The Area Blow Up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals

In this paper we investigate the"area blow-up"set of a sequence of smooth co-dimension one manifolds whose first variation with respect to an anisotropic integral is bounded. Following the ideas introduced by White in (J. Differential Geom., 2016), we show that this set has bounded (anisotropic) mean curvature in the viscosity sense. In particular, this allows to show that the set is empty in a variety of situations. As a consequence, we show boundary curvature estimates for two dimensional stable anisotropic minimal surfaces, extending the results of (Invent. Math., 1987).


Introduction
Consider a sequence (M i ) i of m-dimensional varieties in a subset Ω ⊂ R m+1 with mean curvature bounded by some h < ∞ and such that the boundaries have uniformly bounded measure in compact sets: i.e. Z is the smallest closed subset of Ω such that the areas of the M i are uniformly bounded as i → ∞ on compact subsets of Ω \ Z.
In the recent paper [12], White finds natural conditions implying that Z is empty. These results are useful since if Z is empty, then the areas of the M i are uniformly bounded on all compact subsets of Ω. It follows that, up to subsequences, M i will converge in the sense of varifold to a limit varifold of locally bounded first variation.
The main point of [12] is to show that the set Z belongs to the class of (m, h)-sets. The notion of (m, h)-set is a generalization of the concept of an m-dimensional, properly embedded submanifold without boundary and with mean curvature bounded by h 1 . In particular these sets satisfy a maximum principle which often allows to show that they are empty.
The aim of this paper is to extend the aforementioned results proven in [12] to codimension one manifolds (or, more in general, to co-dimension one varifolds) which are stationary with respect to a parametric integrand F .
Referring to Section 2 below for more details and definitions we simply recall here that a parametric integrand is a even map F : Ω × R m+1 → R + which is one homogeneous, even and convex in the second variable. For a smooth m-dimensional manifold M ⊂ R m+1 with normal ν M we define for every open set Ω ⊂ R m+1 A smooth manifold is then said to be F -stationary in Ω (resp. F -stable) if for every ϕ t (x) = x + tg(x) one-parameter family of diffeomorphisms (for t small enough) generated by a vector field g ∈ C 1 c (Ω, R m+1 ). In this setting our main result reads as follows, see Theorem 3.4 for the more general statement and Definition 3.1 for the definition of (m, h)-sets with respect to a given integrand F : Beside its intrinsic interest, our main motivation for Theorem 1.1 is that, in contrast to the case of the area functional, for manifolds which are stationary with respect to parametric integrand, no monotonicity formula is available, [1]. In particular, a local area bound of the form is not know to hold true. This prevents, a priori, the possibility to establish the convergence of the rescaled surfaces M x,r = (M − x)/r in order to study the local behavior of a stationary surface. Note that, for (isotropic) minimal surface, (1.1) is a trivial consequence of the monotonicity formula. Using Theorem 1.1, we can prove boundary curvature estimates for two dimensional F -stable surfaces, see also Theorem 4.1 for a more general statement: Let Ω ⊂ R 3 be uniformly convex, F be a uniformly elliptic integrand and let Γ ⊂ Ω be a C 2,α embedded curve. Let M be an F -stable, C 2 2-dimensional embedded surface in Ω such that ∂M = Γ. Then there exist a constant C > 0 and a radius r 1 > 0 depending only on F, Ω, Γ such that sup p∈Ω dist(p,Γ)<r 1 where A M is the second fundamental form of M . Furthermore the constants are uniform as long as Γ, Ω and F vary in compact subsets of, respectively, embedded C 2,α curves, uniform convex domains and uniformly convex C 2 integrands.
Let us conclude this introduction with a few remarks on the proof of the main results. To prove Theorem 3.4, we follow the proof of White in [12], and we aim to show that if the blow up set is not an (m, h)-set, than one can provide a vector field yielding a negative first variation. This vector field is what in [9] is called an F -decreasing vector field and its construction seems to be possible only in codimension one, which is the reason for our restriction to this setting. The proof of the boundary curvature estimates will easily follow from [10], once we can show that the mass density ratios H 2 (M ∩ B r (x)) r 2 are bounded. In the interior we can rely on the extended monotonicity formula for 2-dimensional varifolds with curvature in L 2 (note that by stability one easily proves that locally |A| ∈ L 2 ). At the boundary we perform a rescaling argument and we use our assumption on Ω to show that that the area blow up set of the sequence of rescaled surfaces must be contained in a wedge. Since Theorem 3.4 implies that this is a (2, 0)-set, a simple maximum principle argument shows that it is empty, yielding the desired bound.
