the singular set was introduced, and the result was obtained through a relatively simple covering of each singular stratum (and by making explicit the link between singular set and stratification).
for any ε > 0. It is assumed that the dimension of the domain is greater than p: we have already noticed, however, that this is the only interesting case.
• p = 2: in 2017, Naber and Valtorta [NV17] improved the estimate (1.2) for 2minimizing maps, removing the dependence on the parameter ε: assuming that m > 2, then the singular set of a 2-minimizing map with energy bounded by Λ satisfies Vol(S(u) ∩ B 1 (0)) ≤ C(M, N , Λ)r 3 ; (1.4) and thus the upper Minkowski content of S(u) is bounded by a constant C. Moreover, in the same article they showed that S(u) is actually (m − 3)-rectifiable. The main idea to prove both the Minkowski estimate and rectifiability was to replace the simple covering argument of [CN13] with a more refined one, which makes use of a suitable version of Reifenberg Theorem.
In this article, we adopt the same technique used in [NV17]: after giving an appropriate definition of singular stratification (adapted to the case of p-minimizing maps), we exploit the very same version of Reifenberg Theorem developed in [NV17, Theorems 3.3 and 3.4] to build a controlled covering of each stratum. Notice that analogous results (still exploiting this technique) are available for approximate harmonic maps [NV16] and for Q-valued energy minimizers [HSV17].

Definitions and notation
From now on, N will always be an n-dimensional compact, smooth Riemannian manifold without boundary, while M is a m-dimensional smooth Riemannian manifold. For convenience, we can assume it is isometrically embedded in a Euclidean space R N -thanks to the well known Nash embedding Theorem (see [Nas54]). For any p ∈ (1, ∞), one can then define the Sobolev space W 1,p (M, N ) as the space of maps u ∈ W 1,p (M, R N ) such that u(x) ∈ N for almost every x ∈ M. Remark (Relation between p and m). If p > m, by the Sobolev embedding theorem one has that W 1,p (M, N ) ֒→ C 0 (M, N ). Consequently, this case is not particularly interesting for our purposes, since the singular set is empty. More generally, in the case ⌊p⌋ ≥ m (thus also when p is an integer and p = m), it is a known fact that p-minimizers have no singular points (see for example [NVV14,Theorem 2.19]). Thus, from now on, we will implicitly assume p < m.
The aim of this paper is to study the singular set of minimizers of the p-energy. We now give the main definitions involved. Definition 1.1 (p-energy). If u ∈ W 1,p (M, N ), with (M, g) Riemannian manifold of dimension m, and N ֒→ R N , then we define the p-energy of u as where ω g is the volume form associated to the metric g. Recall that, in a local coordinate chart, we have: (1.6) ω g = det g dx 1 · · · dx m , (1.7) where the scalar product ·, · N is the Riemannian scalar product in the target N . Definition 1.2 (Singular set of a map). If u ∈ W 1,p (M, N ), we define its singular set as S(u) . = {x ∈ M | u is not continuous at x}. (1.8) Assumptions on the domain. Since the problem of studying S(u) can be treated as a local problem, from now on we assume the setting to be the following: u is defined on an open domain Ω of R m ; Ω contains a ball BR(0) for someR ≥ 2; and we study the intersection S(u) ∩ B 1 (0). We won't be more precise than this about the actual size we need for BR(0): this will be the result of a collection of assumptions we will gradually make in the next sections. Notice that, given this assumption, the p energy E p (u) actually assumes the simpler form for any map u ∈ W 1,p (Ω, N ).
Remark. When the domain is a generic smooth Riemannian manifold (M, g) of dimension m, the problem can be reduced to the Euclidean setting by suitably choosing a coordinate chart around a given point (see [HL87,Section 7]). For any point q ∈ M, a system of normal coordinates is defined on a geodesic ball B M ε (q) around q; in these coordinates, the metric g is represented by g ij (q) = δ ij at the point q. By smoothness, up to lowering the radius ε, we can assume that |g − Id| is small in B M ε (q). Under this assumption, one can give lower and upper bounds on the p-energy where φ : B M ε (q) → B ε (0) ⊂ R m is the coordinate chart. Through this procedure, one can show that suitable versions of the main results we use (ε-regularity, monotonicity formula) still hold (locally) in the case of non-flat manifolds (details and explicit computations can be found in [HL87, Section 7] and [Xin96, Section 2.2]). Notice that the quantitative results we prove will now have constants depending on the injectivity radius of M, due to the fact that we needed to choose normal coordinates on a geodesic ball.
We are interested in studying minimizers of the p-energy, i.e. maps which minimize the functional E p among those with the same boundary datum. More precisely, we give the following definitions: Definition 1.3. Let u ∈ W 1,p (Ω, N ). We say that: 1. u is a p-energy minimizing map if E p (u) ≤ E p (v) for any compact set K ⊂ Ω and for any v ∈ W 1,p (Ω, N ) such that u Ω\K ≡ v Ω\K (more precisely: u = v almost everywhere in Ω \ K).
2. u is a weakly p-harmonic map if it is a critical point of the p-energy functional with respect to variations in the target manifold, i.e.: for any ξ ∈ C ∞ c (Ω, R N ), it holds: where Π N is the nearest-point projection onto N , defined on a tubular neighborhood of N itself.
3. u is a stationary p-harmonic map if it is weakly p-harmonic and it is a critical point of the p-energy functional with respect to compact variations in the domain. Explicitly: let Φ = {φ t } t∈I be any smooth family of diffeomorphisms of Ω, with I open interval containing 0; assume that φ 0 ≡ id Ω , and that there exists a compact set K ⊂ Ω such that φ t Ω\K = id Ω\K for any t ∈ I; then It is clear that any p-energy minimizing map is (both weakly p-harmonic and) stationary p-harmonic. The results we achieve in this article will always be proved for p-energy minimizers. We now briefly survey the standard and classical tools available when dealing with p-minimizers: these techniques will be then expanded in Sections 2 and 3.

