Soap films with gravity and almost-minimal surfaces

Motivated by the study of the equilibrium equations for a soap film hanging from a wire frame, we prove a compactness theorem for surfaces with asymptotically vanishing mean curvature and fixed or converging boundaries. In particular, we obtain sufficient geometric conditions for the minimal surfaces spanned by a given boundary to represent all the possible limits of sequences of almost-minimal surfaces. Finally, we provide some sharp quantitative estimates on the distance of an almost-minimal surface from its limit minimal surface.


Introduction
In the study of soap films under the action of gravity, one is interested in surfaces with small but non-zero mean curvature spanned by a given boundary. Indeed, as explained in section 2 below, the mid-surface M of a film of thickness 2 h > 0 satisfies in first approximation the equilibrium condition where H M is the mean curvature of M with respect to the unit normal ν M , e 3 is the vertical direction, and κ −1 is the capillary length of the film, defined by κ := g ρ σ . (1.2) Here, ρ is the volume density of mass for the film solution, σ denotes the surface tension of the film (with dimensions Newton per unit length), and g is the gravity acceleration on Earth. The interest for this equation lies in the fact that it correctly encodes several physical properties which are missed by the minimal surface equation H M = 0, e.g. the fact that actual soap films cannot be formed under arbitrary large scalings of the boundary curve.
In this setting, the first question one wants to answer is whether minimal surfaces are a good model for their small mean curvature counterpart. In this paper, we provide a general sufficient condition on the boundary data to ensure the validity of this approximation. When the model minimal surface is smooth and strictly stable, we also provide quantitative estimates for almost-minimal surfaces in terms of their total mean curvature. Since formal Notice that it is not necessary that Γ is contained into a convex set, or into a mean convex set, for the condition to hold. On the right, another set of circles defining a boundary Γ which does not satisfy accessibility from infinity. Indeed, there is no way to touch the smaller circle with an acute wedge containing the larger ones. statements require the introduction of a few concepts from Geometric Measure Theory, we present for the moment just an informal and simplified version of our main results.
Theorem. Let Γ be a compact, orientable (n − 1)-dimensional surface without boundary in R n+1 , and let {M j } j be a sequence of compact, orientable n-dimensional surfaces in R n+1 with boundaries Γ j = f j (Γ) for maps f j converging in C 1 to the identity map, and such that (denoting by H n the n-dimensional Hausdorff measure in R n+1 ), Assume that Γ has the following two properties: Finiteness and regularity of the Plateau problem: There are finitely many minimal surfaces {N i } i spanned by Γ, possibly including in the count "singular" minimal surfaces, whose singularities are anyway located away from Γ.
Under these two assumptions, we have the following conclusions: No-bubbling: There exists a single minimal surface N i such that M j → N i as j → ∞, in the sense that there exist open sets {E j } j with smooth boundary such that Here |E| denotes the (n + 1)-dimensional volume of E ⊂ R n+1 .
Strong convergence and sharp estimates: If in addition Γ j = Γ, N i has no singularities, and N i is strictly stable, in the sense that, for a positive constant λ, (where |A N i | is the Hilbert-Schmidt norm of the second fundamental form of N i ֒→ R n+1 ), and if for some p > n we have a uniform bound then there exist smooth functions u j : N i → R with u j = 0 on ∂N i and u j C 1 (N i ) → 0 as j → ∞ such that and the following sharp estimates hold: for a constant C = C(N, p).
Remark 1.1. As shown by simple examples (see Figure 3.2), if accessibility from infinity fails then bubbling can occur in the convergence of {M j } j . In particular, {M j } j could converge to a smooth minimal surface with multiplicity 2, and some pieces of the limiting minimal surface could not be part of any minimal surface spanned by the whole Γ.
Remark 1.2. In the case M j is the boundary of an open set (and thus, necessarily, Γ = ∅), and M j has almost-constant (non-zero) mean curvature, then the occurrence of bubbling is unavoidable, and its description has been undertaken in various papers, see e.g. [BC84,Str84,CM17,DMMN17,KM17,DM17]. From this point of view, the fact that we can avoid bubbling under somehow generic assumptions on the boundary data Γ is a remarkable rigidity feature of Plateau's problem.
Remark 1.3. The finiteness and regularity assumption is well-illustated in the case when Γ consists of two parallel unit circles in R 3 , having centers on a common axis. The idea here is that, depending on the distance between the circles, there should be at most five "generalized" minimal surfaces spanned by Γ (see Figure 3.1): two parallel disks, two catenoids (one stable, the other unstable), and two singular catenoids. Each singular catenoid is formed by attaching a smaller disk to two catenoidal necks so that the disk floats at mid distance from the two boundary circles, and the necks form three 120-degrees angles along the circle. Notice that the floating circle does not count as a boundary curve, but rather as a curve of "singular" points. Observe that accessibility from infinity trivially holds in this case, while the validity of the finiteness and regularity assumption (which is formally introduced in section 3.4) is not obvious, although it seems quite reasonable to expect it to be true. If that is the case, the compactness theorem indicates that a sequence of smooth almost-minimal surfaces spanned by Γ (or with boundaries converging to Γ) must converge to one of these five minimal surfaces, without bubbling. Actually, a simple additional argument can be used to exclude that the singular catenoids are possible limits, see Remark 4.1.
Remark 1.4. Both estimates (1.4) and (1.5) are sharp. When p = ∞, (1.4) generalizes to arbitrary minimal surfaces the fact that an almost-minimal surface bounded by a circle deviates from a flat disk at most linearly in the mean curvature times the area of the disk. The interest of (1.5) is that the L 2 -norm of the mean curvature appears as the dissipation of the area along a mean curvature flow with prescribed boundary data, see for example Huisken [Hui89] and Spruck [Spr07]. Moreover, we notice the close relation between (1.5) and the main result from [DPM14], which addresses the problem of proving global stability inequalities for smooth, area-minimizing surfaces. Finally, we remark that the bound on H M j L p (M j ) for p > n is needed to enforce the graphicality of M j over N i via Allard's regularity theorem. If one knows a priori that M j is a graph over N i , then (1.4) can be proved for every p ≥ 2 with p > n/2 (for example, p = 2 works for two and three dimensional surfaces); see Theorem 5.1 in section 5 below.
The paper is organized as follows. In section 2 we discuss the equilibrium conditions for soap films with gravity, and derive (1.1) under appropriate conditions. An interesting outcome of this discussion is the idea, based on physical grounds, of formulating Plateau's problem as a singular capillarity problem. Section 3 consists in part of a preliminary review of the necessary concepts from Geometric Measure Theory, and in part of a precise formulation of our two main assumptions. In section 4 we give a precise statement and the proof of our main compactness result, see Theorem 4.1. Finally, in section 5, we explain the reduction to graph-like surfaces, and prove various sharp convergence estimates, see Theorem 5.1. These last results show that on graph-like surfaces one can work with a very weak notion of almost-minimality deficit, a fact that will likely prove useful in future investigations.