Organization of the paper. The paper is organized as follows: in Section 2 we recall some preliminary results and definitions and we compute the explicit formula for the first variation of a smooth manifold. In Section 3 we give the definition of (m, h)-sets, we show some of their properties and we prove Theorem 3.4, from which Theorem 1.1 readily follows. In Section 4 we prove Theorem 4.1, which implies Theorem 1.2.
Acknowledgements. The work of G.D.P. is supported by the INDAM-grant "Geometric Variational Problems".

Notation and preliminaries
We work on an open set Ω ⊂ R m+1 and we set B r (x) = {y ∈ R m+1 : |x − y| < r}, B r = B r (0) and B := B 1 (0). We will denote m-dimensional balls by B m r (x) and we set B m r = B m r (0) and B m = B m 1 . We also let S m be the unit sphere in R m+1 . For a matrix A ∈ R m+1 ⊗ R m+1 , A * denotes its transpose. Given A, B ∈ R m+1 ⊗ R m+1 , we define A : Varifolds. We denote by M + (Ω) (respectively M(Ω, R n ), n ≥ 1) the set of positive (resp. R n -valued) Radon measures on Ω. Given a Radon measure µ, we denote by sptµ its support. For a Borel set E, µ E is the restriction of µ to E, i.e. the measure defined by [µ E](A) = µ(E ∩ A). For an R n -valued Radon measure µ ∈ M(Ω, R n ), we denote by |µ| ∈ M + (Ω) its total variation and we recall that, for all open sets U , If η : R m+1 → R m+1 is a Borel map and µ is a Radon measure, we let η # µ = µ • η −1 be the push-forward of µ through η. An m-varifold on Ω is a positive Radon measure V on Ω × S m which is even in the S m variable, i.e. such that We will denote with V m (Ω) the set of all m-varifolds on Ω. Given a diffeomorphism ψ ∈ C 1 (Ω, R m+1 ), we define the push-forward of V ∈ V m (Ω) with respect to ψ as the varifold ψ # V ∈ V m (ψ(Ω)) such that for every Φ ∈ C 0 c (G(ψ(Ω))). Here d x ψ(x) is the differential mapping of ψ at x and denotes the m-Jacobian determinant of the differential d x ψ(x) restricted to the mplane ν ⊥ , see [7,Chapter 8].
Integrands. The anisotropic (elliptic) integrands that we consider are C 2 positive functions which are even, one-homogeneous and convex in the second variable, i.e. and . We will denote with D 1 F (x, ν) and D 2 F (x, ν) respectively the differential of F in the first and in the second variable. Denoting with {e (2.1) Note that by one-homogeneity: An integrand F is said to be uniformly elliptic on a set Ω if there exists a constant λ > 0 such that Given x ∈ Ω, we will denote by F x the "frozen" integrand We define the anisotropic energy of V ∈ V m (Ω) as For a vector field g ∈ C 1 c (Ω, R m+1 ), we consider the family of functions ϕ t (x) = x + tg(x), and we note that they are diffeomorphisms of Ω into itself for t small enough. The anisotropic first variation is defined as It can be easily shown, see [5, Appendix A], that see for instance [3,Section 3] or [6,Lemma A.4]. We will often omit in the sequel the dependence on F of the matrix B F (x, ν). Moreover let us note the following useful fact: We say that a varifold V ∈ V m (Ω) has locally bounded anisotropic first variation if δ F V is a Radon measure on Ω, i.e. if |δ F V (g)| ≤ C(K) g ∞ , for all g ∈ C 1 c (Ω, R m+1 ) with spt(g) ⊂ K ⊂⊂ Ω. Notice that, by Riesz representation theorem, we can write in Ω. In this case, by the Radon-Nikodym theorem, we can decompose δ F V in its absolutely continuous and singular parts with respect to the measure V : Notice that by the disintegration theorem for measures, see for instance [4,Theorem 2.28], we can write where µ x ∈ P(S m ) is a (measurable) family of parametrized non-negative even probability measures. We define for V -a.e. x ∈ Ω .