Basic tools and references: an overview
Some essential references to keep in mind will be the work of Schoen and Uhlenbeck [SU82], for fundamental results regarding 2-harmonic maps, and its generalization to generic p, due to Hardt and Lin [HL87].
Monotonicity formula. Given a map u ∈ W 1,p (Ω, N ) and a ball B r (x) ⊂ Ω, in Definition 2.3 we define a quantity θ(x, r) which will turn out to be scale invariant and monotone in r. This takes the same form of the normalized energy introduced in [SU82] for 2-harmonic maps and in [HL87] for p-harmonic maps (see also [Sim96, Section 2.4]): indeed, we'll consider the function (1.14) In Theorem 2.6 we'll show that an explicit formula for d dr θ(x, r) is available. As a consequence, it will be clear that θ(x, ·) is non-decreasing, and that θ(x, r) = θ(x, s) with r > s if and only if u is 0-homogeneous in a ball of controlled radius centered at x. Making this statement quantitative will be a key idea: is small enough, then u will be close to be 0-homogeneous in B cr (x); if this happens at k + 1 points sufficiently far away from each other, then u will be close to be invariant along a k-plane in a ball B cr (x 0 ).
We will turn this principle in a precise statement in Corollary 3.6; other version of the same idea will be used throughout the article.
Quantitative stratification. Based on ideas contained in [SU82], the following notion of singular stratification for a W 1,p (Ω, N ) map can be introduced: = {x ∈ Ω | any tangent map to u at x is at most k-symmetric}; (1.15) here a tangent map is an L p limit of blow ups, and k-symmetric means 0-homogeneous and invariant along a k-plane. For p = 2, it was proved in [SU82] that dim H S k (u) ≤ k. In Definition 3.1 we give a quantitative version of this definition, based on the ones appearing in [CN13] for p = 2 and [NVV14] for p generic: given three parameters k, η and r we look at points x such that "in B r (x), u is not η-close to be k-invariant". We'll say that any such point belongs to the stratum S k η,r (u). Another way to read the Heuristic Principle is that: if x 0 ∈ S k η,r (u), then the points in B cr (x 0 ) for which θ(x, r) − θ(x, s) is small must lie close to a k-plane. ε-regularity. The link between the study of singular sets ond the study of singular stratifications is an ε-regularity result (again from [SU82;HL87]): if θ(x, r) is sufficiently small, then u is regular in B r 2 (x). With a simple compactness argument, one can generalize this result to prove that for some η we have S(u) ⊂ S m−⌊p⌋−1 η,r (u) for any r (see Proposition 3.3): thus the study of S(u) reduces to the study of the (m − ⌊p⌋ − 1) th singular stratum.

Acknowledgements
I am deeply grateful to Daniele Valtorta for the precious suggestions he gave me while writing this article; and before that, for sharing his knowledge and ideas about this topic.

Preliminaries
As in the case p = 2, p-harmonic maps satisfy (in a weak sense) some suitable Euler-Lagrange equations, which are stated in the next two theorems (see for example [Mos05, Sections 3.1 to 3.3] for a complete treatment of the case p = 2; the same computations work for the case p ∈ (1, ∞)). Theorem 2.1 (External variations). If u is weakly p-harmonic, then it satisfies (weakly) the equation where A is the second fundamental form of the embedding N ֒→ R N . Explicitly, we have It is easy to show that the projection on the unit sphere u : , S N −1 whenever p < N , and in that case it satisfies Equation (ExtEL) classically out of the origin; as a consequence, one can immediately show that u is weakly p-harmonic in B N 1 (0). A key fact is that the second fundamental form of the sphere can be explicitly computed: It is then an easy exercise to show that the equation actually holds; both sides of the equation turn out to be equal to −(N − 1)

Theorem 2.2 (Internal variations). If u is stationary p-harmonic, then for any
A crucial tool in the study of (p-)harmonic maps is the normalized energy of a map in a fixed ball of the domain. Definition 2.3 (Normalized energy). Let u be a W 1,p (Ω, N ) map. Let ψ ∈ C ∞ c ([0, ∞)) be a non-increasing function supported in [0,R). For all x ∈ B 1 (0) and r > 0 small enough, we define the normalized p-energy as the function The definition we give here is actually not the standard one (although it has already been used in several articles on the argument): one recovers the usual definition (present for example in [SU82; CN13; NV17]) by taking ψ = ξ [0,1] (which is not an admissible choice in our definition).
Remark (Motivation for the definition). As we will see in Theorem 2.6 (and all the subsequent results), a condition of the type θ ψ (x, r) − θ ψ (x, s) = 0 with r > s gives much more information on u with our definition, rather than with the standard one: for example, one can only deduce that u is 0-homogeneous in the annulus {y : s < |y − x| < r} if ψ = ξ [0,1] , while we obtain 0-homogeneity in a whole ball around x if we choose ψ such that ψ ′ < 0 in an interval (0,t).
A first useful property of the normalized energy is that it is scale invariant: = {y ∈ R m | x + ry ∈ Ω} ⊃ B 1 (0).
(2.4) Theorem 2.5 (Scale invariance). If u ∈ W 1,p (Ω, N ), x ∈ B 1 (0) and r > 0, the following identity holds: As a further consequence, if also w ∈ Ω−x r and τ > 0 is small enough, (2.6) Theorem 2.6 (Monotonicity formula). Let u be a stationary p-harmonic map, and ψ a smooth function as before. Fix x ∈ B 1 (0) and r > 0 smaller than dist(x, ∂Ω). Then θ ψ (x, ·) has a derivative at r and the following equality holds: Remark. As we already mentioned in the remark at page 4, this statement is not true if the domain has a non-zero curvature: however, up to transforming the domain as we did in the aforesaid remark, it is indeed true that e Cr θ(x, r) is monotone, with C only depending on m and p (see [HL87, Section 7] and [Xin96, Theorem 2.7]). As it is easily seen, this modification does not affect our computations.
Proof. We'll proceed in two steps.
Step 1. We first consider the case x = 0, r = 1; the general case will then follow by scale invariance. In particular, we have to prove the following identity: The key idea is to find a suitable vector field to plug into the Euler-Lagrange equation (IntEL): thus, we consider the following one: A simple computation gives, for 1 ≤ i, j ≤ m, Then, with this choice of Y , the integral appearing in Equation (IntEL) reads: this follows by a straightforward computation, and by the fact that: (2.9) Now by Equation (IntEL) the integral in (2.8) is zero; hence Equation (2.7) follows easily, just by taking the derivative of θ ψ (0, ·) at r = 1 (and changing the order of integral and derivative): Step 2. Consider now the general case: arbitrarily fix x ∈ B 1 (0) andr > 0. By scale invariance, we know that θ ψ [u](x, r) = θ ψ [T x,r u](0, 1) for all r in a neighborhood ofr. Hence in particular d dr (2.11) Notice that by Step 1 we have information about the quantity d ds θ ψ [T x,r u](0, s) at s = 1, which is not directly the information we seek, but is really close. Indeed, a simple computation (which involves nothing more than the definition of T x,r ) shows that the two quantities are related by (2.12) Thus we have: (2.13) By performing the change of variables w = x+ry, we obtain exactly the desired result.
Corollary 2.7. Let ψ be a smooth function as before; define: (2.14) Then, for any p-stationary map u, for any x ∈ B 1 (0) and 0 < s < r < dist(x, ∂Ω), we have: Definition 2.8 (Assumptions on ψ). From now on, we'll think of ψ ∈ C ∞ c ([0, ∞)) as a fixed function, satisfying for some fixed numbers 0 < t a < t b and ξ > 0 (see Figure 1). Moreover, since this will be needed in Section 6, we'll actually assume 2 < t a < t b : this choice will be better explained in the remark at page 24. The function in Figure 1 can be thought as a valid one.
Such a choice of ψ is mostly justified by computational reasons; it's worth noting, however, that enlarging the value of t b is heuristically equivalent to looking at smaller balls in the domain, thus exploiting again the local nature of the problem.
A straightforward consequence of the monotonicity formula is the following: Corollary 2.9. Let 0 < s < r; let u ∈ W 1,2 (Ω, N ) be a stationary harmonic map, and x ∈ B 1 (0). Take ψ ∈ C ∞ c ([0, ∞)) as in Definition 2.8. If Moreover, by simple geometric considerations, if a map is 0-homogeneous with respect to different points, then it is invariant along the affine subspace generated by those points: Corollary 2.10 (Rigidity). Let 0 < s < r; let u ∈ W 1,p (Ω, N ) be a stationary harmonic map, and take ψ ∈ C ∞ c ([0, ∞)) as in Definition 2.8. Let 0 ≤ k ≤ m be an integer. If there exist k + 1 points {x i } k i=0 such that: , and 0-homogeneous at any point of L.

