Minimal graphs of arbitrary codimension in Euclidean space with bounded 2-dilation

For any $\Lambda>0$, let $\mathcal{M}_{n,\Lambda}$ denote the space containing all locally Lipschitz minimal graphs of dimension $n$ and of arbitrary codimension $m$ in Euclidean space $\mathbb{R}^{n+m}$ with uniformly bounded 2-dilation $\Lambda$ of their graphic functions. In this paper, we show that this is a natural class to extend structural results known for codimension one. In particular, we prove that any tangent cone $C$ of $M\in\mathcal{M}_{n,\Lambda}$ at infinity has multiplicity one. This enables us to get a Neumann-Poincar$\mathrm{\acute{e}}$ inequality on stationary indecomposable components of $C$. A corollary is a Liouville theorem for $M$. For small $\Lambda>1$(we can take any $\Lambda<\sqrt{2}$), we prove that (i) for $n\leq7$, $M$ is flat; (2) for $n>8$ and a non-flat $M$, any tangent cone of $M$ at infinity is a multiplicity one quasi-cylindrical minimal cone in $\mathbb{R}^{n+m}$ whose singular set has dimension $\leq n-7$.


Introduction
It has been a central aim of the theory of minimal graphs in Euclidean space to derive conditions under which an entire n-dimensional minimal graph, that is, a graph defined on all of R n , of codimension m, that is, sitting in R n+m , is affine linear. This is the famous Bernstein problem. Bernstein himself proved it for two-dimensional entire minimal graphs in R 3 . For codimension 1, but in higher dimensions, this holds for n ≤ 7 through successive efforts of Fleming [28], De Giorgi [17], Almgren [1] and culminating with Simons [49]. However, it is no longer true for n ≥ 8 by an example of Bombieri, De Giorgi and Giusti [7]. If we assume, however, in addition, that the graph has a bounded gradient, then this holds for any dimension n by a result of Moser [44]. This is the so-called weak version of the Bernstein Theorem.
Research on the Bernstein theorem was a crucial motivation for the great development of geometric measure theory. It is well known that an entire codimension 1 minimal graph Σ in R n+1 is area-minimizing, i.e., the current associated with Σ is a minimizing current. Fleming [28] proved that any tangent cone C Σ of Σ at infinity is a singular area-minimizing cone in R n+1 , which implies that it is a stable minimal hypersurface with multiplicity one. De Giorgi [17] further showed that C Σ is cylindrical, i.e, C Σ isometrically splits off a factor R.
In higher codimension, Almgren [3] derived sharp codimension 2 estimates for the singular sets of minimizing currents. In [18,19,20], De Lellis-Spadaro developed a new approach to the regularity of minimizing currents and could in particular rederive Almgren's structure theorem. In general, however, minimal graphs of higher codimension are not minimizing. Nevertheless, some general structural results about minimal graphs in higher codimension are available. Utilizing the graph property, it is possible to study the structure and rigidity of minimal graphs of arbitrary codimension under some conditions but without a minimizing assumption.
In this paper, we approach this issue via studying tangent cones of minimal graphs at infinity. First of all, we need some condition on minimal graphs to guarantee that they have Euclidean volume growth. For that purpose, we now introduce the concept of bounded k-dilation. As we shall see, this condition provides a natural generalization of the bounded slope condition that has been used in many other papers about higher codimension Bernstein theorems.
Let f : M 1 → M 2 be a locally Lipschitz map between Riemannian manifolds M 1 , M 2 . For an integer k ≥ 1, f is said to have k-dilation ≤ Λ for some constant Λ ≥ 0 if f maps each k-dimensional submanifold S ⊂ M 1 to an image f (S) ⊂ M 2 with H k (f (S)) ≤ ΛH k (S) where H k denotes k-dimensional Hausdorff measure. We note that this condition is the more restrictive the smaller k (up to the constant Λ). For k = 1, it simply means that f is Λ-Lipschitz. This is, of course, a strong condition. In this paper, we shall explore the case k = 2, that is, consider maps, or more precisely locally Lipschitz minimal graphs given by f : R n → R m with bounded 2-dilation. But even for n = m = 2, minimal graphs have not necessarily bounded 2-dilation for their graphic functions, as the 2-dimensional minimal graphs given by (Re e z , −Im e z ) : R 2 → R 2 . We should note, however, that these examples have zero k-dilation for k ≥ 3. Therefore, we cannot hope for a good theory on tangent cones of minimal graphs at infinity when we only restrict some k-dilation for k ≥ 3.
For any constant Λ ≥ 0, let M n,Λ denote the space containing all the locally Lipschitz minimal graphs over R n of arbitrary codimension m ≥ 1 with 2-dilation of their graphic functions ≤ Λ. Here, the codimension m is bounded by a constant depending only on n, Λ using a result of Colding-Minicozzi [14] (which will be explained later), and this is the reason why the notation M n,Λ does not contain m. De Giorgi [16] obtained the regularity of codimension 1 locally Lipschitz minimal graphs (see also Moser [43]), while high codimensional locally Lipschitz minimal graphs may really have singularities. We can already provide the following geometric intuition.
(2) The product of Euclidean space R ℓ and any minimal graphical hypersurface in R n+1 is contained in M n,Λ .
However, having a bounded Lipschitz constant is a much stronger condition than having bounded 2-dilation. For bounded domains Ω ⊂ R n , we have constructed many examples of n-minimal graphs over Ω of 2-dilation ≤ 1 with arbitrary large slope, where they do not live in any (n + 1)-dimensional Euclidean subspace [23].
Having thus sketched the basic setting, we can explain the two main objectives of this paper. The first objective is the development of the theory of minimal graphs of arbitrary codimension in Euclidean space with uniformly bounded 2-dilation of their graphic functions. The principal aim is to understand the geometric structure, including multiplicity and stability, of such minimal graphs at infinity when the graphic functions are allowed to grow faster than linearly. Without Euclidean volume growth, geometric measure theory cannot say much about the possible limits of minimal graphs at infinity. Therefore, our first crucial issue will be to derive Euclidean volume growth from bounded 2-dilation. Importantly, their tangent cones at infinity have multiplicity one (see Theorem 1.1 below), which plays an essential role in establishing the Neumann-Poincaré inequality on stationary indecomposable components of the tangent cones.
This general theory prepares the ground for the second objective, to study the rigidity of minimal graphs of arbitrary codimension in Euclidean space without the assumption of the bounded slope. Under the condition of bounded 2-dilation for graphic functions, we prove a Liouville theorem for minimal graphs (see Theorem 1.2) with Neumann-Poincaré inequality, which generalizes the classical Liouville theorem obtained by Bombieri-De Giorgi-Miranda [8]. Concerning the Bernstein theorem, we prove that there exists a constant Λ > 1 such that for n ≤ 7 each M ∈ M n,Λ is flat, for n ≥ 8 and any non-flat M ∈ M n,Λ , any tangent cone of M at infinity is a quasi-cylindrical minimal cone (see Theorem 1.3). Here, the dimension 7 is sharp by the counter-examples by Bombieri-De Giorgi-Giusti [7], and a quasi-cylindrical minimal cone is exactly a cylinder in the codimension one case (see the definition below). Moreover, the constant Λ can be arbitrarily chosen < √ 2. The constant √ 2 is not essential for our whole theory but plays a role in making the volume functional subharmonic (see Corollary 7.1).
Let us now be more specific. In [12], Cheng-Li-Yau estimated the codimension for each minimal cone in Euclidean space via its density. Colding-Minicozzi [14] proved the dimension estimates for coordinate functions on more general minimal submanifolds of Euclidean volume growth. In §3, using the result of [14], we prove that every M ∈ M n,Λ has Euclidean volume growth with the density bounded by a constant depending only on n, Λ (see Lemma 3.2 for details). In particular, M lives in a Euclidean subspace with the codimension bounded by a constant depending on n, Λ. Without the condition of bounded 2-dilation, of course minimal graphs need not have Euclidean volume growth, like the 2-dimensional minimal graph by (Re e z , −Im e z ) : R 2 → R 2 that we have already mentioned above (see Remark 3.1 for details).
Let v denote the slope of a minimal graph M over R n defined by detg ij , where g ij dx i dx j is the metric of M induced from the ambient Euclidean space R n+m . In fact, v =ṽ • γ, whereṽ is a natural function in the Grassmannian manifold G n,m and γ stands for the Gauss map. The slope describes how far M is from the fixed nplane R n . We will explain its geometric meaning later in detail from the perspective of the Grassmannian manifold. For 0 ≤ Λ < 1 and M ∈ M n,Λ with codimension m ≥ 1, the correspondingṽ is convex in G n,m , and then v =ṽ • γ is subharmonic. This leads to flatness of M proved by M.T. Wang [50] under the bounded slope condition. When Λ > 1, the functionṽ in general is not convex. In fact, our class M n,Λ is much richer when Λ > 1 is larger.
If M ∈ M n,Λ is minimizing, then it is not difficult to prove the multiplicity one of the tangent cone of M at infinity. However, the multiplicity one holds without the minimizing condition. Moreover, we can show the stablity of the tangent cone in some small neighborhood via the slope function v in §4. Theorem 1.1. Let M be a locally Lipschitz minimal graph over R n of codimension m ≥ 1 with bounded 2-dilation of its graphic function. Then any tangent cone C of M at infinity has multiplicity one. Moreover, if a tangent cone of C contains a line perpendicular to the n-plane {(x, 0 m ) ∈ R n × R m | x ∈ R n }, then it is a cylindrical stable minimal cone in an (n + 1)-dimensional Euclidean subspace. Remark 1.1. We do not know whether the cylindrical stable minimal cone obtained in Theorem 1.1 is minimizing. Even we have no a priori dimensional estimates on its singular set of the cylindrical stable minimal cone.