We will say that a varifold V ∈ V m (Ω) has mean curvature H F (x) in L 1 ( V , R m+1 ) if it has locally bounded anisotropic first variation and in the representation (2.6), we have σ = 0. In this case one can easily check that Furthermore we will say that In particular we say that a varifold V ∈ V m (Ω) has anisotropic mean curvature bounded by h( Remark 2.1. Since all norms are equivalent on finite dimensional spaces, the above definition coincides with the classical one. However the above formulation has the advantage of being coordinate independent, namely if Φ : and it satisfies .
We conclude this section by computing the first variation formula for the varifold induced by a manifold with boundary and by providing an explicit formula for its F mean curvature Proposition 2.1. Let M ⊂ R m+1 be an oriented C 2 m-manifold M with boundary, and let where ν x is the normal to M at x. Then Here A is the second fundamental form 2 of M defined by and we are adopting the convention in (2.1).
Note that (2.10) gives Moreover, by (2.10) and the homogeneity of where for any orthonormal basis Hence, if e i is the standard orthonormal basis of R n and we adopt Einstein convention 2 Note that by this sign convention the second fundamental form is positive definite for a convex set with respect to the outer normal.
where B is evaluated at (x, ν x ) and in the last equality we used that ν, B * D ν g = 0 due to (2.4). Note that B * g is tangent to M (again by (2.4)), hence by the divergence theorem Hence, if we set where in the last equality we have used the one-homogeneity of D 1 F . Furthermore where A ℓj = D τ j ν, τ ℓ is the second fundamental form of M . Combining (2.14), (2.15) and (2.16), we get (2.10) since where in the last equality we have used that, by (2.1)), Remark 2.2. Let us record here the following consequence of the above computations: , then B * X = 0 and thus, by (2.12), (2.13) we get X is what is called an F -decreasing vector filed in [9, Proposition 1] and it will play a crucial role in the proof of our main theorem.

(m, h)-sets
In this section, following [12], we define (m, h)-sets and we prove that the areablow up set of a sequence of varifolds with bounded curvature is an (m, h)-set. Roughly speaking an (m, h)-set is a set which can not be touched by manifolds with F H F , ν greater than h, i.e. they satisfy H F ≤ h in the viscosity sense. This can be phrased in several ways, as the following proposition shows.
Proposition 3.1. Given a closed set Z ⊂ R m+1 , then the following three statements are equivalent. ( where the second term in the left hand side is intended to be zero when where ν int. is the interior normal to N . We can now give the following definition (i) ⇒ (ii): Suppose Z fails to have property (ii), we will show that also property (i) cannot be satisfied by Z. Following the argument in [12, Lemma 2.4], we can construct a function f ∈ C ∞ (Ω, R) such that f | Z attains its maximum at a unique Up to translation, rotation and multiplication of f by |Df (p)| −1 , we can assume without loss of generality that p = 0 and Df (p) = e m+1 . It is easy to verify that there exists an open neighborhood U ∋ p such that where ν Σ 0 (p) denotes the unit normal to Σ 0 at the point p.