An ε-regularity result
Combining Theorems 2.5 and 3.1 of [HL87], we get the following Hölder-regularity result for p-minimizers: Theorem 2.11 (ε-regularity). There exist two constants ε 0 = ε 0 (m, N , Λ, p) and α(m, N , Λ, p) such that the following holds. Let u ∈ W 1,p (Ω, N ) be a minimizer for . This result will be the key argument that connects the singular set to the singular stratification (see Proposition 3.3).
Notice that the situation gets even better when dealing with 2-harmonic maps: by standard elliptic regularity arguments one can get C ∞ regularity instead of Hölder regularity (see [SU82;Sch84]).

Quantitative stratifications
Let u ∈ W 1,p (Ω, N ), and assume L ∈ G k (R m ) is a k-linear subspace. We denote by |∇ L u| or | ∇u, L | the quantity where {v 1 , . . . , v k } is any orthonormal basis of L.
Definition 3.1 (Quantitative stratification). Let x ∈ B 1 (0) and r > 0 small enough for B r (x) to be contained in Ω. Fix k ∈ {0, . . . , m} and a parameter η > 0. We say that u is When this condition holds for some η and k, we'll generically refer to it as "almost invariance". We define the singular k th stratum of u, with scale parameter r and closeness parameter η the subset of B 1 (0) defined by (3.5) for any given η and k.
Remark. Notice that this definition is slightly different from the one used in [NVV14] and originally introduced in [CN13]: indeed, in those papers u is defined to be "almost k-invariant" if it is (quantitatively) close in L p to a map which is 0-homogeneous and k-invariant. The two approaches can be shown to be equivalent (see for example [HSV17, Proposition 3.11]); we won't explore this path further.
In the following proposition we show that the definition of almost invariance given above implies an "almost 0-homogeneity" condition at a smaller scale (and this is why we didn't require such a condition in the very definition of almost invariance). Not only: given a ball B r (x) where u is (δ, k)-invariant for a suitable δ, all the points in B 1 2 r (x) are both almost invariant and almost 0-homogeneous with respect to this smaller scale. Proposition 3.2. Let η > 0 be a fixed parameter. There exists a constantγ(m, N , Λ, p, η), with 0 <γ < 1 2 , such that the following holds. Defineδ for some L ∈ G k (R m ). Then for any y ∈ B 1 2 r (x) there exists a radiusγr ≤ r y ≤ 1 2 r such that u satisfies the following almost invariance and almost 0-homogeneity conditions in B ry (y): Br y (y) θ(y, r y ) − θ y, 1 2 r y < η. (3.8) Proof. By scale invariance, it is sufficient to prove the statement for x = 0, r = 1. Choosē γ so thatγ Consider a point y ∈ B 1 2 (0); assume, by contradiction, that for all i such that 2 −i ≥γ we have Then we should have: a contradiction. Thus for any y ∈ B 1 2 (0) we have a radius r y ∈ γ, 1 2 for which (3.8) holds. Moreover, by (3.9) we also have: which concludes the proof.
The bridge between singular stratification and singular set of a p-harmonic map is given by the following proposition, which strongly relies on the ε-regularity Theorem 2.11. This result can be seen as a quantitative version of the following known fact ([HL87, Theorem 4.5]): a p-minimizing map which is 0-homogeneous and invariant along a (m − ⌊p⌋)-linear subspace must be constant.
Argue by contradiction: assume that for all i ∈ N there exists a p-minimizing map u i with (3.14) Up to precomposing with a rotation of the space, we can assume L i = L for all i, for some affine subspace L. By Proposition 3.2, we have The maps T x i ,α i r i u i are p-minimizing, and they are uniformly bounded in W 1,p (B 1+ε (0), N ) for some ε (by compactness of N and by the bound on the p-energy); thus, up to subsequences, they weakly converge to a mapũ ∈ W 1,p (B 1+ε (0), N ). By [HL87;Luc88], the convergence is actually strong in W 1,p (B 1 (0), N ), andũ is a p-minimizer in B 1 (0). But now by strong W 1,p (B 1 (0), N ) convergence and by (3.15), (3.16) we have soũ is p-minimizing, (m−⌊p⌋)-invariant on B 1 (0) and 0-homogeneous on B 1 (0). By [HL87,Theorem 4.5], this implies thatũ is constant on B 1 (0): thus in particular θ[ũ](0, ·) ≡ 0 in (0, 1). However, by the fact that x i ∈ S(u i ), and by the ε-regularity Theorem 2.11, we have: Since we'll make great use if compactness and limiting arguments, we will need an effective notion of "points in general position" which is preserved when passing to the limit.
We say that a set of points S spans ̺-effectively a given k- (3.20) As we wanted, the notion of ̺-general position passes to the limit: We are now ready to state a first precise version of what we called "Heuristic Principle" in Section 1.2. This makes Corollary 2.10 quantitative; the proof is a simple compactness argument, based on Lemma 3.5 and on the (already used) fact that weak W 1,p limits of p-minimizing maps are actually strong limits, and are p-minimizers themselves. Corollary 3.6 (Quantitative rigidity). Let u ∈ W 1,p (Ω, N ) be a p-minimizing map with energy bounded by Λ. Let 0 ≤ k ≤ m be an integer. Fix the constants η, p, γ, ̺ > 0. There exists a constant ε > 0 (depending on m, N , Λ, η, p, γ, ̺) suich that the following holds. If

The results: precise statements
The main result we will prove is the following. Theorem 3.7 (Singular strata). Let u ∈ W 1,p (Ω, N ) be a p-energy minimizing map with energy bounded by Λ. Let η > 0 and 1 ≤ k ≤ m. There exists two constants C 1 and δ 0 depending on m, N , p, Λ, η such that for any r > 0 Moreover, for any η > 0 and any 0 ≤ k ≤ m, the stratum S k η (u) is k-rectifiable. The proof will be achieved in Section 7, exploiting all the tools developed in Sections 4 to 6.
Notice that, thanks to Proposition 3.3, we obtain the following crucial corollary, which is actually the main goal we wanted to achieve: volume estimates and structural information for the singular set. Corollary 3.8 (Singular set). Let u ∈ W 1,p (Ω, N ) be a p-energy minimizing map with energy bounded by Λ. There exists a constant C 2 (m, N , Λ, p) such that for any r > 0 (3.23)