In fact, we prove a somewhat stronger version than Theorem 1.1, where we do not need to require that the graphs are entire. The proof of multiplicity one is somewhat difficult because of the complex interplay between geometry and analysis for submanifolds of high codimensions. Our strategy therefore consists in using the structure of the minimal surface system to treat the higher codimension case as a perturbation of the codimension one case with error terms including some quantities from the other codimensions. The key idea is projecting the minimal graph M to a hypersurface M ′ in a suitable (n + 1)-dimensional Euclidean subspace. In general M ′ is no more minimal, but from M we can get effective estimates up to a set of arbitrary small measure in the scaling sense.
In 1969, Bombieri-De Giorgi-Miranda [8] showed a Liouville theorem for solutions to the minimal surface equation via interior gradient estimates (see also the exposition in chapter 16 of [32]). For high codimensions, M.T. Wang proved a Liouville type theorem for minimal graphs with positive graphic functions under the area-decreasing condition [51]. This condition also means that the graphic functions have 2-dilation bounded by Λ < 1.
Let B r denote the ball in R n+m with the radius r and centered at the origin. Let B r denote the ball in R n with the radius r and centered at the origin. Inspired by Bombieri-Giusti [9], we establish the Neumann-Poincaré inequality on stationary indecomposable components of tangent cones of minimal graphs at infinity, and get Harnack's inequality for positive harmonic functions on the components. Then we can get the following Liouville theorem without the subharmonic functions in terms of the gradient functions on minimal graphs in §6, which generalizes Bombieri-De Giorgi-Miranda's result in [8], and improves Wang's result in [51]. Theorem 1.2. Let M = graph u be a locally Lipschitz minimal graph over R n of codimension m ≥ 2 with bounded 2-dilation of u = (u 1 , · · · , u m ). If for each α ∈ {2, · · · , m}, and then M is flat.
In the proof of Theorem 1.2, we need a De Giorgi type result, i.e., every tangent cone of M at infinity is a cylinder if it lives in an (n + 1)-dimensinoal Euclidean subspace. Using this, we can reduce the problem to the codimension 1 case with the Neumann-Poincaré inequality. Then we can finish the proof by the regularity of codimension 1 Lipschitz minimal graphs from De Giorgi [16].
According to this introduction, people have an essentially complete understanding of the classical case of codimension 1 Bernstein theoerm. The question then naturally arises what we can say for higher codimension.
We can now briefly review the Bernstein type theorems for minimal graphs of bounded slope of codimension m ≥ 2. Bounded slope condition is an adequate generalization of bounded gradient to higher codimension. For m ≥ 2, Chern and Osserman [13] showed that any 2-dimensional minimal graph of bounded slope in R 2+m is flat, which was generalized in [33](for m = 2) and [36] without bounded slope. Barbosa and Fischer-Colbrie proved this for 3-dimensional minimal graphs of bounded slope in [6] [27]. Recently, Assimos-Jost [5] proved a Bernstein type theorem for minimal graphs of bounded slope in codimension m = 2. For dimension ≥ 4 and codimension ≥ 3, this no longer holds, by an example of Lawson and Osserman [38]. However, Bernstein type theorem holds for small slope v such as Simons [48], Hildebrandt-Jost-Widman [34], Jost-Xin [35], Jost-Xin-Yang [37]. In particular, any minimal graph of slope ≤ 3 in Euclidean space is flat [37].
But we may also ask whether there exist other natural conditions that ensure a higher codimension Bernstein theorem. We point out that when v =ṽ • γ ≤ 3, v need no longer be convex on the Grassmannian manifold, the target manifold of the Gauss map of M. Note that v = 9 in Lawson-Osserman's example mentioned above. Hence, if every minimal graph M ∈ M n,Λ is flat, then Λ must be small. Without the conditions (1.1)(1.2) of Theorem 1.2, we can study the structure of tangent cones of minimal graphs at infinity for small Λ > 1 using Theorem 1.1 and the Neumann-Poincaré inequality. Now let us introduce a concept 'quasi-cylindrical' for studying unbounded slope case. For an integer 1 ≤ k ≤ n and a k-varifold V in R n+m , V is said to be highcodimensional quasi-cylindrical (quasi-cylindrical for short) if there are a countably (k −1)-rectifiable set E in R n , and there is a countably 1-rectifiable normalized curve Note that 'quasi-cylindrical' is simply 'cylindrical' for m = 1. Hence, quasi-cylindrical varifolds can be seen as a generalization of cylindrical varifolds in the case of high codimensions in Euclidean space. Theorem 1.3. There exists a constant Λ > 1 such that if M is a locally Lipschitz minimal graph over R n of codimension m ≥ 1 with 2-dilation of its graphic function bounded by Λ, then either M is flat, or M is non-flat with n ≥ 8. Furthermore, for non-flat M, any tangent cone of M at infinity is a multiplicity one quasi-cylindrical minimal cone in R n+m whose singular set has dimension ≤ n − 7.
In the codimension one case, Theorem 1.3 for n ≤ 7 has been proved by Simons [49], and the dimension 7 is sharp from [7]. So Theorem 1.3 is a generalization in high codimension. Actually, our proof is based on Simons' result that any stable minimal regular hypercone C in R k is flat for k ≤ 7, where Simons' result holds allowing that C has singularities in some sense, see [52]. For the case n ≥ 8 of Theorem 1.3, the singular set of dimension ≤ n − 7 is obtained through stable minimal hypercones combining Theorem 1. In general, the constant Λ in Theorem 1.3 can be arbitrarily chosen < √ 2, and we do not know whether √ 2 is sharp, though it appears naturally for the subharmonicity of log v. In fact, we can find a slightly weaker condition for the Bernstein theorem in all dimensions in the situation of bounded slope (see Theorem 7.1).

Preliminaries
Let R k denote the Euclidean space for each integer k ≥ 1, and 0 k denote the origin of R k . Let B k r (x) be the ball in R k with the radius r and centered at x ∈ R k , and B r (x) = B n r (x) for convenience. Let B r (x) be the ball in R n+m with the radius r and centered at x ∈ R n+m . We denote B r = B r (0 n ), B r = B r (0 n+m ) for convenience. We always use D to denote the derivative on R n . For any subset E in R n , for any constant 0 ≤ s ≤ n we define H s (E) to be the s-dimensional Hausdorff measure of E. Let ω k denote the k-dimensional Hausdorff measure of B k 1 (0 k ). We use the summation convention and agree on the ranges of indices: Let M be an n-dimensional smooth Riemannian manifold, and M → R n+m be an isometric immersion. Let ∇ and∇ be Levi-Civita connections on M and R m+n , respectively. Here, ∇ is induced from∇ naturally. The second fundamental form B M on the submanifold M is defined by B M (ξ, η) =∇ ξ η − ∇ ξ η = (∇ ξ η) N for any vector fields ξ, η along M, where (· · · ) N denotes the projection onto the normal bundle NM (see [55] for instance). Let e 1 , · · · , e n be a local orthonormal frame field near a considered point in M. Let |B M | 2 denote the square norm of B M , i.e., |B M | 2 = n i,j=1 |B M (e i , e j )| 2 (This notation should not be confused with that of the ball B r ). Let H denote the mean curvature vector of M in R n+m defined by the trace of B M , i.e., H = n i=1 B M (e i , e i ). This is a normal vector field. M is said to be minimal if H ≡ 0 on M. More generally, M has parallel mean curvature if ∇H ≡ 0.
2.1. Grassmannian manifolds and Gauss maps. We will study minimal submanifolds in ambient Euclidean space. The target manifolds of the Gauss map of minimal submanifolds are Grassmannian manifolds. For convenience of later application, let us describe the geometry of the Grassmannian manifolds. In R n+m all the oriented n-subspaces constitute the Grassmann manifold G n,m , which is the Riemannian symmetric space of compact type SO(n+m)/SO(n)×SO(m), where SO(k) denotes the k-dimensional special orthogonal group for each integer k. G n,m can be viewed as a submanifold of some Euclidean space via the Plücker embedding. The restriction of the Euclidean inner product on M is denoted by w : G n,m × G n,m → R where P is presented by a unit n-vector e 1 ∧ · · · ∧ e n , Q is presented by another unit n-vector f 1 ∧ · · · ∧ f n , and W = e i , f j is an (n × n)-matrix. It is well-known that with O an orthogonal matrix and 0 ≤ µ 2 i ≤ 1 for each i. Putting p min{m, n}, then at most p elements in {µ 2 1 , · · · , µ 2 n } are not equal to 1. Without loss of generality, we can assume µ 2 i = 1 whenever i > p. For a unit vector ξ = i a i e i ∈ P , let ξ * denote its projection into Q, i.e., Certainly, the matrix ( l e i , f l e j , f l ) i,j has eigenvalues µ 2 1 , · · · , µ 2 n . Hence, we can introduce the Jordan angles θ 1 , · · · , θ n between two points P, Q ∈ G n,m defined by which are actually critical values of the angle between a nonzero vector ξ in P and its orthogonal projection ξ * in Q as ξ runs through P (see Wong [54] or Xin [55] for further details).