If we denote with d(x) the signed distance function from Σ 0 , we also deduce that would not be the maximum of f | Z . We deduce that for every λ > 0 the function g λ (x) := (e λd(x) − 1) satisfies g λ (x) ≤ 0 for every x ∈ Z ∩ B r (p). Fix a non negative cut off function ϕ ∈ C ∞ c (B r (p)) with ϕ(x) = 1 on B r 2 (p) and consider for every λ > 0 the function By the above considerations f λ restricted to Z attains its maximum in p and by direct calculations we have that for every x ∈ B r 2 (p) . Evaluating the previous derivatives in p and implementing (3.3), we get 2 (e m+1 ⊗ e m+1 ) ij . By homogeneity of F , we have F m+1,m+1 (p, e m+1 ) = 0, and combining the previous equation with (3.1), we deduce that there exists λ 0 > 0 such that for all λ > λ 0 We conclude that f λ fails the condition (i) for λ chosen sufficiently big, showing that Indeed, for every v = e m+1 , the strict convexity of F implies that F m+1,m+1 (p, v) > 0 and we can compute lim (ii) ⇒ (i): Suppose Z fails to have property (i), we will show that this implies Z does not satisfy property (ii). Similarly to the previous step, we can make use of the argument of [12, Lemma 2.4] and assume without loss of generality that f ∈ C ∞ (Ω, R), f | Z attains its maximum at a unique point p ∈ Z (f (x) < f (p) for every x ∈ Z), the super-level set {x : f (x) ≥ a} is compact for every a ∈ R, there exist r > 0 and δ > 0 small enough such that f (x) < f (p) − δ for all x ∈ B r (p) and where the right hand side is intended to be zero when Df (p) = 0.
If |Df | (p) = 0, Z fails to have property (ii) since trivially Hence, we are reduced to consider the case Df (p) = 0, i.e. the case in which there exists v 0 ∈ S m such that This is done by relaxation. Up to a translation of Z by p and considering f − f (p) we may assume without loss of generality that p = 0 and f (0) = 0. We can fix M > 0 with M ≥ sup{|f (x)| + |Df (x)| : x ∈ B 2r (0)}. Furthermore, for λ > 0 we define the smooth auxiliary function Observe that, by the stated properties of f , for every x ∈ Z and every y / which implies that for λ big enough x is far enough from x λ . Since y λ ∈ B ( M λ ) 1 4 (x λ ), then as λ → +∞ we get x λ − y λ → 0 and consequently also y λ → 0. For each couple (x λ , y λ ) we distinguish two cases: First case: x λ = y λ . Since y → g λ (x λ , y) admits a global maximum in y λ we have D y g λ (x λ , y λ ) = Df (y λ ) = 0 and D 2 y g λ (x λ , y λ ) = D 2 f (y λ ) ≤ 0. By convexity of F , it holds F ij (y, v) ≥ 0 for every (y, v) ∈ Ω × S m , hence Passing this inequality to the limit for λ → +∞ we get which contradicts (3.4). Second case: x λ = y λ . As before y → g λ (x λ , y) admits a global maximum in y λ , hence The function f λ |Z admits its maximum at x λ because for every Thanks to (3.4), for λ sufficiently large, we deduce that We conclude that Z fails to have property (ii).
Let φ ∈ C ∞ c (B r (p)) be non negative, φ = 1 on B r 2 (p) and |Dφ| < 4 r . By Sard's theorem there is a regular value c of f with 0 < 4 By the choice of c and φ and thanks to (3.5), we compute Hence 0 is a regular value off |B r (p) and therefore 0 is a regular value off on the whole set. Sincef = f on U := B r 2 (p), we infer that U ∩ {f ≤ 0} = U ∩ {f ≤ 0} and we conclude that the relatively closed set N := {f ≤ 0} has the claimed properties. Proof of Lemma 3.3. Fix a smooth proper function u : Ω → R with u < 0 on N . We define the signed distance function d defined as Given r > 0, as before we fix a non negative function φ ∈ C ∞ c (B r (p)), with φ = 1 on U := B r 2 (p). It is now straightforward to check that, choosing r small enough, the function f (x) := φ(x)d(x) + (1 − φ(x))u(x) has the claimed properties.
Remark 3.2. In Proposition 3.1 above, we may replace (ii) with the following equivalent condition: for some a 0 ∈ R, a 1 ∈ S m and A ∈ R (m+1)×(m+1) and if P | Z has a local maximum at p, then (3.6) Indeed, the fact that (ii) implies (ii)' is immediate. For the converse, let f as in (ii) and p a local maximum of f | Z . Consider for any ε > 0 the paraboloid Since f ∈ C 2 , for every ε > 0 there exists r ε > 0 such that Then P ε | Z attains its local maximum in p. Moreover we compute Letting ε → 0 in (3.6), we deduce the inequality in (ii) for f in p.