Soap films with gravity
Due to gravitational forces, surfaces with small but non-zero mean curvature arise naturally in the study of soap films hanging on a wire. This effect is usually neglected in the mathematical literature, leading to an exclusive focus on minimal surfaces. The resulting model describes correctly the physical situation of small soap films. However, as noticed by Defay and Prigogine, "gravitational forces [...] play a dominant role in determining the shapes of macroscopic surfaces"; see [DP66,]. The typical length scale which separates small films from large films is given by the capillary length κ −1 = σ/ρg, introduced in (1.2). For a solution of soap in water at room temperature, the values of the surface tension and of the density are, respectively, σ ≃ 0.03N/m and ρ ≃ 10 3 kg/m 3 , while g ≃ 9.81N/kg is Earth's gravity, so that the length-scale κ −1 is of order of 1.7mm. The deviation of a soap film with gravity from its limit minimal surface is expected to be O(h κ) where h is the average width of the film. For typical soap films, we are in the perturbative region, since we usually have h ≃ 10 −3 mm ≃ 10 −3 κ −1 .
Idealizing the wire frame as a smooth curve Γ without boundary in R 3 , and the soap film as a smooth surface M bounded by Γ, if we neglect gravity then we are led to modeling soap films as minimal surfaces, i.e. surfaces with vanishing mean curvature This condition is derived from balancing the atmospheric pressures on the two sides of the film with the Laplace pressure induced by surface tension [You05,Lap06]. Denoting by σ the surface tension, if S is a small neighborhood of x ∈ M , with outer unit co-normal ν M S with respect to M , then the tension on S is given by Here, H M denotes the mean curvature vector to M , which, once the choice of a unit normal ν M to M is specified, defines a scalar mean curvature H M appearing in (2.1) through the equation H M = H M ν M . If the atmospheric pressures on the two sides of the film are assumed to be equal, as it is the case if we ignore gravity, then the Laplace pressure must vanish, and we find (2.1). Let us recall that (2.1) can also be derived by the principle of virtual works, as first done by Gauss [Gau30], by taking as the total energy of the film the area of M times σ, namely E(M ) = σ H 2 (M ) .
(2.3) Equation (2.1) fails in describing macroscopic soap films in two ways: (i) For a given contour Γ, the minimal surfaces spanned by t Γ, for a scaling factor t > 1, are simply obtained by scaling the minimal surfaces spanned by Γ. This is evidently not the case for real soap films, where there is a competition between the capillary length κ −1 and the length-scale of the boundary curve Γ in determining if a soap film is produced at all. From this point of view, H M = 0 fails completely at describing the macroscopic length-scales at which soap films are actually formed. Equation (1.1), namely H M = κ 2 h ν M · e 3 + O(h 2 ), does not have this problem. Indeed, the solvability of a prescribed mean curvature equation H M = f with ∂M = Γ requires a control on the size of f in terms, for example, of H 2 (M Γ ) −1/2 , where M Γ is the area-minimizing surface spanned by Γ; see, e.g., the papers by Duzaar and Fuchs [DF90,DF92]. In particular, the solvability of (1.1) with boundary condition ∂M = Γ depends on the relative sizes of κ 2 h (which measures the physical properties of the soap solution) and of the length-scale of Γ.
(ii) Equation (2.1) is invariant under rotations, while the effect of gravity is definitely anisotropic. For example, a soap film M hanging from a circular frame Γ of radius r should be exactly a flat disk if Γ is contained in a vertical plane, whereas it should possess a non-trivial curvature if Γ is in horizontal position, with average vertical deviation from the flat disk of order r 2 H M . This deviation is observable depending on the length scale of Γ and on κ. In the case of soap bubbles, where H M = 0 is replaced by H M constant, a deviation is experimentally observed and is substantial; see [CDTR +  In order to take the effect of gravity into account, one might be tempted to add to the surface tension energy functional a term corresponding to the potential energy of the film, namely, to consider in place of (2.3), with ρ * denoting surface density of mass. While this would be correct for a solid elastic slab, or a rubber sheet, for a fluid it is clearly incorrect. In fact, it would amount to replace H M = 0 with the equation H M (x) = κ 2 x 3 , which would incorrectly predict that a soap film hanging from a perfectly planar wire contained in a vertical plane should have curvature and lie out of the plane! In [DP66, Section I.4], Defay and Prigogine explain how the effect of gravity should be modeled by balancing pressures. One needs to consider the finite thickness of the film, bounded by two different interfaces, and to take into account the difference in hydrostatic pressures on the two faces caused by the gravitational pull. We now put into equations this idea, and formulate a PDE for the problem. The resulting PDE, see (2.8), justifies (1.1), which, in turn, appears in the literature when M is axially symmetric and very close (in a C 1 -sense) to a plane; see e.g. [dGBWQ03, Equation (2.5)].
Consider a smooth two-dimensional surface M bounded by a smooth curve Γ in R 3 , and oriented by a unit normal ν M . Here M plays the role of an ideal surface lying inside the film. Given a smooth function α defined on M , we denote its graph over M by The two interfaces of the soap film are described by graphs M (α) and M (−β) for positive functions α and β. Up to replacing M with M ((α−β)/2), and then setting ψ := (α+β)/2, we can actually assume that the interfaces are M (ψ) and M (−ψ), where ψ is a smooth positive function on M . However, it does not seem that the symmetric parametrization is always the most convenient, so we shall argue in terms of α and β.