We also note that the µ 2 i can be expressed as where λ i has explicit meaning in studying graphs of high codimensions (We will explain it later). The distance between P and Q is defined by . Let E iα be the matrix with 1 in the intersection of row i and column α and 0 otherwise. Then, sec θ i sec θ α E iα form an orthonormal basis of T P G n,m with respect to the canonical Riemannian metric g on G n,m (compatible to (2.4)). Denote its dual frame byω iα . Then g can be written as Denoteω βα = 0 for β ≥ n + 1.
where w is defined in (2.1). The Jordan angles between P and P 0 are defined by In [35], we already have the largest geodesic convex subset B JX (P 0 ), which is defined by sum of any two Jordan angles being less than π 2 for any point P ∈ B JX (P 0 ). It is easily seen that T 2,1 = B JX (P 0 ). Hence, the distance function from P 0 is convex on T 2,1 , but is no longer convex on T 2,Λ when Λ > 1.
Combining (2.10)(2.11), it follows that (see (3.9) in [56] for instance) (2.12) Hess logṽ(·, P 0 ) = g + Let M be an n-dimensional smooth submanifold in R n+m . Around any point p ∈ M, we choose an orthonormal frame field e i , · · · , e n+m in R n+m , such that {e i } are tangent to M and {e n+α } are normal to M. Let {ω 1 , · · · , ω n+m } be its dual frame field so that the metric on M is i ω 2 i and the Euclidean metric in R n+m is The Levi-Civita connection forms ω ab of R n+m are uniquely determined by the equations where a, b = 1, · · · , n + m. Moreover, we have the equations where h α,ij = ∇ e i e j , e n+α are the coefficients of the second fundamental form B M of M in R n+m . The Gauss map γ : M → G n,m is defined by via the parallel translation in R n+m for every p ∈ M. We also have (2.14) Up to an isotropic group SO(n) × SO(m) action, we can assume ω i n+α = γ * ω iα at p (see section 8.1 in [55] for instance). Combining (2.13), we obtain By the Ruh-Vilms theorem [46], the mean curvature of M is parallel if and only if its Gauss map γ is a harmonic map. Now we define a function which will play a basic role in this paper. Using the composition formula, in conjunction with (2.12), (2.14) and (2.15), and the fact that τ (γ) = 0 (the tension field of the Gauss map vanishes [46]), we can deduce the following important formula (see also Lemma 1.1 in [27] or Prop. 2.1 in [50]).
Proposition 2.1. Let M be an n-dimensional smooth submanifold in R n+m with parallel mean curvature. Then at any considered point p

2.2.
Varifolds and currents from geometric measure theory. Let us recall Almgren's notion of varifolds from geometric measure theory (see [40][48] for more details), which is a generalization of submanifolds. For a set S in R n+m , we call S nrectifiable if S ⊂ S 0 ∪F 1 (R n ), where H n (S 0 ) = 0, and F 1 : R n → R n+m is a Lipschitz mapping. More general, we call S countably n-rectifiable if S ⊂ S 0 ∪ ∞ k=1 F k (R n ), where H n (S 0 ) = 0, and F k : R n → R n+m are Lipschitz mappings for all integers k ≥ 1. By Rademacher's theorem, a countably n-rectifiable set has tangent spaces at almost every point. Suppose H n (S) < ∞. Let θ be a positive locally H n integrable function on S. Let |S| be the varifold associated with the set S. The associated varifold V = θ|S| is called a rectifiable n-varifold. θ is called the multiplicity function of V . In particular, the multiplicity of |S| equal to one on S. If θ is integer-valued, then V is said to be an integral varifold. Associated to V , there is a Radon measure µ V defined by µ V = H n θ, namely, For an open set U ⊂ R n+m , V is said to be stationary in U if Here, div S Y is the divergence of Y restricted on S. When we say an n-dimensional minimal cone C in R n+m , we mean that C is an integral stationary varifold with support being a cone. One of the most important properties of the stationary varifold V is that the function is monotone non-decreasing for 0 < ρ < ρ 0 with ρ 0 ≤ d(x, ∂V ) and x ∈ R n+m . By Rademacher's theorem, we can define the derivative ∇ V on V for Lipschitz functions almost everywhere (see Definition 12.1 in [48] for instance).
Let {V j } j≥0 be a sequence of integral stationary n-varifolds with the Radon measure µ V j associated to V j satisfying for each W ⊂⊂ U.
By compactness theorem of varifolds (see Theorem 42.7 and its proof in [48]), there are a subsequence V j ′ and an integral stationary n-varifold V ∞ such that V j ′ converges to V ∞ in the varifold (Radon measure) sense.
Let us recall Sobolev inequality on a stationary varifold V in U. Michael-Simon [42] proved the following Sobolev inequality on V (actually, for general submanifolds of mean curvature type). There is a constant c n > 0 depending only on n such that holds for each Lipschitz function f with compact support in U. Recently, Brendle [10] obtained the sharp Sobolev constant for minimal submanifolds of codimensions ≤ 2 (see [41] for the relative version).
Let D n (U) denote the set including all smooth n-forms on the open U ⊂ R n+m with compact supports in U. Denote D n (U) be the set of n-currents in U, which are continuous linear functionals on D n (U). For each T ∈ D n (U) and each open set W in U, one defines the mass of T on W by with |ω| U = sup x∈U ω(x), ω(x) 1/2 . Let ∂T be the boundary of T defined by ∂T (ω ′ ) = T (dω ′ ) for any ω ′ ∈ D n−1 (U). For T ∈ D n (U) , T is said to be an integer multiplicity current if it can be expressed as where S is a countably n-rectifiable subset of U, θ is a locally H n -integrable positive integer-valued function, and ξ is an orientation on S, i.e., ξ(x) is an n-vector representing the approximate tangent space T x S for H n -a.e.
x. Let f : U → R p be a C 1 -mapping with p ≥ n, and f * ξ denote the push-forward of ξ, which is an orientation of f (S) in R p . We define f (T ) ∈ D n (R p ) by letting is an integer multiplicity current in R p .
Let |T | denote the varifold associated with T , i.e., |T | = θ|S|. If both T and ∂T are integer multiplicity rectifiable currents, then T is called an integral current. Federer and Fleming [26](see also 27.3 Theorem in [48]) proved a compactness theorem (or referred to as a closure theorem): a sequence of integral currents T j ∈ D n (U) with M(T j ) and M(∂T j ) uniformly bounded admits a subsequence that converges weakly to an integral current. For an integer k ≥ 1, we recall that an integral current T ∈ D k (R n+m ) is decomposable in U (see ) if there exist integral currents T 1 , T 2 ∈ D k (U) with T 1 U, T 2 U = 0 such that for any W ⊂⊂ U. Here, T 1 , T 2 are called components of T U. On the contrary, T is said to be indecomposable in U. If T is decomposable (indecomposable) in any open W ⊂⊂ R n+m , then we say T decomposable(indecomposable) for simplicity.
All n-dimensional minimal graphs in R n+1 are area-minimizing, which is not true for the higher codimension case in general. Bombieri-Giusti [9] proved that every codimension one area-minimizing current in Euclidean space is indecomposable, and the following uniform Neumann-Poincaré inequality holds on any area-minimizing hypersurface Σ in R n+1 . There is a constant c n > 0 depending only on n such that for any x ∈ Σ, r > 0, any Lipschitz function f on B r (x) The uniform Neumann-Poincaré inequality plays a significant role in the mean value inequality for superharmonic functions. As applications, they got several impressive results for minimal graphs of codimension 1. Note that (2.23) does not hold for all minimal hypersurfaces; for instance, the catenoid is a counterexample.
Let κ 1 , · · · , κ n denote the singular values of the Jabobi matrix df at any considered differentiable point of f . We let |Λ 2 df | be the 2-dilation of f defined by In the case f being a locally Lipschitz map from an open Ω ⊂ R n into R m , its graph defines a locally Lipschitz submanifold M in R n+m . Let Ω * be the largest subset of Ω such that f is C 1 on Ω * , and regM = {(x, f (x))| x ∈ Ω * }. regM is called the regular part of M. Now we have the usual Gauss map from regM. Let {E 1 , · · · , E n+m } be the standard orthonormal basis of R m+n . At each point in regM its image n-plane P under the Gauss map is spanned byf 1 , · · · ,f n with where f α i = ∂f α ∂x i . Let θ 1 , · · · , θ n be the Jordan angles between E 1 ∧ · · · ∧ E n and f 1 ∧ · · · ∧f n . Let ξ denote an orientation of M defined by In particular, ξ is continuous on regM, and ξ(x) represents the tangent space T x M as a unit n-vector for each x ∈ regM. We denote [|M|] ∈ D n (Ω × R m ) as the n-current associated with M and its orientation ξ. Let λ i = tan θ i as (2.3), then λ 1 , · · · , λ n are the singular values of df at each point x ∈ Ω. Namely, λ 2 i are eigenvalues of the matrix α ∂f α ∂x i ∂f α ∂x j . Hence, the Gauss image of any point in regM is spanned by orthonormal vectors 1 Suppose that the n-plane P 0 in (2.6) is spanned by E 1 , · · · , E n . Recalling (2.7), we have a conclusion: Let u = (u 1 , · · · , u m ) be a locally Lipschitz (vector-valued) function on an open Ω ⊂ R n . Let g ij = δ ij + m α=1 ∂ x i u α ∂ x j u α , and (g ij ) be the inverse matrix of (g ij ) for almost every point in Ω. Let M be the graph of the function u, which is countably n-rectifiable. We can define the slope function of M by on Ω. Note that we also see u, v as the functions on M by letting u(x, u(x)) = u(x) and v(x, u(x)) = v(x) for every x ∈ Ω. If the varifold associated with M is stationary, then all the coordinate functions are weakly harmonic on M (see [15] for instance), i.e., all the x 1 , · · · , x n and u 1 , · · · , u α are weakly harmonic on M. Namely, for any Lipschitz function φ on Ω with compact support on Ω, there holds We call M a minimal graph if and only if the varifold associated with M is stationary. Unlike minimal graphs of codimension one, minimal graphs of higher codimensions may be only Lipschitz as in the examples by Lawson-Osserman [38]. For simplicity, we say that the graph M has 2-dilation bounded by Λ, if its graphic function u : Ω → R m has 2-dilation bounded almost everywhere by Λ. Namely, |Λ 2 du| ≤ Λ H n -a.e. on Ω.