The following is our main theorem. The proof is based on (the proof of) the maximum principle of Solomon and White for varifolds which are stationary with respect to an anisotropic integrand, see [9].
Then the area-blow up set Z := {x ∈ Ω : lim sup k→∞ V k (B r (x)) = +∞ for every r > 0 } is an (m, h)-set in Ω with respect to F .
Proof. We first observe that Z is a closed set. Indeed, given {x n } n∈N ⊂ Z, such that x n → x ∈ Ω, then, for every r > 0, there exists n big enough such that B r/2 (x n ) ⊂ B r (x). We deduce that lim sup which implies that x ∈ Z and consequently that Z is closed. Assume now that Z is not an (m, h)-set. Hence due to Proposition 3.1 there is a smooth function f : Ω → R and a point p ∈ Ω ∩ Z such that f | Z has a unique local maximum at p, Df (p) = 0 and (ii) fails. After translation by p and rotation and scaling of f we may assume that p = 0, f (p) = 0 and Df (p) = −e m+1 . The contradiction then reads Let us define the vector field Firstly note that X(x), Df (x) = F (x, Df (x)) hence X is pushing along "outside" the level sets {f ≤ t}. Furthermore Moreover, by (2.17) where H F (x) is the F -mean curvature of a level set {f = t}. Now we want to show how this vector field can be used to derive the contradiction to (3.7). First fix a radius r > 0 and δ > 0 such that for all x ∈ B 2r (0) (3.12) and |Df (x)| ), which combined with (3.12), gives the following estimate on B 2r (0) (3.14) By assumption we have Z ⊂ {f ≤ 0} and Z ∩ {f = 0} = {0}, hence there exists η 1 > 0 such that f (x) < −η 1 for all x ∈ Z \ B r (0). Now we fix a non-negative cut off function ϕ(x) supported in B 2r (0) with ϕ(x) = 1 on B r (0). For 0 < η 2 < η 1 to be chosen later, we define the function

Now we consider the vector field
Then we have Hence for every a we have We analyze the three terms separately. Note that |III| ≤ C1 B 2r \Br∩{f ≥−η 1 } . Since by the choice of r and Concerning II we have due the uniform convexity of F there is a constant c F where, for v, w ∈ S m , we set dist RP m (v, w) := min{|v + w|, |v − w|}, (3.16) and we introduced the function We conclude taking into account (3.13) It remains to estimate I. By (3.12), (3.11) and the C 2 regularity of F , there exists a constant C F ≥ 0 such that Taking additionally into account that {η ≥ η 2 }∩B 2r ∩Z = ∅ and (3.13), we conclude Combing all the estimates for I − III we have Observe that 0 ≤ η • f ≤ 2η 2 on the set {f < η 2 } and η ′ = 1 on the set {η > 0}. Let us consider the polynomial For a fixed µ ≥ 0 its minimum is obtained in t min. = 2C F µ c F and takes the value Hence if µ ≤ 2η 2 with η 2 > 0 sufficient small, p(µ, t) is non-negative i.e. for such a choice of η 2 we have is an open neighbourhood of 0 and 0 ∈ Z, we conclude that contradicting the assumption (3.7) and proving the theorem.

Consequences of Theorem 3.4.
By repeating the arguments of [12], we can now derive several properties of area blow-up sets (and more in general of (m, h)sets).

Proposition 3.5.
Let Ω ⊂ R m+1 be open, (F k ) k be a sequence of anisotropic integrands, and (Z k ) k be a sequence of (m, h k )-subset of Ω with respect to the integrand F k . Suppose that F k converges uniformly on compact subsets of Ω to some integrand F , Z k converges in Hausdorff distance to a closed set Z and h k → h, then Z is an (m, h)-subset of Ω with respect to the integrand F .
Proof. We will prove that the condition (ii)' in Remark 3.2 holds. Let be a paraboloid that realizes its maximum on Z in p ∈ Ω. Let r > 0 such that B r (p) ⊂⊂ Ω. For any ε > 0 and k sufficient large, the map P ε (x) := P (x) − ε |x − p| 2 2 realizes a strict local maximum on Z k ∩ B r (p) along a sequence of point p k ∈ Z k ∩ B r (p), such that p k → p.