Given x ∈ M , and with reference to Figure 2.1, at equilibrium, the pressure p(x + ) at where H M (α) is the scalar mean curvature of M (α) with respect to the unit normal pointing outside the film, p 0 is the atmospheric pressure, and σ is the surface tension. The pressure where H M (−β) is the scalar mean curvature of M (−β) with respect to the unit normal pointing outside of the film. Subtracting the two equations, we obtain The difference between p(x − ) and p(x + ) is the hydrostatic pressure (2.7) Combining (2.5), (2.6) and (2.7) we obtain the equation for minimal surfaces with gravity (2.8) If |∇α| and |∇β| are sufficiently small at x, and we consider the mid-surface parametrization, then we can assume that locally α ≡ β ≡ h, where h is a small positive constant. Denoting by {κ 1 , κ 2 } the principal curvatures of M , and stressing the smallness of h by requiring 0 < h < max{|κ 1 |, |κ 2 |} −1 , we thus obtain and (2.8) is readily seen to imply We now explain how (2.8) can be derived from energy considerations. The idea is treating the problem of a soap film hanging from a wire frame as a capillarity problem. We model the wire frame as a solid δ-neighborhood of an idealized curve Γ, setting We model the soap film as a set E ⊂ A δ with very small volume ε = |E|, and, following Gauss' treatment of capillarity theory, we define its energy as see Figure 2.2. Here γ ∈ (−1, 1) is a dimensionless parameter taking into account the ratio between the surface tension on the liquid-air interface, and the surface tension on the liquid-solid interface along the wire frame walls. Assuming that E is a smooth critical point of this energy, the Euler-Lagrange equations boil down to the equilibrium condition where H E denotes the scalar mean curvature of ∂E with respect to the outer unit normal to E, and λ is a Lagrange multiplier associated to the volume constraint. Equation (2.9) is coupled with Young's law, Under the assumption that ε H 2 (S) ≪ δ and that δ is sufficiently small in terms of the local and global geometric properties of Γ, it is reasonable to expect the existence of critical points E described by means of midsurfaces M spanned by Γ. More precisely, we consider critical points E corresponding to surfaces M with ∂M = Γ in the sense that, for every x ∈ M ∩ A δ we can find r > 0 such that In this case, (2.9) computed at y = x + α(x) ν M (x) = x + and at (2.11) Notice that our sign conventions on scalar mean curvatures have been such that Subtracting the two equations we deduce indeed the validity of (2.8) as a consequence of the equilibrium condition for Gauss' capillarity energy. Notice that the full set of equilibrium conditions is expressed by considering Young's law together with the two equations (2.11), or with the single equation (2.9), rather than by (2.8) alone. Here the role of (2.8) is stressed because, as explained above, it clearly motivates the study of surfaces with small mean curvature.
In summary, we have seen in this section how surfaces with prescribed boundary and small mean curvature, such as the ones described by equation (2.8), or by its approximation (1.1), arise naturally in the study of soap films hanging from a wire. More generally, the use of capillarity theory to model soap films provides an additional, more physical, point of view on the long-debated issue of prescribing boundary data in the mathematical formulation of Plateau's problem; see [Har04, Dav14, HP16, DPDRG16, DLGM17, GLF17, DLDRG17, ABP17, FK18, DR18] for the most recent developments on this venerable question. Leaving a more complete discussion of this last point to a forthcoming paper, we focus here on a first problem raised by this approach, namely understanding the relation between almost-minimal and minimal surfaces.

Almost-minimal surfaces
Let Γ be a compact (n−1)-dimensional surface in R n+1 without boundary. Motivated by the study of surfaces obeying (1.1), we now consider the general question of understanding the relation between the minimal and the almost minimal surfaces spanned by Γ. The question we want to address is the following: In the class of surfaces spanned by Γ is the family of minimal surfaces rich enough to describe all the possible limits of almost-minimal surfaces?
(3.1) Theorem 4.1 answers affirmatively to this question under the assumptions that Γ is accessible from infinity and spans finitely many minimal surfaces without boundary singularities. The statement of the theorem is actually quite delicate, as it involves several choices and assumptions. In the following paragraphs we shall address these points. In § 3.1 we propose various ways of measuring the almost-minimality of a surface, while in § 3.2 we review two notions of convergence for smooth surfaces arising in Geometric Measure Theory. In § 3.3 we discuss our geometric assumption on the connected components of Γ, and in § 3.4 we make precise the idea that Γ spans at most finitely many minimal surfaces.
3.1. Measuring almost-minimality. Directly motivated by the equation for minimal surfaces with gravity (1.1), we shall consider the uniform deficit as our chief option to measure almost-minimality. But depending on other possible applications of almost-minimal surfaces, the family of integral deficits may be more relevant. For example, δ 2 (M ) definitely plays a role in the study of the gradient flow defined by Plateau's problem, see [Hui89,Spr07]. At the weaker end of the spectrum, and closer to the point of view usually adopted when discussing Paley-Smale sequences in variational problems, one may consider the duality deficits This last definition is motivated by the tangential divergence theorem, stating that if M is a smooth compact n-dimensional surface with boundary Γ, then Here ν M Γ is outer unit co-normal to Γ with respect to M , and div M X is the tangential divergence of X with respect to M , that is An interesting fact is that on surfaces M that are a priori known to be graphs over strictly stable minimal surfaces, the duality deficit δ −∞ (M ) already controls the area deficit, see Theorem 5.1.

3.2.
Convergence of smooth surfaces. In order to provide a better insight into question (3.1), we need to discuss possible notions of limit for a sequence of smooth surfaces.
To introduce the relevant ideas, let us consider a sequence {M j } j of smooth oriented n-dimensional surfaces such that Geometric Measure Theory provides two canonical ways to discuss the convergence of such a sequence {M j } j . Both approaches require the identification of each M j as a linear functional on a space of test functions, or, equivalently, as a Radon measure on a suitable finite dimensional space. The first approach, the theory of currents, allows to transfer the spanning information ∂M j = Γ to a generalized limit surface. The second approach, the theory of varifolds, allows to infer from δ ∞ (M j ) → 0 the existence of a limit surface that is minimal, again in a generalized sense. A subtlety lies in the fact that the generalized limit surface in the varifold sense may be larger that its counterpart in the sense of currents.
The viewpoint of currents. We see each oriented surface M j in (3.4) as a linear continuous functional M j on the space D n (R n+1 ) of smooth, compactly supported n-dimensional differential forms, equipped with the standard topology of test functions. More precisely, if M j is oriented by a continuous choice of a unit normal vector field ν M j , we set where, given ν ∈ S n , ⋆ν denotes the simple unit n-vector corresponding to the n-dimensional plane ν ⊥ oriented by ν, and the duality between n-vectors and n-covectors appears under the integral. Let us recall that ⋆ν M j induces a smooth orientation τ Γ on Γ (that is, a smooth field of simple unit (n − 1)-vectors defining and orienting the tangent planes to Γ) in such a way that Stokes' theorem holds where dω is the exterior differential of the (n − 1)-form ω. In this setting, it is quite natural to define the "boundary" of M j as the linear continuous functional defined on D n−1 (R n+1 ) by setting Of course, Stokes' theorem (3.5) implies that if Γ is oriented by the orientation τ Γ induced by the choice of ν M j then ∂ M j = Γ . The second and the third condition in (3.4) and the compactness theorem for Radon measures imply the existence of a linear continuous functional T on D n (R n+1 ) such that, up to extracting subsequences, Is the linear functional T still represented by the action on forms of an oriented surface with boundary, like the functionals M j are? A deep theorem of Federer and Fleming [FF60] gives a positive answer, provided that we introduce a suitable class of generalized surfaces with boundary. The key notion here is that of a rectifiable set. We say that a Borel set N ⊂ R n+1 is locally H n -rectifiable if, up to a H n -null set, N can be covered by countably many Lipschitz images of R n into R n+1 , and if H n (N ∩ B R ) < ∞ for every R > 0. If N is locally H n -rectifiable, then N has a tangent plane almost-everywhere, in the sense that for H n -a.e. x ∈ N there exists an n-dimensional linear subspace T Analogously to the smooth setting, such a vector field ν N will be called an orientation of the rectifiable set N . Coming back to (3.7), the Federer-Fleming compactness theorem shows the existence of a locally H n -rectifiable set N , of a Borel measurable orientation ν N , and of a function α ∈ L 1 loc (H n N ; Z) (an integer-valued multiplicity on N ) such that T = N, ⋆ν N , α , i.e.