From the interior regularity theorem of Morrey, u is smooth at the differentiable points. Then from (2.28), u satisfies the following minimal surface system at differentiable points of u. By [45] (or [38]), (2.29) is equivalent to at differentiable points of u.

Volume estimates for minimal graphs
Let M be a locally Lipschitz minimal graph of the graphic function u = (u 1 , · · · , u m ) over B R of codimension m ≥ 1 in R n+m . Then the induced metric of the graph is g ij = δ ij + ∂u α ∂x i ∂u α ∂x j , and we denote v = (det(g ij )) 1 2 . Let D denote the derivative on R n , and ∇ denote the Levi-Civita connection of the regular part of M. Lemma 3.1. Suppose that u has 2-dilation bounded by Λ. Then we have a volume estimate: Proof. For proving (3.1), we only need to consider the ball B r with B 2r ⊂ B R × R m . Without loss of generality, we can assume u α (0) = 0 for each α. Let u α r be a function on B 2r defined by For any δ ∈ (0, r], we define a non-negative Lipschitz function η on R n given by We can check which is obvious when m = 1. Combining (3.2) and (3.3), one has For each considered point in B (1+δ)r \B r , we can assume Moreover, there is an orthonormal matrix (a αβ ) so that ∂u α ∂x j = a αj λ j = a αj tan θ j (We let a αj = 0 for j ≥ m + 1). Hence, for each α Under the condition of the 2−dilation bounded by Λ, namely, tan where c and c ′ depend on Λ and n. Using Cauchy inequality, from (3.5) we get Thus, This completes the proof.
If M is an entire minimal graph, the estimation in (3.1) of Lemma 3.1 can be independent of the codimension m.
Lemma 3.2. Let M = graph u be a locally Lipschitz minimal graph over R n of codimension m ≥ 2 with sup R n |Λ 2 du| ≤ Λ. Then there is a constant C n,Λ ≥ 1 depending only on n, Λ such that M is contained in some affine subspace of dimension ≤ C n,Λ and for any ball Proof. From Colding-Minicozzi (Corollary 1.4 in [14]), M must be contained in some affine subspace V of the dimension where c n = 3n n−1 2 n+3 e 8 and In particular, for p = n, M is flat. Let V be the linear space spanned by vectors in V and vectors in the n-plane {(x, 0 m ) ∈ R n × R m | x ∈ R n }. Then V has dimensioñ p ≤ p + n. Hence up to an isometric transformation of R m , M can be written as a graph over R n in Rp with the graphic function w satisfying sup R n |Λ 2 dw| ≤ Λ. From Lemma 3.1 and (3.8), we have which implies p ≤ c 2 n c 2 n,Λ . Therefore, from Lemma 3.1 again, we get We complete the proof by the monotonicity of r −n H n (M ∩ B r (x)) on r.
The constant c n,Λ in (3.1) or C n,Λ in (3.7) depends on Λ in general. Even, without the bounded 2-dilation, minimal graphs have much faster volume growth in view of the following remark.
is a special Lagrangian submanifold in R 4 , and in particular, minimal. Let φ = Re e z = e x cos y. Then Dφ = (Re e z , −Im e z ). By a straightforward computation, which implies In other words, M has volume growth strictly larger than Euclidean.
Let Ω be an open set in B R ⊂ R n . For each α = 1, · · · , m, let M α be the graph in Ω × R defined by By a diagonal argument, it is clear that for each α. At any C 1 -point of u, the unit normal vector of M α can be written as where E j is a unit constant vector in R n+1 with respect to the axis x j . For each fixed α, let W denote a bounded open set with n-rectifiable ∂W in R n+1 such that there is a constant γ > 0 satisfying (3.18) H n (∂W ∩ B r (x)) ≥ γr n for any x ∈ ∂W, any r ∈ (0, 1).
Let Y be a measurable vector field in Ω defined by By parallel transport, we obtain a Lipschitz vector field in Ω × R, still denoted by Y . Let∇ denote the Levi-Civita connection of R n+1 . Then from (2.28) and the co-area formula, Let ν ∂W be the outward unit normal vector to the regular part of ∂W . Since Y, E n+1 = 0 a.e. on Ω × R, we may denote Y, E n+1 = 0 on ∂W . By the definition of Y , Y, ν ∂W is well-defined H n -a.e. on ∂W . From (3.18) and Ambrosio-Fusco-Pallara (in [4], p. 110), letting ε → 0 in (3.22) implies Recalling (3.17), we have

Cylindrical minimal cones from minimal graphs
Let M k be a sequence of n-dimensional locally Lipschitz minimal graphs over B R k in R n+m of codimension m ≥ 1 with R k → ∞, and the graphic functions e. for some constant Λ > 0. We may suppose that |M k | converges in the varifold sense to a minimal cone C in R n+m with 0 n+m ∈ C. From Lemma 3.1, the multiplicity of C has a upper bound depending only on n, m, Λ.
, u k has uniformly linear growth. Then the minimal cone C has multiplicity one from Lemma 10.1, which completes the proof of Theorem 1.1. Now we assume that there is a point y * = (0 n , y * ) ∈ sptC with 0 m = y * ∈ R m . Without loss of generality, we assume y * = (1, 0, · · · , 0) ∈ R m , then y * = E n+1 . Let Proof. For any regular point x ∈ sptC y * , the tangent cone of C y * at x is an n-plane with constant integer multiplicity by constancy theorem (see Theorem 41.1 in [48]).
Let T x C y * denote the tangent space of C y * at x, which is an n-plane with constant multiplicity. We represent the support of T x C y * by an n-vector τ 1 ∧ · · · ∧ τ n with orthonormal unit vectors τ i . Since sptC y * splits off a line ty * isometrically, then From the construction of C y * , there is a sequence of minimal graphs Σ k in R n × R m such that |Σ k | converges to C y * in the varifold sense. Here, Σ k is a rigid motion of M k . Let ξ k be an orientation of Σ k for each k (see (2.25)). Since |Σ k | converges in the varifold sense to C y * , then there is a sequence r k → ∞ such that 1 r k Σ k ∩ B r 2 k (y * ) converges to T x C y * in the varifold sense. Hence, up to a choice of the subsequence, there is a sequence of regular points Let λ 1,k , · · · , λ n,k be the singular values of the matrix Du k at (x k 1 , · · · , x k n ) with λ j,k ≥ 0 for all j = 1, · · · , n, and e j,k = 1 for each integers 1 ≤ j ≤ n and k ≥ 1. Then {e j,k } n j=1 forms an orthonormal basis of T x k M k (up to a permutation of λ 1,k , · · · , λ n,k and a rotation of R n ), and we can choose ξ k (x k ) = e 1,k ∧ · · · ∧ e n,k . From (4.1), the assumption |Λ 2 du k | ≤ Λ a.e. and ξ k (x k ) → τ 1 ∧ · · · ∧ τ n , we get λ 1,k → ∞, n j=2 λ j,k → 0 as k → ∞, and then τ 1 ∧ · · · ∧ τ n = E 2 ∧ · · · ∧ E n+1 represents the orientation of T x C y * . Note that sptC y * splits off the line {tE n+1 | t ∈ R} isometrically. So there is an (n − 1)-dimensional cone C * in R n such that sptC y * can be written as This completes the proof. Let and (g ij k ) n×n be the inverse matrix of (g k ij ) n×n . Let π * denote the projection from R n+m into R n by For any For studying the multiplicity of the cone C, we only need to prove it at any regular point of sptC. Hence, it suffices to prove the following lemma. Proof. From the proof of Lemma 4.1, we can assume sptC = {(x 1 , · · · , x n , y 1 , · · · , y m ) ∈ R n × R m | x 1 = 0, y 2 = · · · = y m = 0}, or else we have finished the proof by Lemma 10.1 in the Appendix II. For each k, let x k = (x, u k (x)) ∈ R n × R m be a vector-valued function on B R k , and τ j,k be a tangent vector field of M k in R n+m defined by then det ( τ i,k , τ j,k ) = v 2 k a.e. for each k. Hence, the orientation of M k can be written as with |ξ k | = 1 a.e.. From Lemma 22.2 in [48], for any compact set K in R n+m , we have Since ξ k has the expansion (4.6) with |a k,j 1 ,··· ,jn | ≤ 1, then from (4.5) we have With |Λ 2 du k | ≤ Λ a.e., there are a sequence of positive numbers ε k → 0 (as k → ∞) and a sequence of open sets Then for any small constant 0 < ε < 1 we have Let π * denote the projection from R n+m into R n+1 by Since |M k | converges to the cone C, then M k ∩ K converges to sptC ∩ K in the Hausdorff sense for any compact set K in R n+m . Thus, for all the sufficiently large k and r ∈ (0, 2]. In particular, We claim that there is a constant k 0 > 0 such that for all k ≥ k 0 and all the C r (z) ⊂ C 2 (0 n+1 ), C r (z) \ M * k has only two components.
Assume that C r (z) \ M * i k has at least 3 components for a sequence i k → ∞. Without loss of generality, we assume Since M k converges locally to sptC in the Hausdorff sense, then C r (z)∩M * k converges to the n-dimensional ball where π ′ denotes the projection from R n+1 to R n by π ′ (x 1 , · · · , x n+1 ) = (x 1 , · · · , x n ). From the argument of (3.6), there is a constant c Λ > 0 depending only on n, m, Λ such that From (3.24) for Y k and (4.15), we have From (4.8), the above inequality contradicts to lim k→∞ H n (G k ) = 0, and we have proven the claim.