Since Z k are (m, h k )-subset of Ω, we can apply the characterization (ii)' in Remark 3.2 to P ε to deduce that Passing to the limit as k → ∞ and ε → 0, we obtain Corollary 3.6. Let Ω ⊂ R m+1 be open and Z ⊂ Ω be an (m, h)-set with respect to the anisotropic integrand F . Consider a sequence r k ց 0 and a point p ∈ Ω ∩ Z such that Then Z ∞ is an (m, 0)-set of R m+1 with respect to the frozen integrand F p (ν) := F (p, ν).
Proof. It is straight forward to check that for every r > 0 and q ∈ Ω Z − q r is an (m, rh)-set with respect to the integrand F q,r (x, ν) := F (q + rx, ν).
By Proposition 3.5, Z ∞ is an (m, 0)-subset of the integrand Proof. If Z = ∅ there is nothing to prove. Assume that Z = ∅ and suppose by contradiction that Z = M . Since Z is closed, there exists B r (q) ⊂ Ω \ Z with q ∈ M and p ∈ Z ∩ B r (q). For a sequence of positive numbers λ k ց 0 consider Due to the regularity of M , we have that M k \ B r λ k ( q−p λ k ) converges in Hausdorff distance to a half plane H of T p M . Hence, passing to a subsequence, Z k → Z ∞ in Hausdorff distance, with Z ∞ ⊂ H and 0 ∈ Z ∞ . After a rotation O, we may assume that H = {x ∈ R m+1 : x m+1 = 0, x 1 ≥ 0}. By corollary 3.6 we have that Z ∞ is an (m, 0)-subset of R m+1 with respect to the frozen integrandF (ν) := F (p, Oν). Now consider the function . Observe that f takes a strict local maximum at 0 on H, hence f | Z∞ has a strict local maximum in 0, but this contradicts the characterization (ii) of Proposition 3.1, since D 2F (e 1 )(e 1 ⊗ e 1 + e m+1 ⊗ e m+1 ) > 0.
For the sake of completeness we prove also the anisotropic counterpart of the "classical" constancy theorem for varifolds. The reader may compare it with [7,Theorem 8.4.1] for the proof in the isotropic setting.
Proof. The strategy of the proof is similar to the one for the area functional, compare [7,Theorem 8.4.]. To simplify the presentation, we divide the proof in two steps: Step 1) if M is a plane, i.e. M = {x m+1 = 0}, and Ω = B 2r (0), then the conclusion of the proposition holds on B r (0).
Since η = 1 on M and B F (x, ν) is even in the second variable, the first variation formula reads Hence V is constant on M ∩ B r (0). This concludes the proof of step 1.

Proof of
Step 2: Fix any p ∈ M ∩ spt(V ) and 0 < r < dist(x, ∂Ω) such that the following holds: there is a C 2 function Φ : where ν x is the normal vectorfield to M .

Boundary curvature estimates
In this section we prove the following theorem which easily implies Theorem 1.2.

Recall that a set Ω is strictly
It easily follows by (2.11) that a uniformly convex set is strictly F -convex in in sufficiently small balls. Moreover the constants C and r 1 are uniform as long as Ω, Γ and F vary in compact classes 3 .
We start with the following simple lemma. Then the "area-blow up set" Z := {x ∈ R m+1 : lim sup j→∞ µ j (B r (x)) = +∞ for every r > 0} 3 For a family of curves Γα this amounts also in asking that all the considered curves should be "uniformly" embedded: Proof. Up to consider as new sequence of measures µ j |B 1 , we can assume that spt(µ j ) ⊂ B 1 . We claim that there exists a sequence of cubes {C i } i∈N with side length l i such that (i) C i+1 ⊂ C i for all i ∈ N; (ii) l i = 2 1−i ; (iii) lim sup j→∞ µ j (C i ) = +∞ for all i ∈ N. We will prove this claim by induction on i. We remark that C 0 exists: it is enough to consider a cube containing B 1 , for instance B 1 ⊂ [−1, 1] m =: C 0 , so that we have lim sup j→∞ µ j (C 0 ) = +∞.