Moreover, as a simple by-product of (3.6), we see that the limit current T has still boundary Γ, in the sense that ∂T = Γ , or, more explicitly: (3.10) The viewpoint of varifolds. The next question is if the rectifiable set N , found by taking the limit of {M j } j in the sense of currents, is minimal, at least in some generalized sense. The starting point is the tangential divergence theorem applied on M j to fields supported away from Γ, which yields Notice that, since δ ∞ (M j ) → 0, the right-hand side of (3.11) converges to zero as j → ∞.
To pass to the limit on the left-hand side we adopt the following point of view. Let us set where ν 1 ≡ ν 2 if and only if ν 1 = ±ν 2 , and denote by [ν] the ≡-equivalence class of ν ∈ S n . The point (x, [ν]) ∈ G n identifies the (unoriented) n-dimensional affine plane orthogonal to ν and passing through x in R n+1 . Given The definition is well-posed, as the right-hand side is invariant when exchanging ν with −ν. In this way Given that δ ∞ (M j ) → 0, the above argument shows that V, ϕ X = 0 for every X compactly supported in the complement of Γ. We then ask the question whether the varifold V can be associated to a generalized surface, and to what extent this surface is minimal. Another deep theorem, this time due to Allard [All72], provides the following answer: there exists a locally H n -rectifiable set N and a function θ ∈ L 1 loc (H n N ; N) (a non-negative integral multiplicity on N ) such that V is represented by N and θ, in symbols V = var (N, θ), in the sense that (3.14) As noticed, under the assumption (3.4), we have V, ϕ X = 0 whenever sptX ∩ Γ = ∅.
In other words, the varifold V = var (N, θ) is minimal on R n+1 \ Γ (or stationary, in the common terminology of Geometric Measure Theory), in the sense that Two remarks are in order: (i) The rectifiable set N arising in the varifold convergence is in general larger than the rectifiable set N obtained by taking the limit of {M j } j in the sense of currents. The typical example is obtained by considering M j = B 1 ∩ (K/j) (for j → ∞) where K is a fixed catenoid. In this case the limit in the sense of currents is trivial, N = ∅, because the two sheets of the catenoid cancel out in the limit due to their opposite orientations; at the same time, if the limit is taken in the sense of varifolds, N is equal to a unit disk with multiplicity θ = 2. For an example with fixed boundary data, see Example 3.4 below. From this point of view, answering question (3.1) partly amounts to determine conditions under which this ambiguity between the two limits, one taken in the sense of currents and the other in the sense of varifolds, does not occur ; (ii) Coming back to the generalized minimal surface condition (3.15), in the next classical example we notice how this condition allows one to include in the theory of minimal surfaces non-smooth examples that are actually physically relevant.
Example 3.1. Let Γ = Γ 1 ∪ Γ 2 be given by two parallel circles in R 3 with centers on a same axis. We can construct generalized minimal surfaces on R 3 \ Γ as multiplicity one varifolds var (N i ) := var (N i , 1), associated to the rectifiable sets D 1 and D 2 are two disks spanned by Γ 1 and Γ 2 resp. ; K 3 and K 4 are the catenoids (one stable, the other unstable) spanned by Γ ; K 5 and K 6 are two catenoids meeting at a 2π/3-angle along a circle Γ 3 lying on the midplane between Γ 1 and Γ 2 , centered along the same axis ; D 7 is the disk spanned by Γ 3 ; K 8 and K 9 are another pair of catenoids meeting at a 2π/3-angle along a circle Γ 4 lying on the midplane between Γ 1 and Γ 2 , centered along the same axis, with the radius of Γ 4 smaller than the radius of Γ 3 ; D 10 is the disk spanned by Γ 4 .
We claim that the var (N i )'s are generalized minimal surfaces. Since N 4 and N 5 are not smooth, we need to check carefully if they satisfy (3.15). By applying the tangential divergence theorem separately on the three minimal surfaces K 5 , K 6 and D 7 , we find that The sum of the above three co-normals is identically zero by the 2π/3-angle condition imposed on K 6 and K 7 , and so (3.15) holds, thus showing that N 4 is minimal. The minimality of N 5 follows analogously. We also notice that every integer valued combination satisfies (3.15), and is thus a possible limit for a sequence {M j } j satisfying (3.4) with Γ = Γ 1 ∪ Γ 2 . If such a limit arises with i q i ≥ 2, we speak of bubbling. In fact, an additional subtlety lies in the fact that varifolds of the form V = q 1,1 var (D 1 ) + q 1,2 var (D 2 ) + 5 i=2 q i var (N i ) with q 1,1 = q 1,2 (3.17) satisfy (3.15), and thus can arise as limits of almost-minimal surfaces (and indeed do so, see Example 3.5 below, if the mean curvature deficit is sufficiently weak). A limit like (3.17) is qualitatively worse than a limit of the form (3.16), in the sense that D 1 and D 2 alone do not span the whole Γ, but just some of its connected components.
3.3. A geometric assumption: accessibility from infinity. Given x ∈ Γ, we say that Γ is accessible from infinity at x if there exist a unit vector e and an angle θ ∈ [0, π) such that where Γ co denotes the convex envelope of Γ. Notice that if (3.18) holds at a given x then every minimal surface N spanned by Γ is automatically contained in the wedge centered at x which appears on the right hand side of (3.18).