Remark. Here, the stable sptV means that sptV is stable outside its singular set.
Proof. Let M = regV be the regular part of V . Let ν M denote the unit normal vector of M in R n+1 . Since M is open in regV by Allard's regularity Theorem [2], then for any point q ∈ M, there is a constant r = r q such that B 2r (q) ∩ sptV ⊂ M. Let q k ∈ M k with q k → q. Let λ 1,k , · · · , λ n,k be the singular values of Du k with λ 1,k ≥ · · · ≥ λ n,k ≥ 0. From Allard's regularity Theorem [2] and Lemma 4.2, M k ∩ B 7 4 r (q k ) converges to M ∩ B 7 4 r (q) smoothly. Then lim For a point z ∈ B r (q) ∩ M, let z k ∈ B r (q) ∩ M k be a sequence of points with z k → z such that M k is smooth at z k for each k. Let {ν α k } m α=1 be a local orthonormal frame of the normal bundle NM k on B r (q k ) ∩ M k such that (h k α,ij ) 2 = 0.
With the Cauchy inequality and the above limit, we obtain Let φ be a smooth function in M with compact support and sptφ ∩ sptV ⊂ M. From (4.33) and the covering lemma, we have on sptφ ∩ M k for some sequence of positive numbers ε k with lim k→∞ ε k = 0. Then with the Cauchy inequality, we have Let B M denote the second fundamental form of M in R n+1 . Letting k → ∞ in the above inequality implies We complete the proof.
Let M be a locally Lipschitz minimal graph over R n of codimension m ≥ 1 with bounded 2-dilation of its graphic function. From Lemma 3.1 and the compactness theorem for integral varifolds (Theorem 42.7 in [48]), we can suppose that a minimal cone C is a tangent cone of M at infinity. Namely, there is a sequence r k → ∞ such that | 1 r k M| converges in the varifold sense to C in R n+m with 0 n+m ∈ C. Combining Lemma 4.2 and Theorem 4.1, we can get Theorem 1.1 immediately. which means ∂T W = 0. Let W ′ be an open set in W such that sptV ∩ W ′ is contained in the regular part of sptV . From Allard's regularity theorem, M k ∩ W ′ converges to sptV ∩ W ′ smoothly as V has multiplicity one from Lemma 4.2. Hence, for any p ∈ sptV ∩ W and any ε ∈ (0, d(p, ∂W )), there is an orientation ξ of sptV ∩ B ε (p) such that
As a corollary, we immediately have the following corollary. In [9], Bombieri-Giusti proved that any codimension one minimizing current in Euclidean space is indecomposable, and established a Neumann-Poincaré inequality on such currents. Inspired by their ideas in [9], we introduce a concept 'stationary indecomposable' for integral currents associated with stationary varifolds as follows.
Definition 5.1. Let T be an integral current such that |T | is a stationary varifold. We say T stationary decomposable in an open set W if there are two components T 1 , T 2 of T W such that |T 1 |, |T 2 | are stationary varifolds in W . On the contrary, we say T stationary indecomposable in W . Furthermore, T 1 is said to be a stationary indecomposable component of T W if T 1 is a component of T W , and T 1 is stationary indecomposable in W .
By the above definition, for an integral current S with stationary |S|, if S is indecomposable, then S is stationary indecomposable.
Remark 5.2. We do not know yet whether limits of stationary indecomposable currents are still stationary indecomposable. Hence, the coefficient δ T in (5.5) depends on the current T .
Proof. Let us prove (5.5) by contradiction. Suppose that there is a sequence of open U k ⊂ B 2 with (n − 1)-rectifiable sptT ∩ ∂U k such that (5.6) Then all T ± k are integer multiplicity currents. Without loss of generality we can assume H n−1 (T ∩ ∂B 2 ) < ∞, or else from co-area formula we consider a sequence of balls B 2−s k for some sequence 0 < s k → 0 with H n−1 (T ∩ ∂B 2−s k ) < ∞. Hence with (5.6), M(∂T ± k ) are uniformly bounded independent of k. Clearly, M(T + k W ) + M(T − k W ) = M(T W ). By Federer-Fleming compactness theorem, there are two integer multiplicity currents T + * , T − * with sptT ± * ⊂ sptT such that T ± k converges weakly to T ± * as k → ∞ up to a choice of a subsequence.

For any open
which implies For any |ω ′ | B 2 ≤ 1, ω ′ ∈ D n−1 (B 2 ), sptω ′ ⊂ W , from (5.6) we have (5.11) Since T has multiplicity one on sptT ∩ W for any open W ⊂ B 2 from Corollary 5.1, T ± * has multiplicity one on its support. From the co-area formula, for almost all 1 < t < 2, we have With (5.4) and (5.6), for almost all 1 < t < 2, we get (5.14) which implies M(T ± k B t ) > 0 for any t > 1 and any k > c n . Then we solve the above differential inequality and from (5.10) we get For any small fixed ε > 0 and any integer k ≥ 0, there is a collection of balls Let η l be a Lipschitz function on B 3 with 0 ≤ η l ≤ 1 such that η l = 0 on B r l (x l ), , where∇ denotes the Levi-Civita connection of R n+m . Set η k,ε = N k,ε l=1 η l ∈ C 1 . Then η k,ε = 0 on a neighborhood of sptT ∩ ∂U k ∩ B 2 and Let φ be a smooth function with compact support in B 2 . Let ∇ T denote the Levi-Civita connection of the regular part of |T |. Since |T + k | is a multiplicity one stationary n-varifold in U k , then every position function x i is weakly harmonic on sptT + k for each i = 1, · · · , m + n (see [15] for instance). Hence With (5.17), it follows that From Lemma 3.1, there is a constant c n,Λ ≥ 1 depending only on n, Λ so that Letting ε → 0 in the above inequality implies Up to a choice of a subsequence, we can assume that |T ± k | converges to |T ± * | in the varifold sense as k → ∞. With (5.6), we have for each i = 1, · · · , n + m. In other words, T + * is stationary, and similarly T − * is also stationary. Combining (5.9)(5.12)(5.15), we conclude that T is stationary decomposable in B 2 . It is a contradiction. This completes the proof.
Using Lemma 5.2, we can prove a Neumann-Poincaré inequality on stationary indecomposable components of limits of minimal graphs.
<f + t} for all s > 0 and t ∈ R. From Sard's theorem, for almost all t, ∂U ± s,t is C 1 in B s outside S. In particular, ∂U ± s,t is (n − 1)-rectifiable for almost all t. Without loss of generality, we assume H n (U + 1,0 ) ≤ H n (U − 1,0 ). Then H n (U + 1,t ) ≤ H n (U − 1,t ) for any t ≥ 0. From Lemma 5.2, we have . Using now the co-area formula, and then Using (3.1), we complete the proof.
Let T ∈ M n,m,Λ,B 3 with 0 n+m ∈ sptT. Then H n (sptT ∩ B r ) ≥ ω n r n for every r ∈ (0, 3). From (3.1), the exterior ball sptT ∩ B r admits volume doubling property. Namely, there is a constant c n,m,Λ ≥ 1 depending only on n, m, Λ such that (5.26) H n (sptT ∩ B 2r ) ≤ c n,m,Λ ω n 2 n r n ≤ c n,m,Λ 2 n H n (sptT ∩ B r ) for every r ∈ (0, 1]. From the Sobolev inequality [42], nonnegative subharmonic functions on stationary varifolds admit the mean value inequality on sptT (see [32] for instance). Since Neumann-Poincaré inequality (5.23) holds on a stationary indecomposable component T of T, by De Giorgi-Nash-Moser iteration (see [43] [44], or [39], or Theorem 3.2 in [21] for instance) there holds the mean value inequality for superharmonic functions on sptT . Hence, we get Harnack's inequality for weakly harmonic functions on sptT as follows. where ∆ T is the Laplacian of the regular part of sptT , Θ T > 0 is a constant depending on n, m, Λ, T .

A Liouville theorem for minimal graphs of bounded 2-dilation
For codimension 1, De Giorgi [17] proved that any limit of non-flat minimal graphs over R n in R n+1 is a cylinder. For arbitrary codimensions, we have the following splitting.
Let S denote the singular set of ∂W . By the strong maximum principle, ∂U ∩ ∂W is a closed subset in S.
For any β ≥ 0, let H β ∞ be a measure defined by for any set W in R n+1 , where ω β = π β/2 Γ( β 2 +1) , and Γ(r) = ∞ 0 e −t t r−1 dt is the gamma function for 0 < r < ∞. From Lemma 11.2 in [30], if H β (W ) > 0, then H β ∞ (W ) > 0. From the argument of Proposition 11.3 in [30] and (6.3), there is a point q ∈ ∂U ∩ ∂W \ {0 n+1 } and a sequence r k → 0 such that Up to choosing the subsequence, we may assume that Γ q,k converges to a closed set Γ * in the Hausdorff sense. Let S * be the singular set of ∂W * . If y k ∈ S q,k and y k → y * ∈ ∂W * , then it's clear that y * is a singular point of ∂W * by Allard's regularity theorem and multiplicity one of ∂W * , which implies lim sup k→∞ S q,k ⊂ S * . With ∂U ∩ ∂W ⊂ S, it follows that Γ * ⊂ S * . Analog to the proof of Lemma 11.5 in [30], we have Let us continue the above procedure. By dimension reduction argument, there are a 2-dimensional open cone V 0 ⊂ R 2 with |∂V 0 | minimal, an open cone V ⊂ V 0 × R ⊂ R 3 with |∂V | minimal, a sequence of open sets V i , W i (obtained from scalings and translations of U, W , respectively) such that ∂V 0 has an isolated singularity at the origin, and [|W i |] converges to [|V 0 × R n−1 |], [|V i |] converges to [|V × R n−2 |] in the current sense. Since Σ is a minimal graph over W , then Σ is smooth stable. From Theorem 2 of [47] by Schoen-Simon (see also Lemma 11.1), we get ∂V = ∂V 0 × R. It's well-known that a smooth 1-dimensional minimal surface(geodesic) in S 2 is a collection of circles of radius one, which implies that ∂V is a collection of planes through 0 3 ∈ R 3 . Hence ∂V = ∂V 0 × R. It's a contradiction. We complete the proof.