Proof of the Inductive step: Let C the collection of the dyadic cubes that are obtained by dividing C i into 2 m+1 sub-cubes with half side length. Suppose Since there are only 2 m+1 of these cubes, there exists j 0 ∈ N and K > 0 such that But this contradicts the assumption, since We consequently can find a cube C i+1 ⊂ C i satisfying the properties (i), (ii) and (iii). As a consequence we obtain a decreasing sequence of dyadic closed cubes with nonempty intersection, i.e. there exists x ∈ ∞ k=0 C i . Since for every r > 0 there exists i ∈ N such that C i ⊂ B r (x), we have lim sup j→∞ µ j (B r (x)) = +∞.
This implies that x is in the area blow up set. Finally since x must be in the support of infinitely many µ j , we have x ∈ B 1 . This concludes the proof of this lemma.
The next proposition ensures that we have a local bound on the mass ratio, indeed assuming the contrary the varifolds associated with M x,r = M − x r would have unbounded masses. If Z is the area blow up set for this sequence, we can exploit our F convexity assumption together with the Hopf lemma to show that Z is contained in a wedge, this contradicts the fact that it is an (m, h)-set.
Step 2: There exists a radius 0 < R 0 ≤ R such that for every p ∈ ∂Ω the rescaled domain Ω−p R 0 and the rescaled manifold M −p R 0 satisfy the conditions of Assumption 4.1.
By a classical covering argument, one can show that Step 1 and Step 2 together imply Proposition 4.3.
Proof of Step 1: Assume the conclusion 4.1 does not hold in B 1 , then there exists a sequence M k , r k , p k satisfying (1) Γ k := ∂M k ⊂ ∂Ω with uniformly bounded C 2,α -norm; (2) p k ∈ Γ k ∩ B 1 , 0 < r k < 1 k and H m (M k ∩ B r k (p k )) r m k > k. We denote with γ k the projection of Γ k onto the plane {x m+1 = 0}, i.e.
where G Φ (x) := (x, Φ(x)) is the graph map of Φ. Up to subsequences, and performing if necessary a rotation of B 2 , we may assume that (3) there exists x 0 ∈ B 1 such that p k = (x k , Φ(x k )) → p 0 = (x 0 , Φ(x 0 )); (4)ν k (x k ) → e m , whereν k (x) denotes the normal of γ k in the plane {x m+1 = 0} at the point x ∈ γ k . To set up the contradiction we need the following additional construction: Consider r 0 > 0 small enough so that for all k ∈ N we have where A γ k denotes the second fundamental form of γ k . This is possible since we Furthermore we fix an orthonormal basis t 1 , . . . , t m spanning T q ∂Ω = ν ⊥ q ∼ R m i.e. R m+1 = T q ∂Ω × span ν q . We will write (x, x m+1 ) for points in T q ∂Ω × span ν q .
These are the non-parametric functionals associated to the image of the parametrized surfaces x → q + rx + ru(x)ν q . Let L r be the Euler-Lagrange operators for F r . By strict convexity of F , planes are the unique minimizers for the frozen integrand ν ∈ S m → F (q, ν). With respect to F r this implies that the constant functions u are the unique minimizers of F 0 , and in particular L 0 u = 0 for every constant function u. The convexity of F translates into the ellipticity of the linearization of L r around the constant u 0 = 0. Hence the implicit function theorem implies the existence of δ q , R q > 0 such that, for every couple of scalar functions f, g with f C 0,α < δ, g C 2,α < δ q , U ⊂ B 2 and r ≤ R q , the boundary value problem has a unique solution u ∈ C 2,α (U, R) satisfying u C 2,α (U,R) ≤ C( f C 0,α + g C 2,α ).
The size of R q , δ q only depends on the C 2,α norm of F . Hence by compactness there exist R 1 , δ 1 > 0 such that δ q > δ 1 and R q > R 1 for all q ∈ ∂Ω ∩ B 2R . Let H F (q) as before denote the anisotropic mean curvature of ∂Ω with respect to the inner normal ν(q). Fix 0 < R 0 ≤ R 1 such that