Definition 3.2. We say that Γ is accessible from infinity if, for each connected component Γ m of Γ, the set of points x ∈ Γ m such that Γ is accessible from infinity at x has positive H n−1 -measure. Example 3.4 (Negative answer to (3.1) and bubbling with uniform deficit). Consider two concentric disks S 1 and S 2 contained inside a same plane, and bounded by circles Γ 1 and Γ 2 , see Figure 3.2. Set Γ = Γ 1 ∪ Γ 2 , so that N = S 1 \ int(S 2 ) is definitely a minimal surface spanned by Γ. Also, choose orientations on S 1 , S 2 and Γ in such a way that the spanning condition holds for the associated currents, that is ∂ N = ∂( S 1 − S 2 ) = Γ . We construct a sequence of surfaces M j by slightly bending S 1 and S 2 in the radial direction, and then connecting the two pieces with a catenoidal neck, see Figure 3.2. Evidently, this can be arranged so that In particular: (3.20) Bubbling is possible even when Γ is accessible from infinity if a weak notion of deficit is used. Here M j is the surface of revolution obtained by rotating the one-dimensional profile on the right, B εj (Γ 1 ) denotes an ε j -neighborhood of the circle Γ 1 , and M * j is the part of M j lying outside B εj (Γ 1 ). We take ε j such that M j intersects ∂B εj (Γ 1 ) in three circles, and so that the H Mj is uniformly small on M j \ M * j . The limit surface counts one copy of K, and two copies of the disk filling Γ 1 .
On the other hand, the currents M j satisfy because the two copies of S 2 appearing in the limit come with opposite orientations, and hence the corresponding currents cancel out. For this simple boundary curve Γ, we thus have a negative answer to (3.1): indeed, as shown by (3.20), the limit of the {M j } j cannot be described only in terms of minimal surfaces spanned by Γ (which indeed is not spanning S 2 ). In this example the bubbling phenomenon occurs, as part of the limit surface has multiplicity 2. Observe also that Γ is not accessible. Indeed, (3.18) cannot hold at any x ∈ Γ 2 . Finally, the example can be easily generalized to the situation when S 1 and S 2 are two smooth, bounded, simply connected orientable minimal surfaces S 1 and S 2 , spanned by curves Γ 1 and Γ 2 , with S 2 ⊂ S 1 .
Example 3.5 (Bubbling under accessibility from infinity with very weak deficit). As in Example 3.1, let Γ consist of two parallel disks Γ 1 and Γ 2 with centers on a same axis, so that Γ is accessible from infinity. We can give a negative answer to question (3.1) if a too weak notion of almost-minimality deficit is used, arguing along the following lines. Consider a catenoid K spanned by Γ, and construct a sequence M j by slightly deforming K outwards while keeping the boundary data at Γ 2 , sharply turning around along Γ 1 , going all the way towards the center of Γ 1 , turning again downwards with a small catenoidal neck, and then almost filling Γ 1 with a disk; see Figure 3.3. Denoting by M * j the part of M j lying at distance at most ε j from Γ 1 , by suitably selecting ε j → 0 as j → ∞, we entail We claim that Thus the limits in the sense of varifolds and currents do not agree (we observe bubbling), while an almost-minimality deficit goes to zero (although this is indeed the weakest possible deficit in our scale). To show that δ −∞ (M j ) → 0, we fix a vector field X compactly supported away from Γ and with |∇X| ≤ 1. We fist notice that If Γ * j is the component of the boundary of M j \ M * j that is not Γ 2 , then by our choice of ε j we find , where we have used |∇X| ≤ 1 and X = 0 on Γ to deduce: (i) that |X| ≤ ε j on Γ * j ; and, (ii) that |X| ≤ diam(M j ) on M j . Since H 1 (Γ * j ) → 3 H 1 (Γ 1 ) by construction, we have proved our claim.
3.4. Finiteness and regularity of the Plateau problem. The second main assumption we shall consider is that Γ spans finitely many minimal surfaces. This is an idea that has to be formulated with great care, because of the singularities that minimal surfaces can exhibit.
Let Γ be an (n−1)-dimensional compact smooth surface without boundary. As discussed in § 3.2, any varifold V = var (N, θ) corresponding to a compact H n -rectifiable set N in R n+1 and to a function θ ∈ L 1 (H n N ; N) such that can arise as a possible limit of almost minimal surfaces. Possible limits V have two other important properties: (i) As a consequence of (3.23), the support of V is bounded: indeed, an application of the monotonicity identity implies that spt V is contained in the convex hull of Γ, see [Sim83, Theorem 19.2]; (ii) Given our assumptions on M j , V has bounded first variation, in the sense that sup N θ div N X dH n : X ∈ C 1 c (R n+1 ; R n+1 ) , |X| ≤ 1 < ∞ .

In particular, by differentiation of Radon measures, (3.23) is always extended to
where µ * is singular with respect to H n N , and where ν is a Borel unit vector field. Fully understanding the regularity of sptV when (3.23) holds is a major open problem in Geometric Measure Theory. What is known on this specific problem is the following. Define (for any compact set N ) the sets of regular and singular points of N as We further divide Reg(N ) into Reg • (N ), the set of regular points of interior type (i.e., N ∩B ρ (x) is diffeomorphic to an n-dimensional disk), and into Reg b (N ), the regular points of boundary type. Now, let V = var (N, θ) be such that (3.23) holds, and consider any open set A such that θ is constant on A ∩ N . Then Allard's regularity theorem [All72] shows that There is also a boundary regularity theorem [All75], showing the existence of ε(n) > 0 such that if θ = 1 on A∩N and H n (N ∩B ρ (x)) ≤ (1+ε(n)) ω n ρ n /2 for some x ∈ A∩N ∩Γ, then N ∩ B ε(n)ρ (x) is diffeomorphic to a half-disk. The application of Allard's boundary regularity theorem can be quite deceptive. With reference to the notation of Example 3.1, it suffices to take N = D 1 ∪ K 3 with θ ≡ 1 to construct an example of V solving (3.23), with N \ Γ = Reg • (N ), and with Γ 1 = Σ(N ). Notice also that a similar example holds even in the "smoother" case when the measure µ * considered in the extension (3.24) of (3.23) actually agrees with H n−1 Γ, and when ν is H n−1 -a.e. orthogonal to Γ; that is to say, when (3.24) takes the more geometric form (3.25) Indeed, if the distance between the circles Γ 1 and Γ 2 in Example 3.1 is such that K 3 meets with D 1 along Γ 1 at a 120-degrees angle, then adding up the unit conormals of D 1 and K 3 on Γ 1 we obtain a unit vector such that (3.25) holds, but still the boundary regularity theorem cannot be applied at any point of Γ 1 , as N \ Γ = Reg • (N ) and Γ 1 = Σ(N ). Summarizing, the analysis of almost-minimal surfaces spanned by Γ unavoidably leads to consider minimal varifolds in R n+1 \ Γ, but, in turn, these objects are only partially understood. Our compactness theorem will thus be conditional to assuming a rather precise structure for minimal varifolds in R n+1 \ Γ. Namely, we shall require the possibility of decomposing them as linear combinations, with integer coefficients, of finitely many, unit density, connected pieces N i with unit conormals ν co i along finite unions Γ (i) of connected components of Γ (in particular, each piece N i may just be spanned by part of Γ); when removing its singular set and Γ, each piece N i is disconnected into at most finitely many smooth connected components. As explained in Proposition 3.8 below, these assumptions hold in the fundamental case when Γ is a graph over a convex surface.