Remark 6.1. Cheeger-Naber [11] showed the Minkowski content estimation on the quantitative singular sets of stationary varifolds. Using it we can simplify the proof of Lemma 6.1. Namely, without dimension reduction argument, we immediately have the following conclusion in Lemma 6.1: there are a point x * ∈ ∂U ∩ ∂W , a 2-dimensional open cone V 0 ⊂ R 2 with |∂V 0 | minimal, and an open cone V ⊂ V 0 × R ⊂ R 3 with |∂V | minimal such that ∂V 0 has an isolated singularity at the origin, and 1 From Proposition 5.1 and Lemma 6.1, we can obtain a Liouville type theorem for minimal graphs as follows.  Proof. Assume that u has bounded 2-dilation by a constant Λ > 0. From (6.9), there are a constant Θ > 0 and a sequence of numbers r k → ∞ such that (6.10) sup Recalling that [|M|] ∈ D n (R n+m ) is the n-current associated with M and its orientation (2.25). From Lemma 5.1, we can assume that [| 1 r k M|] converges weakly as k → ∞ to a multiplicity one current T = 0 with 0 ∈ sptT and ∂T = 0. Moreover, the varifold associated with sptT is a minimal cone in R n+m with the vertex at the origin. Let x = (x 1 , · · · , x n+m ) be the position vector in R n+m . From (6.8), we get (6.11) sup Suppose that T is stationary decomposable in B 1 . Let T ′ be a stationary component of T B 1 . Then from ∂T ′ = 0 in B 1 , we conclude that sptT ′ is a truncated cone. In particular, 0 ∈ sptT ′ . Hence we have (6.12) M(sptT ′ ∩ B r ) ≥ ω n r n for any r ∈ (0, 1].
From Lemma 3.2, we have If T ′ is stationary decomposable in B 1 , then we consider a stationary component T ′′ of T ′ B 1 . Clearly, sptT ′′ is a truncated cone and 0 ∈ sptT ′′ . Combining (6.12)(6.13), the procedure of decomposition will cease after finite times. Hence, there is a collection of indecomposable stationary components T 1 , · · · , T l of T B 1 , where l is a positive integer ≤ C n,Λ . In particular, all sptT 1 , · · · , sptT l are truncated cones.
From Proposition 5.1 and (6.11), we get x n+α ≡ 0 on the truncated cone sptT k for each α ∈ {2, · · · , m} and k ∈ {1, · · · , l} as x n+α is weakly harmonic in sptT k ∩ B 1 . In particular, the varifold associated with sptT is a minimal cone living in an (n+1)dimensional Euclidean space R n+1 . From (6.10), we conclude that (6.14) sup From Lemma 6.1, it follows that Note that [| 1 r k M|] ⇀ T and sptT is a cone. Then sptT can be written as an entire graph over R n with the graphic function (φ, 0, · · · , 0), where φ is 1-homogenous on R n . Then (6.16) graph is a (Lipschitz) minimal graph in R n+1 . Therefore, the regularity theorem of De Giorgi [44] implies that φ is linear, and then sptT is flat. Since T has multiplicity one on sptT , then Allard's regularity theorem yields the proof.

Bernstein theorem for minimal graphs of bounded slope
For a domain Ω ⊂ R n , let M = graph u be a smooth minimal graph over Ω of codimension m ≥ 2. Let g ij = δ ij + m α=1 ∂ i u α ∂ j u α , and (g ij ) be the inverse matrix of (g ij ). Let v be the slope function of M defined by Lemma 7.1. Let λ 1 , · · · , λ n be the singular eigenvalues of Du at any point of Ω. If λ 1 ≥ · · · ≥ λ n ≥ 0 and λ 2 1 λ 2 i ≤ 2 + λ 2 i for all i ≥ 2, then Moreover, when the above inequality becomes an equality, Proof. From (2.17), we have (see also the decomposition (4.16) in [24]) Without loss of generality, we assume λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 0. Let for every x, y, z ∈ R with mutually distinct i, j, k. Then (7.5) From a direct computation, we have Then we consider a function We further conclude that all the eigenvalues of Hess f are non-negative, then it follows that From (7.4), we have (7.9) i,j,k mutually distinct Moreover, by the assumption for i ≥ 2 and λ 1 ≥ 1 we have From the Cauchy inequality, we have (7.11) (2 + λ 2 i )h 2 i,ij + h 2 j,ii + 2λ i λ j h i,ji h j,ii ≥ 0. Substituting (7.9)(7.11) into (7.3), we have When the above inequality becomes an equality, we clearly have i =j which completes the proof.
As a corollary, we have the following result.
This completes the proof. Now we consider minimal graphs with bounded slope. Lemma 7.3. Let M k = graph u k be a family of Lipschitz minimal graphs over R n of codimension m ≥ 1 with sup k Lip u k < ∞ and |Λ 2 du k | 2 ≤ 2(Lip u k ) 2 |(Lip u k ) 2 −1| a.e. on R n . Let V be the limit of |M k | in the varifold sense. If sptV is a regular cone or there is a regular l-dimensional cone C with sptV = C × R n−l , then sptV is flat.
Proof. Denote M = sptV , which is a Lipschitz minimal graph over R n for some graphic function u = (u 1 , · · · , u m ). From Lemma 10.1 in the appendix II and Allard's regularity theorem, u k converges to u in C 3 -sense at any regular point of u. Hence, we have |Λ 2 du| 2 ≤ 2(Lip u) 2 |(Lip u) 2 −1| a.e. on R n from the assumption of u k . Now we assume l ≥ 2. Since M = C × R n−l , then C lives in an (m + l)dimensional Euclidean space. Up to a rotation, C can be represented as a graph of a 1-homogenous vector-valued function φ = (φ 1 , · · · , φ m ) on R l . Then up to two rotations of R n and R m , there are a constant matrix (c α j ) for j = l + 1, · · · , n and α = 1, · · · , m such that for each α. After a rotation of R n−l , we can further assume for each α = 1, · · · , m, where we let c n+j = 0 for any positive integer j. (7.20), g ij is a function of x 1 , · · · , x l , and v = det g ij can be seen as a function of x 1 , · · · , x l . From Lemma 7.1, we have on the regular part of M. Note that log v is smooth on R l \ {0}. Since log v is 0-homogenous, it achieves its maximum on B l 2 \ B l 1/2 at a point in ∂B l 1 . From the strong maximum principle, v is a constant. With Lemma 7.2, u α is harmonic on R n \ ({0 l } × R n−l ) for each α = 1, · · · , m. Namely, φ α = φ α (x 1 , · · · , x l ) is harmonic on R l \ {0}. Note that φ α is 1-homogenous. Then φ α is harmonic on R l for each α, and it must be affine, i.e., φ α − φ α (0) is linear. From (7.20), it follows that each u α is affine and then M is flat.
For l < 2, M is regular, then M is flat from the above argument. This completes the proof.
Let us prove a Bernstein theorem for minimal graphs with bounded slope. Theorem 7.1. Let M = graph u be a Lipschitz minimal graph over R n of codimension m ≥ 2. If |Λ 2 du| 2 ≤ 2(Lip u) 2 |(Lip u) 2 −1| a.e. on R n , then M is flat.
Proof. Assume that M is not flat. From Lemma 10.1 in the appendix II, there is a sequence r k → ∞ such that | 1 r k M| converges to a minimal cone C of multiplicity one in the varifold sense, where sptC can be rewritten as a graph over R n with a Lipschitz homogenous graphic function u ∞ = (u 1 ∞ , · · · , u m ∞ ) satisfying Lip u ∞ ≤ Lip u ≤ L for some constant L > 0. Then from Allard's regularity theorem, when W is a bounded open set with W ∩ sptC belonging to the regular part of C, W ∩ 1 r k M converges to W ∩ sptC smoothly. Hence we have a.e. on R n .
If sptC is a regular cone, then sptC is flat from Lemma 7.3. Now we assume that there is a singular point q in sptC \ {0 n+m }. We blow C up at the point q, and get a nonflat minimal cone C ′ whose support splits off R isometrically. If u ′ ∞ denotes the graphic function of sptC ′ , then we have a.e. on R n . By dimensional reduction argument, we get a sequence of minimal graphs M k (which are scalings and rigid motions of M) such that |M k | converges to a nonflat minimal cone C * , where sptC * = C * × R l for a regular (n − l)-cone C * . Moreover, the graphic function u * ∞ of sptC * satisfies a.e. on R n . From Lemma 7.3, we get the flatness of sptC * , which is a contradiction. Hence, sptC is flat. Since C has multiplicity one, then Allard's regularity theorem implies that M is flat.
By a contradiction argument, we have the following curvature estimate. In particular, M is smooth.