Reifenberg Theorems and approximating planes
The next step will be to introduce some more advanced techniques which allow us to analyze the behaviour of each singular stratum at every scale r around a point x. In order to state Reifenberg Theorem (in a form which is suited to our context), we first need to recall the definition of Jones' numbers of a measure µ (first appeared in [Jon90]; for a detailed introduction, see [Paj02]): this is a scale-invariant notion which quantifies how close supp(µ) is to be contained in an affine k-space (near a given point). Definition 4.1 (Jones' numbers). Let x ∈ B 1 (0) and 0 < r < 1. Assume µ is a positive Radon measure on Ω. For any k ∈ {0, . . . , m} we define the k-dimensional Jones' Remark. The quantity β k µ is scale invariant in the following sense. Assume µ is defined in a ball B r (x); define the blow up measureμ =μ k x,r on B 1 (0) aŝ Then it is easy to compute that β k µ (0, 1) = β k µ (x, r). The main hypothesis one needs, in order to apply Reifenberg Theorem in its different forms, is a control on the Jones numbers of a suitable measure (e.g., H k restricted to a set). In all the cases, the condition we need takes the following form: Definition 4.2 (Reifenberg condition). Let µ be a positive Radon measure on Ω, and k ∈ {0, . . . , m}. We say that µ satisfies the (k-dimensional) Reifenberg condition with constant δ if for any x ∈ B 1 (0) and 0 < r < 1 we have: As we will clarify in Theorem 4.4, two versions of Reifenberg Theorem are available for our purposes: in one of them (necessary for the rectifiability of a set), one needs to check (k-Reif) on the restriction of the Hausdorff measure to the given set; the other one (necessary for volume estimates) makes use of discrete measures as the following: Definition 4.3 (Measure associated to a disjoint family of balls). Assume C is a (discrete) subset of B 1 (0), and F = {B rx (x)} x∈C is a collection of disjoint balls centered in C, each contained in B 2 (0). For any k ∈ {0, . . . , m}, we define the following measure associated to F : where δ x is the Dirac measure centered at x. Theorem 4.4 (Reifenberg). There exist two constants C R and δ R such that the following statements hold true.
for any x ∈ S and 0 < r < 1. Since we will need it in this form, for the sake of clarity we state here a rescaled version of Theorem 4.4, part (i).
for all w ∈ Br(x) and all 0 < r < 1, then Proof. It suffices to apply Theorem 4.4 to the measureμx ,r . =r −k Tx ,r ♯µ introduced in the remark above: by using the change of variable formula for the integral, and exploiting the scale invariance of β k we obtain the result.
Remark. Notice that the constants δ R and C R for the rescaled version are the same as in Theorem 4.4, and thus only depending on m.

Estimates on Jones' numbers
The key estimate, linking the Jones' numbers of a measure with the normalized p-energy of a p-minimizing map, is given in the following theorem, which we prove in several steps.
Theorem 4.6 (Estimates on β k µ ). Let u ∈ W 1,p (Ω, N ) be a p-energy minimizing map. Fix the following constants: There exists a constant C 3 (m, p, η, σ,r) such that the following estimate holds for any positive Radon measure µ on Ω.
Remark. When this theorem will be used in Section 6, we'll assign a precise value to σ; notice that in order that all the expressions involved are meaningful we could need to enlarge the domain Ω (and thusR) according to σ (see Assumptions on the domain at page 4).
Remark. By scale invariance (of both β k µ and θ), it will be enough to prove the estimate for x = 0 and r = 1. Moreover, since the inequality does not change when multiplied by a constant, we can assume µ is a probability measure on B 1 (0). That is: assuming u is not (η, k + 1)-invariant in Br(0), we will prove that for any measure µ with µ(B 1 (0)) = 1 Summary of the proof. First of all, we find an explicit expression for the Jones' number of a measure. This relies on the fact that the infimum among affine planes which defines β k µ is actually achieved, and the minimal plane has a particularly manageable characterization (Subtheorem 4.6.1). Secondly (and separately from the first point), we use directly the monotonicity formula (MF) to estimate the "radial p-energy" of u in a ball with the quantity θ(·, σ) − θ(·, 1) computed at the center of the ball (Subtheorem 4.6.2). Next, we estimate the p-energy of u along some selected directions v j ∈ S m , exploiting the information we got from the second step; here the explicit expression for β k µ emerges (Subtheorem 4.6.3). Finally, we consider k + 1 selected directions together: we give an upper bound to the p-energy of u along a (k + 1)-plane in terms of θ(·, σ) − θ(·, 1). We then use the fact that u is not (k + 1)-almost invariant at 0 to say that the same p-energy must be at least some fixed amount.
We begin by writing the Jones' number in an explicit way. Some preliminary definitions are needed. Definition 4.7. Let µ be a measure with support in B 1 (0). We define: (i) the center of mass of µ as the point x µ cm ∈ B 1 (0) such that We'll usually drop the superscript µ when it is clear from the context.
Since Q is symmetric and positive-definite, by the Spectral Theorem the associated matrix (which we still denote by Q) admits an orthonormal basis of eigenvectors, with nonnegative eigenvalues. We denote with λ µ 1 , . . . , λ µ m the eigenvalues of Q in decreasing order, and with v µ 1 , . . . , v µ m the respective eigenvectors (pairwise orthogonal and of norm 1), again dropping the superscripts when they are clear; in particular: (4.10) Finally, we denote by V µ k (or V k ) the following affine k-plane: (4.13) We are now ready to characterize β k µ : Subtheorem 4.6.1. Let µ be a measure on B 1 (0). The affine space V µ k achieves the minimum in the definition of β k µ (0, 1). Moreover, (4.14) For a proof of this fact, see [NV17,Lemma 7.4] or [NV16, Subsection 6.1]. This is based on the (visually helpful) fact that the eigenvalues λ k and eigenvectors v k admit the following characterization: • λ 1 is the maximum of B 1 (0) x − x cm , v 2 dµ among vectors v of norm 1, and v 1 is any maximizing vector; • λ k is the minimum of the same operator among all unit vectors orthogonal to v 1 , . . . , v k−1 , and v k is any maximizing vector.
We have thus reduced the problem to showing: or, since the eigenvalues are ordered decreasingly, we need to show even less: The following result gives an estimate involving the difference θ(y, σ) − θ(y, 1). It is really a direct consequence of the monotonicity formula (and our choice of ψ): none of the tools just introduced is required. Recall that ξ and t a are the constants introduced in Definition 2.8. Subtheorem 4.6.2. Let u ∈ W 1,p (Ω, N ) be p-minimizing, y ∈ B 1 (0); assume R 1 , R 2 and R are radii satisfying R 2 ≤ R ta < R 1 . Then the following inequality holds for some constant C 4 (m, R, R 1 , p) (provided both sides are well defined): (4.15) In particular, taking R 2 = 1, R = max{2, t a }, and R 1 > R/t a we also get with C 5 depending only on R 1 , m, p.
Proof. By Theorem 2.6: (4.17) Since z ∈ B R (y) and r ≥ R ta , by the assumptions on ψ made in Definition 2.8 we have: The two integrals can be then separated and we get: (4.19) But now notice that |z − y| −1 ≥ R −1 : inequality (4.15) is proved. The last statement (Equation (4.16)) follows from the fact that B 1 (0) ⊂ B R (y), because y ∈ B 1 (0) and R ≥ 2.
Next, we use the "radial" information we just achieved to estimate the p-energy along the eigenvectors v j . Notice that the eigenvalues λ j also appear in the estimate. Subtheorem 4.6.3. Let u be p-minimizing, and µ a Radon measure on B 1 (0); let {λ j } j , {v j } j be the eigenvalues and eigenvectors of Q µ , as before. There exists a constant C 6 (m, N , p) such that for all j = 1, . . . , m the following holds: (4.20) Proof. Up to performing a translation, we can assume x cm = 0. By definition of the eigenvalue λ j , we have y, v j y dµ(y). (4.21) Fix z ∈ B 1 (0). By taking the scalar product of ∇u(z) with both sides of the previous equality, we obtain: y, v j ∇u(z), y dµ(y); (4.22) moreover, since x cm = 0, we have: (4.23) By subtracting (4.23) to (4.22), and multiplying both sides by |∇u(z)| p 2 −1 , we can write: (4.24) We then take the squared norms of both sides and apply Hölder inequality: (4.25) Now if λ j = 0 the statement is trivial, so we can assume λ j > 0 (since all the eigenvalues are non-negative); we are thus allowed to divide by λ j . Equation (4.25) holds for all z ∈ B 1 (0), thus we can then integrate both sides on B 1 (0) with respect to the Lebesgue measure in the variable z. At this point we get, also using Tonelli's Theorem: (4.26) Hence, a direct application of Subtheorem 4.6.2 gives the desired result. Now, thanks to the last result, we have an upper bound on the p-energy along the (k + 1)plane V k introduced in Equation (4.12); but this is bounded from below by a constant, by the lack of almost invariance in 0. We have all the ingredients to complete the proof of Theorem 4.6.