Definition 3.6 (Finiteness and regularity of minimal varifolds spanned by Γ). Let Γ be a compact (n − 1)-dimensional smooth surface without boundary in R n+1 , and let {Γ m } M m=1 denote the connected components of Γ. We say that Γ spans finitely many minimal surfaces without boundary singularities if there exists a finite family {N i } i of compact H n -rectifiable sets with the following properties: (i) for each i, N i \ Γ is connected, and there exists a finite union Γ (i) = m∈I (i) Γ m of connected components of Γ with and such that for some ν co i : moreover, Reg • (N i ) has finitely many connected components {N i,ℓ } L(i) ℓ=1 such that, for each ℓ, cl (N i,ℓ ) \ Σ(N i ) is an orientable, smooth n-dimensional surface with boundary, whose boundary points are contained in Γ (i) ; (ii) if V = var (N, θ) has bounded support, bounded first variation, and satisfies then there exist q i ∈ N such that Remark 3.7. By Allard regularity theorem and by property (i), for each i, var (N i ) is a minimal varifold in R n+1 \ Γ with constant unit density, and thus we have Notice that we are excluding the possibility that Σ(N i ) intersects Γ: in other words, singularities are allowed, but not up to the boundary. In principle, this is the situation depicted in Figure 3.1. It is not hard, however, to observe soap films with curves of singular points extending up to the wire frame, so we do not expect this assumption to be generic.
The problem of checking Definition 3.6 on some classes of examples, or even in simple explicit situations like the one described in Example 3.1, seems delicate. In the next proposition we address the case of graphs over convex boundaries.
Proposition 3.8. If Ω ⊂ R n × {0} is a bounded connected open set with smooth and convex boundary, and if Γ ⊂ R n+1 is the graph of a smooth function u over ∂Ω, then Γ spans finitely many minimal surfaces in the sense of Definition 3.6.
Proof. Let us assume without loss of generality that 0 ∈ Ω. Let V = var (N, θ) be an integral varifold with bounded support satisfying (3.26). We first prove that sptV is contained in cl (Ω×R), where cl (A) denotes the closure of A ⊂ R n+1 . Indeed let H Ω denote the mean curvature of ∂Ω with respect to the outer unit normal to Ω. Consider the open cylinders K(t) = t (Ω × R) for t > 1. Since the support of V is bounded, for t large enough we have that sptV ⋐ K(t). If t * = inf{t : sptV ⋐ K(t)}, then t * < ∞ and thus there exists x = (x ′ , x n+1 ) ∈ spt V ∩ ∂K(t * ) such that, in the ordering of ν ∂K(t * ) (x) = ν Ω (x * ), x * := (x ′ /t * , 0), the smooth surface ∂K(t * ) touches from above sptV locally at x. Let us assume that x ∈ R n+1 \ Γ. Since ∂K(t * ) is smooth, H ∂K(t * ) (x) · ν ∂K(t * ) (x) = H Ω (x * )/t * ≥ 0, and V is minimal in a neighborhood of x, by the strong maximum principle of Schätzle [Sch04, Theorem 6.2] this is possible only if, locally at x, ∂K(t * ) is contained in sptV . Since sptV is anyway contained in cl (K(t * )), by a continuity argument, and by the connectedness of ∂K(t * ), we obtain ∂K(t * ) ⊂ sptV . This would be a contradiction, since sptV is bounded. Thus it must be that x ∈ Γ, i.e. t * = 1, and sptV ⊂ cl (Ω × R).
The classical area integrand theory (see, e.g. [Giu03, Chapter 1]) implies the existence of a smooth extension of u to the whole Ω, still denoted u, such that G(u) = {(z, u(z)) : z ∈ cl (Ω)} satisfies , properties (i) and (ii) in Definition 3.6 are clearly satisfied by N 1 .
We finally prove that V = q var (G(u)) for some q ∈ N. Since sptV is bounded and contained in the closure of Ω × R, we find that s * = inf s : x n+1 < s + u(z) ∀(z, x n+1 ) ∈ sptV is finite. In particular, s * e n+1 + G(u) touches sptV from above in the ordering of e n+1 . If the touching point x does not belong to Γ, then, again by Schätzle's strong maximum principle we find that s * e n+1 + G(u) ⊂ spt V with s * = 0. But then sptV would have a contact point with ∂Ω×R outside of Γ, where V is minimal, and thus the strong maximum principle would imply ∂Ω × R ⊂ sptV , once again against the boundedness of spt V . The touching point x of s * e n+1 + G(u) and sptV must thus lie on Γ, so that s * = 0, and x n+1 ≤ u(z) whenever (z, x n+1 ) ∈ spt V . An entirely similar argument shows that so that we also have x n+1 ≥ u(z) whenever (z, x n+1 ) ∈ spt V . We have thus proved that G(u) = spt V . The constancy theorem for integral varifolds, [Sim83,Theorem 41.1], implies that V = q var (G(u)) for a constant q ∈ N.

The compactness theorem
We are finally ready to state and prove our main compactness theorem.
Theorem 4.1 (Compactness theorem for almost-minimal surfaces). Let Γ be a smooth (n − 1)-dimensional compact orientable manifold without boundary in R n+1 , and let Γ be the (n − 1)-current corresponding to the choice of an orientation τ Γ on Γ. Assume that Γ is accessible from infinity (see Definition 3.2) and that Γ spans finitely many minimal surfaces without boundary singularities (see Definition 3.6).