Proof. Let us prove (7.22) by contradiction. Suppose that there is a sequence of Lipschitz minimal graphs graph u k over B 2 of codimension m ≥ 2 with Lip u k ≤ L and If u is not smooth at a point q ∈ B 2 , then we blow M up at (q, u(q)), and get a contradiction from Theorem 7.1. Hence M is smooth. From (7.23), there exists a sequence of points p k ∈ B 3 2 such that Note that τ k 2 ≤ 3 2 − |x| for all x ∈ B τ k 2 (p k ). We have (7.25) sup (7.25), after choosing a subsequence, graph w k converges smoothly to a minimal graph with the graphic function w * such that Lip w * ≤ L, |Λ 2 dw * | 2 ≤ 2(Lip w * ) 2 |(Lip w * ) 2 −1| a.e. and |D 2 w * | ≤ 1 on R n . Moreover, (7.26) implies |D 2 w * | (0) = 1 2 . However, this contradicts to Theorem 7.1. We complete the proof.

Quasi-cylindrical minimal cones from minimal graphs
denote the standard basis of R n+m such that E i corresponds to the axis x i . Lemma 8.1. Let M k ∈ M n,m,Λ,B 2 for some constant 0 < Λ < √ 2. Assume that [|M k |] B 2 converges to a stationary varifold V in the varifold sense. If S denotes the singular set of sptV ∩ B 1 , then S has Hausdorff dimension ≤ n − 3.
Proof. We assume that S has Hausdorff dimension > n−3. Then there is a constant β > n − 3 so that β-dimensional Hausdorff measure of S satisfies H β (S) > 0. From Lemma 11.2 in [30], H β (S) > 0 implies H β ∞ (S) > 0 (see (6.4) for its definition with R n+1 there replaced by R n+m ). From the argument of Proposition 11.3 in [30], there are a point q ∈ S and a sequence r k → 0 (as k → ∞) such that Without loss of generality, we assume that 1 r k (V, q) converges to a tangent cone (V * , 0 n+m ) in R n+m in the varifold sense as k → ∞. By the definition of V , there is a sequence of minimal graphs M ′ k in R n+m (rigid motions of M k ) such that |M ′ k | converges to the minimal cone V * in the varifold sense as k → ∞.
Let S * be the singular set of V * . If y k ∈ S k and y k → y * ∈ V * , then it's clear that y * is a singular point of V * by Allard's regularity theorem and multiplicity one of V * , which implies lim sup k→∞ S k ⊂ S * . Analog to the proof of Lemma 11.5 in [30], we have Let us continue the above procedure. By dimension reduction argument, there is a l(l ≤ 2)-dimensional non-flat regular minimal cone C ⊂ R m+k such that there is a sequence of minimal graphs Σ k ∈ M n,m,Λ,B R k (which are scalings and translations gotten from M k ) with R k → ∞ so that Σ k converges to a minimal cone C * in the varifold sense, which is a trivial product of C and R n−l .
From Lemma 11.1 in the Appendix III, l = 1 is impossible. For l = 2, sptC ∩ ∂B 1 (0 m+2 ) is smooth minimal in ∂B 1 (0 m+2 ), hence it is a disjoint union of geodesic circles in a sphere. So sptC * ∩ R n+m is a union of n-planes P 1 , · · · , P j 0 with j 0 ≥ 2 as sptC is non-flat. If there is a unit vector ξ ∈ sptC * with ξ, E i = 0 for each i = 1, · · · , n, then Theorem 4.1 contradicts to that sptC splits off R n−2 isometrically. Therefore, sptC * can be written as a graph over R n , and then sptC * is an n-plane. It is a contradiction. This completes the proof.
Remark. The above dimension estimate is not sharp. We will establish a sharp one through the Bernstein theorem for minimal graphs in Lemma 8.5.
With Lemma 8.1, we can use De Giorgi-Nash-Moser iteration for nonnegative superharmonic functions on the regular part of stationary indecomposable currents as follows. By Besicovitch covering lemma, we can further require that there is a constant c n+m > 0 depending only on n + m so that for each j = 1, · · · , N ε and each n-rectifiable set E ⊂ R n+m . Let η k be a C 2 function on B 2 with 0 ≤ η k ≤ 1 such that η k = 0 on B r k (x k ), η k = 1 on B 2 \ B 2r k (x k ) and where∇ denotes the Levi-Civita connection of R n+m , |λ(∇ 2 η k )| denotes the maximum of the absolution of eigenvalues of the Hessian of η k on R n+m , c is an absolute positive constant. Let e 1 , · · · , e n be a local orthonormal tangent frame field of M.
Since M is minimal, we have (see (5.4) in [22] for instance) where Hess η k denotes the Hessian of η k on R n+m . Set η ε = Nε k=1 η k ∈ C 2 . Then η ε = 0 on a neighborhood of S. Let ∇ M be the Levi-Civita connection of M, and div M be the divergence of M. From (8.7) and the Cauchy inequality, one has (8.8) Then with (8.6) we deduce Similarly, Combining Lemma 3.1 and (8.5), we deduce Let ϕ be a nonnegative Lipschitz function with compact support in B 2 . Then the support of ϕ is in B 2−ε for small ε > 0. With integrating by parts, we have Combining (8.11) and Cauchy inequality, we conclude that which means that f is superharmonic on M in the distribution sense. Now we can follow the argument of De Giorgi-Nash-Moser iteration (see Theorem 3.2 in [21] for instance) and finish the proof. Note that the constant δ * in (8.4) is obtained from the dimension n and the exponent of the Sobolev inequality, which implies that δ * only depends on n.
We suppose that M k converges in the varifold sense to a minimal cone C in R n+m with the vertex at the origin. Since the multiplicity function of sptC is one from Theorem 4.2, we denote sptC by C for simplicity. Without loss of generality, we assume that [|M k |] converges to an integral current T in R n+m . From Lemma 5.1, ∂T = 0 and T has multiplicity one on sptT = sptC. As before (in §4), π * denotes the projection from R n+m into R n by π * (x 1 , · · · , x n+m ) = (x 1 , · · · , x n ), C r (x) = B r (π * (x)) × B r (x n+1 , · · · , x n+m ) denotes the cylinder in R n+m for any x = (x 1 , · · · , x n+m ) ∈ R n+m , and C r = C r (0 n+m ). Lemma 8.2. Suppose that for any regular point q ∈ C there is a sequence of points Proof. For any x ∈ π * (C), there is a point x ∈ C such that π * (x) = x. Then tx ∈ C implies tx = π * (tx) ∈ π * (C). This means that π * (C) is a cone. It is easy to check that π * (C) is closed in R n . For any point q ∈ regC, from the assumption there is a unit vector η q ∈ T q C such that E i , η q = 0 for each integer i = 1, · · · , n. There are a small constant r q > 0 and a local orthonormal tangent frame {e i } n i=1 on C rq (q) ∩ C such that e 1 (z), E i = 0 for any z ∈ C rq (q) ∩ C and i = 1, · · · , n. In other words, e 1 is a C 1 tangent vector field on C rq (q) ∩ C with π * (e 1 (z)) = 0 for any z ∈ C rq (q) ∩ C. After choosing the constant r q > 0 suitably small, for each y ∈ C rq (q) ∩ Γ q there is an integral curve γ y in C rq (q) ∩ C withγ y = e 1 • γ y .
With Proposition 5.1, we can weaken the condition in Lemma 8.2 and get the same conclusion.
Theorem 8.1. If there is a sequence of points q k ∈ M k with lim sup |q k | < ∞ such that lim k→∞ v k (q k ) = ∞, then C is a multiplicity one quasi-cylindrical minimal cone.
Proof. Let ∆ M k and ∇ M k denote the Laplacian and the Levi-Civita connection of M k , respectively. From Corollary 7.1, (8.17) on M k . Let ξ k denote an orientation of M k , namely, T x M k can be represented by the unit n-vector ξ k (x), such that v −1 k = ξ k , E 1 ∧ · · · ∧ E n . For any regular point z ∈ C, there is a constant r z > 0 such that B 2rz (z) ∩ C is smooth. From Allard's regularity theorem, M k ∩ B 3rz/2 (z) converges to C ∩ B 3rz/2 (z) smoothly. Recalling [|M k |] ⇀ T . Let ξ denote the orientation of T , and v −1 T = ξ, E 1 ∧ · · · ∧ E n on regT (the regular part of T ). We extend v T to singT (the singular part of T ) by letting regT ∋y→x ξ(y), E 1 ∧ · · · ∧ E n for any x ∈ singT.
Then from (8.17) we have where ∆ T denotes the Laplacian of T .
From monotonicity of the density and Lemma 3.2, there is a constant c n ≥ ω n depending only on n such that (8.20) ω n r n ≤ M(T B r (x)) ≤ c n r n for any r > 0 and x ∈ sptT . From the proof of Theorem 6.1, we assume that there exist indecomposable multiplicity one currents T 1 , · · · , T l ∈ D n (R n+m ) for l ≥ 1, T 1 , · · · , T l = 0 such that for any open W ⊂⊂ R n+m . From Proposition 8 .1 and (8.19), there are a constant δ * depending only on n, and a constant Θ T depending only on n, m, Λ, T such that for any j, q ∈ sptT j , r > 0 There is a point q ∈ sptT j for some integer j ∈ {1, · · · , l} so that q k → q up to a choice of a subsequence. From lim k→∞ v k (q k ) = ∞, it follows that v −1 T (q) = 0 if q is a regular point of C. Now we assume that q is a singular point of C and v −1 T (q) > 0. Then we blow C up at q, and get a contradiction from (8.18) and Theorem 7.1.
From Allard's regularity theorem, any sequence of points y k ∈ M k converging to a regular point y of sptT satisfies lim k→∞ v k (y k ) = ∞. Recalling Lemma 8.2, we complete the proof.