A collection of structural lemmas
This section is devoted to building a series of "quantitative" geometric results about pminimizing mappings, describing the behavior of some special subsets of S(u). Analogous results for (approximate) 2-harmonic maps can be found in [NV16, Section 4], although stated with some differences.
In Lemmas 5.1 to 5.3, recall that t a , t b are the structural constants introduced in Definition 2.8, describing some particular features of ψ. Moreover, in each of those results we implicitly assume that u ∈ W 1,p (Ω, N ) is a p-energy minimizing map with energy bounded by Λ, and the constants we find are independent of u.
In the first lemma we convey this idea: consider the set of points in B 1 (0) at which θ[u] satisfies a suitable pinching condition; if it spans ̺-effectively a k-dimensional plane L, then for some δ > 0 the stratum S k η,δ (u) lies inside a fattening of L. See Definition 3.4 for the definition of a set effectively spanning a k-plane. Lemma 5.1. Let ̺ 1 , λ 1 , η > 0 and 0 < γ < 1 be (small enough) constants. Define c(γ) .
where t a is introduced in Definition 2.8. There exist constants δ 0 , ε (depending on m, p, N , Λ and on the parameters just introduced) such that the following holds: if the set K . = {y ∈ B cr (x) | θ(y, r) − θ(y, λ 1 r) < ε} (5.1) spans ̺ 1 r-effectively a k-plane L, then S k η,δ 0 r (u) ∩ B cr (x) ⊂ B ̺ 1 r (L). In the proof we drop the subscript 1 on ̺ and λ; it was introduced so that lemma is easier to recall when we need it.
Sketch. Assume without loss of generality that x = 0, r = 1. We are thus assuming that for some τ ≥ δ 0 , with δ 0 depending only on m, p, N , Λ, η, ̺, λ, γ, and for some k + 1dimensional plane V . Denote by y 0 , . . . , y k a set of points of K which ̺-effectively span L.
Step 1. By Subtheorem 4.6.2, with y ∈ K, R 1 = 1, R 2 = λ and max{λ, 1 − γ}t a < R < t a , we have Step 2. Notice that B c (0) ⊂ B R (y) for any y ∈ K and R as in the previous point; then for any such y: DefineL to be the linear subspace associated to L. For any v ∈L of norm 1, we have that Hence, by a standard estimate, As a consequence, if {v 1 , . . . , v k } is an orthonormal basis ofL, thus, along k directions, we have some information that goes in the direction we need (recall that ε is a constant we still have to choose).
Step 3. For points lying out of B ̺ (L) we need to gain another direction of smallness, and we do it by considering the orthogonal direction to L passing through w: we can estimate the quantity using the same technique of the first point, and then exploit the triangle inequality to achieve an estimate on (5.10) More precisely: assume now that τ < ̺ 2 ; define for the sake of simplicity h(z) . = z−π L (z) |z−π L (z)| , where π L is the orthogonal projection onto L. Similarly to what happened in the previous step, we have |∇u| p−2 | ∇u, z − π L (z) | 2 dz ≤ c 6 (m)ε, (5.12) so that In that case, the projection of z onto L can be written as π L (z) = π L (w) + τ v ⊤ . As a consequence, we can estimate (5.14) To estimate the expression in (5.10), we exploit Equations (5.13) and (5.14), the bound on θ(w, τ ) and a Cauchy-Schwarz inequality: Step 4. Putting together the previous steps, we consider V =L ⊕ h(w): a simple computation gives: (5.16) Thus, in order to conclude, we only need to choose τ so that the last term is smaller than η 2 , and then choose ε such that also the sum of the first two pieces is smaller than η 2 .
The upcoming lemma says the following: if we have a set of points that satisfy a suitable pinching condition on θ, and they effectively span a k-subspace L, then all the points of L inherit a (possibly weaker) pinching condition. This can be seen as a further quantitative version of Corollary 2.10 (it is indeed applied to the limit of a contradicting sequence).
There exists a constant δ(̺ 2 , λ 2 , Λ, γ, c) such that the following holds: if θ(y, r) ≤ E for all y ∈ B cr (x) ∩ S (with S ⊂ Ω and E ≤ Λ), and the set spans ̺ 2 r-effectively a k-space L, then we have In the proof we drop the subscript 2 on ̺ and λ; it was introduced so that lemma is easier to recall when we need it.
Proof. Assume x = 0, r = 1. If the statement is false, one can find a sequence of pminimizing maps {u i } i , k + 1 sequences of points {y ij } i∈N in B c (0) ∩ S (with j = 1, . . . , k + 1), and a further sequence {z i } i in B c (0) such that: • {y ij } k+1 j=1 spans ̺-effectively a k-space L (which can be assumed to be the same for all i).
Up to subsequences, {u i } converges in W 1,p to a p-minimizing mapū, y ij →ȳ j ∈B c (0), and z i →z ∈B c (0) ∩ L. Moreover, the set of points {ȳ j } k+1 j=1 still spans L, and we have Finally, a result which states that the lack of almost invariance spreads uniformly along pinched points. This is yet another quantitative rephrasing of the fact that if θ(·, r) − θ(·, λr) = 0 at two different (close) points, then u is invariant along the direction connecting them. Lemma 5.3. Let λ > 0, σ 0 ∈ 0, 1 2 t b , κ 0 ∈ (0, 1). There exists a constant ε such that the following holds. If the following conditions are satisfied by a pair of points x, y: 3. θ(y, r) − θ(y, λr) < ε; Proof. Assume x = 0, r = 1. By contradiction, there exist: a sequence {u i } i of pminimizing maps, a sequence {y i } i of points in B 1 2 t b (0) and a sequence such that: Up to subsequences, they converge, respectively, to a p-minimizing mapū (in W 1,p ), to a point y ∈B 1 2 t b (0) and to a numberσ ∈ σ 0 , 1 2 t b . Moreover, due to Equations (5.20) and (5.21) and by the L p -convergence of gradients,ū is 0-homogeneous with respect to both x and y, thus it is invariant along the direction x − y in B 1 2 t b (0). Again by the fact that u i →ū in W 1,p (Ω, N ), however, we also have that this contradicts the fact that the two left hand sides should be equal (by translation invariance ofū).
Remark (Assumptions on ψ, again). The structural constants t a and t b we introduced in Definition 2.8 were broadly used in these last few lemmas. In practice, the key feature (of ψ) we need for our purposes is the possibility to work handily in the ball B 1 (0) (so for example we require that 1 is an admissible value for c(γ) in Lemma 5.1). In the end, with the choice t a > 2 done in Definition 2.8, we can apply Lemmas 5.1 and 5.2 with c = 1 and Lemma 5.3 with |x − y| < r.