Let {M j } j be a sequence of smooth n-dimensional surfaces, oriented by smooth unit normal vector fields ν M j , and with smooth boundaries Γ j oriented in such a way that, if Assume that Γ j converges to Γ, in the sense that there exist Lipschitz maps f j : Then, there exist an H n -rectifiable set N , and Borel vector fields ν N : N → S n and ν : Γ → S n with Remark 4.1. A point that we are not trying to formalize here is that in situations like the one considered in Figure 3.1, when Σ(N ), if present, is "classical", then one can actually prove that Σ(N ) = ∅, thus concluding that smooth M j 's cannot converge to minimal surfaces with singularities. To illustrate the idea, let Γ 1 and Γ 2 be the circles of Example 3.1, and fix orientations on Γ 1 and Γ 2 in order to define the associated currents Γ 1 and Γ 2 . Suppose by contradiction that as a limit of a sequence M j of almost-minimal surfaces with ∂ M j = Γ := Γ 1 + Γ 2 one obtains the singular minimal surface N = K ∪ K ′ ∪ D obtained by gluing two catenoids K and K ′ to a disk D along the boundary circle Σ = ∂D with a 120-degrees angle. Assign orientations to K, K ′ , and D in such a way that The limit current T of the sequence M j must then satisfy Since T is the limit of currents defined by the M j 's, we also have which implies α 1 = 1 = α 2 and σ 1 + σ 2 + α 3 = 0 , which is impossible, given σ 1 , σ 2 , α 3 ∈ {−1, 1}. A general argument along these lines can be repeated if assuming that a number of odd half-spaces meet along points in Σ(N ).
Before giving the proof of the theorem, we need to introduce some notation. Given an n-dimensional varifold V on R n+1 , that is, a Radon measure on G n = R n+1 × (S n / ≡) as described in section 3, we denote by We denote by V T = var (N, |α|) the induced varifold of T .
Proof of Theorem 4.1.
Step one: We start by discussing the varifold limit of the M j 's. By the area formula and by (4.2) we have (4.8) Setting V j := var (M j ), by the tangential divergence theorem we have for every X ∈ C 1 c (R n+1 ; R n+1 ). In particular, (4.8) and δ 1 (M j ) → 0 imply lim sup while at the same time V j (R n+1 ) = H n (M j ). By (4.1), the supports of the V j 's are contained in a fixed ball, and sup (4.11) Indeed (4.2) implies that if spt X ⊂ R n+1 \ Γ, then spt X ⊂ R n+1 \ Γ j for every j large enough. Thus, δ 1 (M j ) → 0 and (4.9) give for every X ∈ C 1 c (R n+1 \ Γ, R n+1 ), as claimed. Since Γ spans finitely many minimal surfaces without boundary singularities, (4.11) implies the existence of finitely many compact H n -rectifiable sets where, for each i, N i \ Γ is connected, there exist a finite union Γ (i) = m∈I (i) Γ m of connected components of Γ with and a vector field ν co i : (4.14) Moreover, for each i, Reg • (N i ) has finitely many connected components {N i,ℓ } L(i) ℓ=1 such that, for each ℓ, cl (N i,ℓ ) \ Σ(N i ) is an orientable, smooth n-dimensional surface with boundary, whose boundary points are contained in Γ (i) . As noticed in Remark 3.7, (4.14) and Allard's regularity theorem imply H n (Σ(N i )) = 0 . (4.15) In particular, N i is H n -equivalent to Reg • (N i ), so that can rewrite (4.12) as with q i,ℓ = q i for every ℓ = 1, . . . , L(i).
Step two: We now take the limit of the M j 's in the sense of currents. Setting T j := M j , by (4.8), sup j H n (M j ) < ∞, and by the Federer-Fleming compactness theorem [FF60], see also [Sim83,Theorem 27.3]), we have that T j → T in the sense of currents, up to extracting subsequences, where T is an integral current. The C 1 -convergence of Γ j to Γ, T j → T , and ∂T j = Γ j , are easily seen to imply ∂T = Γ . Moreover, it is easily seen that, as Radon measures on R n+1 , since the mass of currents is lower semicontinuous, the weight of varifolds is continuous on sequences with bounded supports, and since T j = H n M j = V j . By (4.12), (4.18) Next we introduce the integral n-current T i,ℓ := T N i,ℓ . Notice that N i,ℓ is a smooth, connected n-dimensional surface, and that thanks to (4.13). By the constancy theorem for integral currents (cf. [Sim83,Theorem 26.27]), we find α i,ℓ ∈ Z and realizations N i,ℓ of N i,ℓ as multiplicity one integral currents such that Since H n (Σ(N i )) = 0, (4.18) implies that Applying the boundary operator in the sense of currents to (4.19), and recalling that ∂T = Γ , we find that (4.20) Recall that cl (N i,ℓ ) \ Σ(N i ) is a smooth surface with boundary, with boundary points contained in Γ (i) . If Γ m is one of the components of Γ (i) , then there is exactly one ℓ such that Γ m ∩ Reg b [cl (N i,ℓ ) \ Σ(N i )] = ∅, and, in correspondence to it, In particular, localizing (4.20) to Γ m , and setting Γ m = Γ Γ m , we have and since Γ m itself is connected, In particular, for suitable σ m i,ℓ ∈ {±1}, we deduce from (4.21) that i,ℓ : Γm⊂cl(N i,ℓ ) σ m i,ℓ α i,ℓ = 1 for every m = 1, . . . , M . (4.22) Step three: We now link T to V . Let V T denote the integral varifold associated with T , that is Noticing that and taking into account that Γ j is converging to Γ in C 1 , V j → V as varifolds, and T j → T as currents, we are allowed to apply White's theorem [Whi09, Theorem 1.2] to deduce the existence of an integral varifold W such that Therefore, it has to be and the integrality condition of W in turn yields that there exist β i,ℓ ∈ N such that Indeed, using a ≡ b mod(2) as a shorthand for saying that a and b have the same parity, (4.25) implies that At the same time so that, taking (4.22) into account, (4.26) holds.
This implies σ(x) ∈ (T x Γ) ⊥ for H n−1 -a.e. x ∈ Γ, as well that We are now ready to prove (4.28). Thanks to (4.26), for every m ∈ {1, . . . , M } we can find p ∈ N ∪ {0} such that i : m∈I (i) and we want to show that it must always be p = 0. Since Γ is accessible from infinity, we can select x 0 ∈ Γ m such that (4.34) holds at x = x 0 , and such that there exists a wedge W (strictly contained in a half-space) with vertex at x 0 and containing Γ co . Up to rigid motions, we assume that x 0 = 0 and that The n-plane π := e ⊥ 1 = {x 1 = 0} is then a supporting hyperplane to Γ co at x 0 = 0. Furthermore, since x 0 = 0 is a point on Γ m ⊂ Γ, the tangent space T 0 Γ is a linear subspace of π. We may assume that T 0 Γ = {x 1 = 0 = x n+1 }. Finally, by the classical convex hull property of minimal surfaces, we have N i ⊂ Γ co ⊂ W for every i. Now, for every i such that m ∈ I (i) , ν (i) := −ν co i (0) is a unit vector in the twodimensional plane (T 0 Γ) ⊥ = {x j = 0 for j = 2, . . . , n}. In the coordinates (x 1 , x n+1 ), thanks to N i ⊂ W , we find that ν (i) points inwards W , and thus that ν (i) = cos θ i , sin θ i for some |θ i | ≤ φ .