Lemma 8.3. Let Σ be a closed co-dimension 1 minimal variety in S n . Suppose Σ is not the totally geodesic S n−1 . Then if n ≤ 6, the cone CΣ is not stable.
Using Lemma 8.3, Simons proved the celebrated Bernstein theorem in Theorem 6.2.2 in [49] with the help of Fleming's and De Giorgi's arguments. In high codimensions, we have the following Bernstein theorem based on Simons' work.
Proof. If C is an entire graph over R n , then C is an n-plane from Theorem 7.1. Hence we can assume that there is a point y * = (0 n , y * ) ∈ C with 0 = y * ∈ R m . Without loss of generality, we assume y * = (1, 0, · · · , 0). From Lemma 4.1 and Lemma 4.2, C t = C − ty * converges as t → ∞ to a minimal cone {(x 1 , · · · , x n , y 1 , · · · , y m ) ∈ R n × R m | (x 1 , · · · , x n ) ∈ C y * , y 2 = · · · = y m = 0}, where C y * is a minimal cone in R n with multiplicity one. From Theorem 4.1, C y * is stable with the dimension n − 1 ≤ 6. With Lemma 8.1 and dimension reduction argument, we get the flatness of C y * by Lemma 8.3. In other words, y * is a regular point of C. Denote C = sptC. Hence, there is a constant r > 0 such that M k ∩ B 2r (y * ) converges smoothly to C ∩ B 2r (y * ) as k → ∞. Let ξ denote the orientation of regC. From the argument in the proof of Theorem 8.1, we get (8.24) ξ, E 1 ∧ · · · ∧ E n = 0 on regC.
There is a ball B δy (y * ) with the radius δ y > 0 such that B δy (y * ) ∩ C is regular everywhere. From Lemma 8.2, π * (B δ (y * ) ∩ C) contains a neighborhood of the origin in π * (C) for any δ ∈ (0, δ y ], and the origin is the regular point of C in particular. Hence C is flat. We complete the proof.
Analogously to the argument in Lemma 8.4, we have the following sharp dimension estimate.
Assume that |M k | B 2 converges to a stationary varifold V in the varifold sense. If S denotes the singular set of sptV ∩ B 1 , then S has Hausdorff dimension ≤ n − 7.
Proof. Let us prove it by following the steps in the proof of Lemma 8.1. We assume that S has Hausdorff dimension > n − 7. By the dimension reduction argument, there is a k(k ≤ 6)-dimensional non-flat regular minimal cone C ⊂ R m+k such that there is a sequence of minimal graphs Σ k = M n,m,Λ,Br k (which are scalings and translations gotten from M k ) with r k → ∞ so that Σ k converges to a minimal cone C * in the varifold sense, which is a trivial product of C and R n−k . From Theorem 7.1, there is a point y * = (0 n , y * ) ∈ C * with 0 = y * ∈ R m . If y * is a singular point of C * , then we blow C * up at y * , and get the contradiction by Theorem 4.1, Lemma 8.1 and Lemma 8.3. If y * is a regular point of C * , then there is a constant δ y > 0 such that B δy (y * ) ∩ C * is regular. From Lemma 8.2, we conclude that the origin is a regular point of C * . It's a contradiction. We complete the proof.
We summarize Theorem 8.1, Lemma 8.4 and Lemma 8.5, and complete the proof of Theorem 1.3.
From (9.5), the elements of Since the inverse matrix of b = g + c * satisfies combining this with (9.10)(9.11), we have (9.13) This completes the proof.
This completes the proof.

Appendix II
Analog to Lemma 4.3 in [21], we have the following multiplicity one convergence for Lipschitz minimal graphs. Let Ω be a domain in R n with countably (n − 1)rectifiable boundary ∂Ω.
Lemma 10.1. Let M k = graph u k be a sequence of Lipschitz minimal graphs over Ω of codimension m ≥ 1 with sup k Lip u k < ∞. Then there are a Lipschitz function u ∞ : Ω → R m with Lip u ∞ ≤ sup k Lip u k , and a multiplicity one n-varifold V in Ω × R m with sptV = {(x, u ∞ (x)) ∈ R n × R m | x ∈ Ω} such that up to a choice of the subsequence |M k | converges as k → ∞ to V in Ω × R m in the varifold sense.
Remark. In Proposition 11.53 of [31], Giaquinta and Martinazzi have already proved the multiplicity one of V in the above lemma. Here, we give an alternative proof for completeness.
Proof. By Arzela-Ascoli theorem, up to a choice of the subsequence, we assume that there is a Lipschitz function u ∞ on Ω with Lip u ∞ ≤ sup k Lip u k . By compactness of varifolds (see [48]), there is an n-varifold V in Ω × R m such that up to a choice of the subsequence, |M k | converges to an integer multiplicity stationary varifold V in Ω × R m in the varifold sense. Let µ V denote the Radon measure associated to V . By monotonicity of the density of V , for any x * ∈ sptV ∩ (Ω × R m ) we have µ V (B r (x * )) ≥ ω n r n for sufficient small r > 0. By the convergence of |M k |, there is a sequence x k ∈ M k with x k → x * . Denote x k = (x k , u k (x k )). Then x k converges to a point x * with π * (x * ) = x * , where π * is defined in (4.3). Therefore, x * = lim k→∞ x k = lim k→∞ (x k , u k (x k )) = (x * , u ∞ (x * )), which implies the support of V Note that for any z ∈ Ω, H n (B r (z k ) ∩ M k ) ≥ ω n r n for all suitably small r > 0 with z k = (z, u k (z)). Since (z, u k (z)) → (z, u ∞ (z)) as k → ∞, from the convergence of |M k | we get µ V (B r (z)) ≥ ω n r n for z = (z, u ∞ (z)). In particular, z ∈ sptV , which implies Now it only remains to prove that V has multiplicity one. Let regV denote the regular part of V in Ω × R m . For any y ∈ regV , let T y V denote the tangent plane of sptV at y. Let ξ 1 , · · · , ξ n be an orthonormal basis of T y V . From Lemma 22. |e k,1 ∧ · · · ∧ e k,n − ξ 1 ∧ · · · ∧ ξ n | 2 = 0, where e k,1 , · · · , e k,n is a local orthonormal tangent frame of M k for each k. We also treat e k,i as a vector on π * (M k ) by letting e k,i (x) = e k,i (x, u k (x)) for each i = 1, · · · , n and k ≥ 1. Let {E i } n+m i=1 denote the standard orthonormal basis of R n+m such that E i corresponds to the axis x i for i = 1, · · · , n + m, v k be a function on R n defined by v −1 k = | e k,1 ∧ · · · ∧ e k,n , E 1 ∧ · · · ∧ E n |. |e k,1 ∧ · · · ∧ e k,n − ξ 1 ∧ · · · ∧ ξ n | 2 v k = 0 with y = (y, u ∞ (y)). Let v ∞ = | ξ 1 ∧ · · · ∧ ξ n , E 1 ∧ · · · ∧ E n | −1 . From (10.4) and (10.5) | e k,1 ∧ · · · ∧ e k,n − ξ 1 ∧ · · · ∧ ξ n , E 1 ∧ · · · ∧ E n | v k ≤ Br(y) |e k,1 ∧ · · · ∧ e k,n − ξ 1 ∧ · · · ∧ ξ n | v k , with the Cauchy inequality we get v k = ω n v ∞ = ω n | ξ 1 ∧ · · · ∧ ξ n , E 1 ∧ · · · ∧ E n | −1 .
With (10.1), we conclude that V has multiplicity one everywhere on sptV . This completes the proof.

Appendix III
Let Λ be a positive constant < √ 2, and M k ∈ M n,Λ for each integer k ≥ 1. From Theorem 7.2, M k is smooth for each k. From (7.13), integrating by parts infers that there is a constant c n,Λ > 0 depending only on n and Λ such that (11.1) M k ∩Bρ(p) |B M k | 2 ≤ c n,Λ ρ n−2 for any p ∈ M k and any ρ > 0. Here, B M k is the second fundamental form of M k in R n+m .
Lemma 11.1. Suppose that |M k | converges to a nontrivial stationary varifold V in the varifold sense. If V splits off R n−1 isometrically, then sptV is an n-plane.
The proof is similar to the argument in the proof of Theorem 2 of [47] by Schoen-Simon. For self-containment, we give the proof here.
Proof. Let us prove it by contradiction. Suppose that there is a varifold T in R m+1 such that sptT is not a line in R m+1 , and sptV = {(x, y) ∈ R n−1 × R m+1 | y ∈ sptT } = R n−1 × sptT.
We write with l ≥ 2, n j positive integers, |p j | = 1 and p 1 , · · · , p l spanning a space of dimension ≥ 2. Let ξ k be the orientation of M k defined in (2.25). Note that T has multiplicity one. For any 0 < ρ ≤ 1/2, from Allard's regularity theorem there is a constant k ρ > 0 such that for each k ≥ k ρ and each x ∈ R n−1 with |x| ≤ 1, where l ′ = l j=1 n l , γ k j (x) are smooth properly embedded Jordan arcs having their endpoints in {x} × {y ∈ |x| < 1 and using the co-area formula yield (11.9) There are an integer l ρ > 1 and a finite sequence of {x ′ j } lρ j=1 ⊂ R n−1 with |x ′ j | < 1 such that lρ j=1 B ρ (x ′ j ) ⊃ B 1 (0 n−1 ) and l ρ ρ n−1 < c ′ n for some constant c ′ n depending only on n. Here, B ρ (x ′ j ) denotes the ball in R n−1 centered at x ′ j with the radius ρ, B 1 (0 n−1 ) denotes the unit ball in R n−1 centered at the origin. Denote z j = (x ′ j , 0 m+1 ).