Covering arguments
In the sequel, if B = B r (x) and k is a constant, we denote by kB the ball B kr (x). We first give two useful definitions of "sets of points satisfying a pinched condition"; we have (more or less) already used both of them in Section 5. Definition 6.1. Let u be a p-minimizing map, x ∈ B 1 (0), r > 0. Assume E, η, λ, δ > 0 are fixed, and S ⊂ B 1 (0). We define If B = B r (x), we also denote by H B , K B the sets H(x, r), K(x, r) respectively.
It is clear that, if all the parameters appearing are fixed, and u is such that θ(y, r) ≤ E for all y ∈ B r (x), then H(x, r) ⊂ K(x, r). As a consequence, whenever H ̺-effectively spans a k-subspace, also K trivially does. Heuristically H should be thought as a set of pinched points at which θ(y, λr) has a value which is close to the maximum possible.
Notations and map of the constants. This will be the context for the whole section: -m, N , p, Λ are fixed as in the previous sections (respectively: dimension of the domain, target manifold, exponent for the energy, upper bound on the p-energy).
-u ∈ W 1,p (Ω, N ) is a p-minimizing harmonic map with p-energy bounded by Λ.
-We let ̺ > 0 be a fixed constant, and r = ̺ for some ∈ N ≥1 . The radius r will be the scale parameter for the singular stratification, up to a constant. The constant ̺ will be arbitrary in the first covering, and will be then suitably selected in the construction of the second covering.
-γ > 0 is a constant used for the pinching condition on θ. It will be arbitrary in the construction of the first covering, then selected in Proposition 6.4.
-δ 0 is the constant introduced in Lemma 5.1 and δ is the constant produced by Lemma 5.2, both depending on ̺, η, γ.
The goal of the upcoming constructions will be to build a "controlled" covering of S. In words, the ultimate goal will be to cover S with balls B satisfying the following: 1. The sum of the k th powers of the radii is bounded by a universal constant.
2. Up to rescaling by a fixed constant, the balls are pairwise disjoint.
3. Either the radius of B is less or equal to the fixed radius r; or the (normalized) p-energy in B is lower than the "maximal initial p-energy" E by a fixed amount δ (E and δ were introduced in the previous list of constants). In the latter case, we say that B satisfies a uniform energy drop condition (see the below Definition 6.2).
This will be achieved in Proposition 6.5. Once we have this, we can then apply the same reasoning to each of the balls where the p-energy drops uniformly (while keeping the other balls as they are). At each step other balls of radius ≤ r are produced, while the p-energy continues to drop uniformly in all the other balls. The procedure lasts a finite number of steps, until there's no energy left: indeed the total initial p-energy was bounded by a fixed constant Λ. This is the content of Section 7.1. Let's give a precise definition of energy drop: Definition 6.2. We say that a ball B satisfies the uniform (λ, δ)-energy drop condition if From now on λ = 1 5 and δ will be fixed (δ as in the Map of the Constants above), so we omit them and simply say "uniform energy drop".
At first, we are only able to reach a partial result: we don't manage to fully get a uniform energy drop condition on the balls of the covering; but we can show that, in each ball, the points for which the p-energy does not drop uniformly lie close to a (k − 1)-plane (so in the end they can be controlled very efficiently). This is the content of the next subsection.

First covering
Recall that r = ̺ is the scale parameter of the singular stratum we are considering. Here ̺ > 0 is a fixed parameter and ∈ N: we are allowed to work with constants which depend on ̺, but not on.

Construction of the first covering
We construct a covering F of S with the following properties: for some x. Mnemonic rule: when the construction of F is complete, the subcovering E is made of balls with radius equal to r, while the subcoverings labeled with G are made of balls with radius greater than r.
The strategy will be to apply inductively the lemmas from Section 5 at different scales. We thus proceed inductively on j ∈ {0, . . . ,}.
Step 2, case A. If H(x, ̺) ⊂ B 1 5 ̺ 2 (V ) for some V ∈ H k−1 (R m ), then B is one of the balls that we want to keep in our final covering F; we define Step 2, case B. If instead B / ∈ G 1 , this means that H(x, ̺) spans 1 5 ̺ 2 -effectively a k-space L(x, ̺). Thus, applying Lemmas 5.1 to 5.3 with the same constants as in Step 1, Case B, we get: . Now we cover S ∩ B ̺ (x) \ G 2 with balls of radius ̺ 2 such that for any pair B = B ′ of such balls we have 1 5 B ∩ 1 5 B ′ = ∅ and 1 5 B ⊂ B ̺ (x) \ G 2 ; define E 2,x such a covering. Define This concludes Step 2.
After the j th step, we have: If j =, then we are done. Otherwise, we proceed in the same fashion. Let B = B ̺ j (x) ∈ E j .
Step j + 1, case A. If H(x, ̺ j ) ⊂ B 1 5 ̺ j+1 (V ) for some V ∈ H k−1 (R m ), then B is one of the balls that we want to keep in our final covering F; we define (6.7) Step j + 1, case B. If instead B / ∈ G j , this means that H(x, ̺ j ) spans 1 5 ̺ j+1 -effectively a k-space L(x, ̺ j ). Thus, applying Lemmas 5.1 to 5.3 with the same constants as in Case B of the previous steps, we get: . Now we cover S ∩ B ̺ j (x) \ h≤j G h with balls of radius ̺ j+1 such that for any pair B = B ′ of such balls we have 1 5 B ∩ 1 5 B ′ = ∅ and 1 Iterating the procedure until, we obtain the desired construction. Definition 6.3. If F = G 0 ∪ · · · ∪ G −1 ∪ E is the covering just constructed, define the following sets of centers: Moreover, if x ∈ C, we'll also denote by r x the radius of the ball centered at x which is contained in F. Notice that and C = C ⊂ C −1 ⊂ · · · ⊂ C 1 ⊂ C 0 = D.
The next step is probably the most important of the whole construction: indeed, we show that we have a control on the k th powers of the radii of the balls in F. Here is where the refined techniques of Section 4 become involved: we use Reifenberg Theorem 4.4 to achieve the final estimate, and Theorem 4.6 to check Reifenberg's hypothesis. Unfortunately, the proof is a bit intricate: we split it in several subtheorems. Remark. From now on, we will assume that ̺ is of the form 5 −κ for some κ ∈ N. This does not affect in any way the general procedure (at some point we will choose ̺ as an arbitrary number smaller than a certain constant) and simplifies a bit some computations. By Reifenberg Theorem 4.4, the estimate (6.14) is achieved if the condition holds for any ball B τ (w) with w ∈ B 1 (0) and 0 < τ < 1 (or 0 < τ < τ max for some τ max , at the only price of worsening the constants involved). Here µ .
whenever one of the following holds: (ii) B s (x) contains a point of C h+1 \ C h and s ∈ 2̺ −h , 2̺ −(h+1) . Remark. We could obviously state the same property with more general constants in front of the radii involved. This is however the form we will need: notice that the "upper bound" for the radius gains a factor 2.
• If s ∈ ̺ −h , 2̺ −h−1 , then: we can cover B s (x)∩supp(µ h ) with a controlled number c 1 (̺, m) of balls centered in C h with radius ̺ −h , so that we obtain: is also bounded by a constant c 2 (̺, m), because balls centered in (C h+1 \ C h ) with radius 1 5 ̺ −h−1 do not contain points of C other then their center. Thus (6.20) By The next step is to prove that the estimate Equation (6.16) actually holds when h = 0. Subtheorem 6.4.2. There exists a constant C 0 (̺, m) such that: for any x ∈ B 1 (0) and s ∈ 1 5 ̺, ̺ , Proof. An argument already used in Subtheorem 6.4.1: if x = y ∈ C 0 , then B 1 5 ̺ (x) and B 1 5 ̺ (y) are disjoint, thus the number of such centers contained in B s (x) is bounded by a constant (the same c 2 (̺, m) as in the previous proof). Thus which is what we needed.
It may seem that, having an inductive step and a base step, we could already get the volume estimate we need. The problem is that we are applying Subtheorem 6.4.1 with an initial constant C in that keeps getting bigger at any step; instead, we would need in the end a universal constant that only depends on ̺ and m, since the number of steps is not fixed a priori, and we don't want our constants to depend on it. Here is where Reifenberg Theorem comes into play. The trick will be to prove that the estimate holds whenever one of the following holds: Then the following estimate is also true: for all x ∈ B 1 (0) and all s ∈ 1 5 ̺, ̺ −h , where C R is the constant appearing in Reifenberg Theorem 4.4.
Step 1. (Application of Reifenberg Theorem) Clearly, if we are able to prove that holds for all w ∈ B 1 (0) and all τ < ̺ −h , then we can exploit the rescaled version of Reifenberg Theorem (Corollary 4.5), and we get exactly the thesis.
Step 2. (Application of the Estimates on β k µ ) Notice that the integral with respect to µ h appearing in Equation (6.26) is actually a sum on y ∈ C h ∩B τ (w). Let y ∈ C h , and consider β k µ h (y, s). Then: • If s ≤ 1 5 r y , then β k µ h (y, s) = 0, because y is the only point of C h contained in B s (y) (and thus any k-plane through y is a best approximating plane for µ); • If s ≥ 1 5 r y , by property 4 of the covering F (specifically Equation (6.3)), u is not 1 2 η, k + 1 -invariant in B 5δ 0 s (y). Thus we can use Theorem 4.6 withr = 5δ 0 and σ = 5 (for example!) and obtain: where now C J depends on m, p and η only.
More compactly, if we define the following function: then for all s > 0 (smaller than a suitable constant) we have: Step 3. By Step 2 and by Tonelli's Theorem, for a fixed h ∈ {0, . . . , − 1} we have: (6.30) Notice that by the triangle inequality Using again Tonelli Theorem, we also switch the two integrals in µ h+1 , thus getting: Now for all the relevant pairs (z, s) (for which W is not 0) the estimate (6.24) holds: Step 4. Let x ∈ D ℓ , so that r x = ̺ ℓ = 5 −κℓ (see the Remark before the statement). The following estimate holds: where the last inequality is a consequence of property 4 of the covering F (specifically Equation (6.2)) and C 7 depends on ̺ and η. Plugging this information into Equation (6.35) (provided that τ ≤ 1 5 ), we get The left hand side is 0 whenever B 2τ (w) contains a single point of C h ; in all the other cases, the assumption (6.24) holds, thus , we have the desired result.
We can finally prove Proposition 6.4.
Proof of Proposition 6.4. The proof is now a simple induction: by Subtheorem 6.4.2 we have an estimate on µ 0 depending on a constant C 0 ; applying Subtheorem 6.4.1 the same estimate holds for µ 1 with C f = C f (C 0 , m, p, ̺); but then we apply Subtheorem 6.4.3 to improve the constant: the estimate now holds for µ 1 with C R (m). So we can repeat the procedure: the final constant for each µ h will still be C R (m).

Second covering
The goal now is to refine the covering in order to find balls which satisfy a clean energy drop; that is, we get rid in some sense of the sets of type H(x, r), where the uniform energy drop does not happen, and which is already bound to lie in the fattening of a (k − 1)-dimensional plane.
Construction of the second covering Consider again the "first covering" F for S. It is split in F = G ∪ E, where F, E and G have the following properties: 1. Balls in E . = E (0) . = E have radius equal to r = ̺; 2. Balls in G .
3. The estimate B∈F r k B ≤ C I (m) holds, where r B is the radius of B. We now refine F inductively, applying at each step a rescaled version of the procedure from Section 6.1.
Step 1. Consider B ∈ G and the associated (k − 1)-plane V B . We cover B ∩ S with balls of radius ̺r B , divided in three subcoverings: E B (with radius equal to r), D B (satisfying an energy drop condition), W B (wild balls on which we have no control). At this point we have a covering of S of this type: and so that (6.46) Step h + 1. Assume that, for some h ≤ − 1, we have a covering of S of the form Consider a ball B ⋆ ∈ E (h) . Applying a rescaled version of the first construction (and of Proposition 6.4) we first find a covering F B ⋆ for S ∩ B ⋆ of the type where G B ⋆ are balls on which the energy drop condition is verified up to a neighborhood of a (k − 1)-plane, E B ⋆ are balls of radius r = ̺, and B∈F B ⋆ r k B ≤ C I (m)r k B ⋆ . Secondly, re-cover each ball of G B ⋆ with a rescaled version of Step 1, thus obtaining Then define All the conditions 1 to 3 are satisfied with h + 1 instead of h; as for the estimates (6.47), we have and (6.55) Thus, as a consequence of this procedure, we have the following. Proposition 6.5. Let u ∈ W 1,p (Ω, N ) be a p-minimizing map with energy bounded by Λ. Let η > 0 be a constant and 1 ≤ k ≤ m. Assume that E ≤ Λ is such that θ(y, 1) ≤ E for all y ∈ B 1 (0) ∩ S. Let S ⊂ S k η,δ 0 r (u) for some r > 0. There exists a finite covering F ⋆ of S with the following properties: (i) All the radii satisfy r x ≥ r; (ii) The k th powers of the radii are controlled by B∈F ⋆ r k B ≤ C II , where C II depends only on m. Proof. Let̺ =̺(m) = 1 2 K 2 (m), where K 2 is the constant introduced in (6.45). Once̺(m) is chosen, also a constantγ(m, p) is fixed by Subtheorem 6.4.3; as a consequence, δ 0 (m, p) gets determined by Lemma 5.1 and the constant δ(m, p) is fixed as well by Lemma 5.2. Assume without loss of generality that r = ̺ for some ∈ N. Perform the construction (of the first covering and then) of the second covering until Step: then W () = ∅ (by the bounds on the radii), so F () = E () ∪ D () ; moreover, by Equation (6.47), (6.57) This proves the proposition, by setting C II (m) = 2K 3 (m).

Proof of the main theorems
We split the main result Theorem 3.7 in two parts, one concerning the estimate on the volume Vol B r (S k η,δ 0 r (u)) ∩ B 1 (0) , and one for the rectifiability of S k η .

Volume estimate
Theorem 7.1. Let u ∈ W 1,p (Ω, N ) be a p-minimizing map with energy bounded by Λ. Let η > 0 and 1 ≤ k ≤ m. There exists a constant C 1 = C 1 (m, N , p, Λ, η) such that for any r > 0 Vol B r (S k η,δ 0 r (u)) ∩ B 1 (0) ≤ C 1 r m−k . (7.1) The proof is a straightforward consequence of the following lemma: Proof. We proceed by induction on i ∈ N. For i = 0, we can simply take F ⋆ 0 = D ⋆ 0 = {B 1 (0)}. Assume then the lemma is true for some i ≥ 0. Consider a ball B 0 ∈ D ⋆ i , and cover it with c 7 (m) balls of radius 1 5 r B 0 (call D i,B 0 this covering); for each of these balls B consider the rescaling of B (and u) through the transformation that maps it into the unit ball. Applying Proposition 6.5 with E = Λ − iδ and S \ i L i , and scaling back to the original B, we find a covering F ⋆ B = E ⋆ B ∪ D ⋆ B with 1. IfB ∈ E ⋆ B , then rB = r; 2. IfB ∈ D ⋆ B , then θ y, 1 5 rB ≤ Λ − iδ − δ for all y ∈B ∩ S; 3.
Thus, by defining we get the needed result.

Rectifiability
We now tackle the problem of the rectifiability of the strata of type S η k . It is clear that we'll need to use the second part of Theorem 4.4; the technique is basically the same we used for the volume estimates, even with some simplifications. Theorem 7.3. Let u ∈ W 1,p (Ω, N ) be a p-energy minimizing map. For any η > 0 and any 0 ≤ k ≤ m, the stratum S k η (u) is k-rectifiable. As we'll see shortly, the result follows easily from this Lemma. Lemma 7.4. Let m, p, Λ, η be fixed. There exist a universal constant κ(m, p, Λ, N , η) with 0 < κ < 1 such that the following holds. Let u be p-minimizing, and let S ⊂ S k η (u) ∩ B 1 (0) be a H k -measurable subset. There exists a H k -measurable subset R ⊂ S with the following properties:

The set S \ R is k-rectifiable.
Before proving this lemma, which requires some effort, we show how it is applied to prove Theorem 7.3.
Proof of Theorem 7.3. By induction, for any j ∈ N there exists a H k -measurable set R j ⊂ S k η (u) such that: • H k (R j ) ≤ κ j H k S k η (u) ; • The set S \ R j is k-rectifiable.
This is easily proved: the step j = 1 comes from the application of Lemma 7.4 to the stratum S k η (u), while the (j + 1) th step descends from the application of the same Lemma to R j . Now we can defineR . = j∈N R j (7.5) HereR has H k -measure zero; andS is the countable union of sets, each of which is countable union of Lipschitz k-graphs; thereforeS itself is a countable union of Lipschitz k-graphs. This means precisely that S k η (u) is k-rectifiable.
Now we turn to prove Lemma 7.4.
Proof. We can assume that H k (S) > 0, otherwise the statement is trivial.
Step 1. Consider the following map: for x ∈ B 1 (0) and r > 0 (small enough), where θ(x, 0) . = lim s→0 θ(x, s). As r tends to 0, the map f r converges pointwise and decreasingly to the constant function f 0 ≡ 0; moreover, all the maps f r are bounded by the constant map Λ, which is integrable with respect to the measure H k S. Now fix a δ > 0. By the Dominated Convergence Theorem, there exists ar > 0 depending on δ such that S f 5r (x) dH k (x) ≤ δ 2 H k (S). (7.8) Consider the following sets: x ∈ S f 5r(δ) (x) > δ (7.9) G δ . = x ∈ S f 5r(δ) (x) ≤ δ = S \ F δ ; (7.10) observe that, since f 5r is nonnegative, we have: this, combined with Equation (7.8), gives H k (F δ ) ≤ δH k (S). (7.12) We claim that, for δ sufficiently small, the set G δ is k-rectifiable; if we manage to show this, then the lemma is proved. In order to prove this claim, we consider a finite covering {Br(x i )} L i=1 of G δ made with balls of the fixed radiusr(δ). It is sufficient to show that for δ small G δ ∩ Br (δ) (x i ) is rectifiable for any i: our main aim will be now to check the applicability of the second Reifenberg Theorem (Theorem 4.4, part (ii)), that gives exactly that result. that inequality: applying Tonelli Theorem (twice), we find: θũ(y, 5s) − θũ(y, s) dµ δ (y) ds s = = C 10 C 1 B 2r (w) r 0 θũ(y, 5s) − θũ(y, s) ds s dµ δ (y). (7.20) Consider for a moment the inner integral; r can simply be bounded by 1. We use basically the same trick we exploited in Proposition 6.4: we get exactly the hypothesis needed for the second part of Reifenberg Theorem: thusG δ is k-rectifiable, and tracing back the steps of the proof this proves the k-rectifiability of G δ .