If {i 1 , . . . , i r(m) } ⊂ {1, . . . , k} is the set of indexes i such that m ∈ I (i) , we define the vectors v 1 , . . . , v 2p+1 by setting so that, by (4.34) applied at x = x 0 = 0, has length ≤ 1. We conclude the proof by showing that, if p ≥ 1, then A proof of (4.36) is in [DR16,Lemma 6.16]. For the reader's convenience and for the sake of clarity, we verbatim repeat the argument used in [DR16]. First, we order the vectors v h in such a way that θ 1 ≤ θ 2 ≤ · · · ≤ θ 2p+1 . For every j ≤ p, set w j := v j + v 2p+2−j . Using simple geometric considerations, one immediately sees that w j is a positive multiple of the vector Since θ j ≤ θ p+1 ≤ θ 2p+2−j , the angle between the vectors w j and v p+1 is so that w j · v p+1 > 0. Then, we can use the Cauchy-Schwarz inequality to estimate This proves (4.36).
Step five: We conclude the proof. By (4.28), for every m = 1, ..., M , adding up over those i such that m ∈ I (i) , we find q i = 1. By exploiting this fact, we find that: • q i ∈ {0, 1} for every i ∈ {1, . . . , k}; in other words, it cannot be q i ≥ 2; • if q i = 1, then q i ′ = 0 for any i ′ = i such that I (i) ∩ I (i ′ ) = ∅: hence, for every m = 1, . . . , M there is one and only one i = i m with m ∈ I (im) and q im = 1; • from (4.25): since q i ∈ {0, 1} for every i, β i,ℓ = 0 for every i ∈ {1, . . . , k} and ℓ ∈ {1, . . . , L(i)}. Thus, if q i = 1 then α i,ℓ = ±1 for every ℓ; if q i = 0 then α i,ℓ = 0 for every ℓ. We can thus argue as follows. We set m 1 := 1, and let i 1 be the only index in {1, . . . , k} such that 1 ∈ I (i 1 ) and q i 1 = 1. Next, let m 2 := min{m ∈ 1, . . . , M : m / ∈ I (i 1 ) }, and let i 2 be the corresponding index. Proceeding inductively, after a finite number h of steps the set {m : m / ∈ I (i 1 ) ∪ · · · ∪ I (i h ) } will be empty. We finally set and claim that N satisfies the conclusions of the theorem. In order to verify (4.6), we define ν : Γ → S n by and use (4.14). Noticing that q i = 0 if i = i r for every r = 1, ..., h, and q i = 1 otherwise, we see that so that var (M j ) → var (N ), which is the second conclusion in (4.7); and as for the first conclusion in (4.7), T ≤ V implies with α ir,ℓ = ±1. Taking into account that H n (Γ ∪ h r=1 Σ(N ir )) = 0, we can now define a Borel orientation ν N : N → S n by setting ν N | N ir ,ℓ := α ir,ℓ ν N ir ,ℓ , where ν N ir ,ℓ is the orientation defining the current N ir,ℓ . With this definition, equation (4.37) reads which implies that M j → N, ⋆ν N , 1 in the sense of currents. This completes the proof of (4.5), thus of the theorem.

Sharp decay estimates
In this last section we refine the conclusions of Theorem 4.1 with sharp quantitative estimates under the additional assumptions that: (i) the boundaries of the surfaces M j are fixed, i.e., we assume Γ j = Γ; (ii) for some fixed p > n, (5.1) and (iii) the limit minimal surface N is classical, that is, Σ(N ) = ∅. Under these assumptions, by combining Allard's regularity theorem [All72] and the implicit function theorem one can show the existence of smooth functions u j : N → R with u j = 0 on ∂N , and such that Assumption (i) is not really needed to parameterize M j over N . Indeed, one could obtain a global parametrization (possibly with non-trivial tangential components) as soon as Γ j converges to Γ in, say, C 1,α , see [CLM16,LM17]. Assumptions (ii) and (iii) are instead needed to have the quantitative regularity estimates of Allard, and the possibility to apply them to the M j 's for proving the graphicality property. We omit the details of the argument leading to the existence of the functions u j , since it has appeared many other times in the literature: for instance, see [FM11,CLM16,LM17,KM17,CM17].
We now collect some formulas concerning the geometry of almost-flat normal Lipschitz graphs over a smooth compact embedded orientable n-dimensional surface N ⊂ R n+1 , and prove a basic C 0 -estimate; see in particular (5.13) below, whose proof should be compared with the argument of [KM17, Section 4]. Let us consider a Lipschitz function u : N → R with u = 0 on ∂N and u C 0 (N ) + Lip(u) ≤ ε small enough depending on N . We set ψ u (p) := p + u(p)ν N (p) , p ∈ N , and let Ψ u := ψ u (N ). We also assume that Ψ u has a distributional mean curvature H Ψu ∈ L 1 (Ψ u ), so that where ν Ψu is the normal to Ψ u induced by ν N through ψ u . By the area formula, it holds for every bounded Borel measurable function g on Ψ u that For every ϕ ∈ C 1 c (N ) and t in a neighborhood of 0, we consider the variation where we denote by π N : B ε 0 (N ) → N the smooth nearest point projection of the ε 0 -neighborhood of N onto N , and where of course we are assuming ε < ε 0 . By the standard first variation formula for the area applied to Ψ u we find that Since π N restricted to Ψ u is the inverse of ψ u , we have We now want to compute H Ψu by using local coordinates. Let us cover N by open sets A ⊂ R n+1 such that at every p ∈ A ∩ N we can define an orthonormal frame {τ i (p)} n i=1 for T p N with ∇ τ i ν N = κ i τ i , where κ 1 ≤ κ 2 ≤ · · · ≤ κ n denote the principal curvatures of N . Setting ∂ i u = ∇ τ i u and Du = (∂ 1 u, ..., ∂ n u) ∈ R n , we find (5.6) Noticing that, on A ∩ N , we find that (ν Ψu • ψ u ) · ν N = 1 1 + |D * u| 2 on A ∩ N .
By (5.5) we also have, again on A ∩ N , Thus, if we test (5.4) with ϕ ∈ C 1 c (A ∩ N ), and then we integrate by parts, we obtain To understand the structure of (5.8), we compute for the ξ-gradient of G, (5.9) and for the z-derivative of G, (5.10) By exploiting u C 0 (N ) + Lip(u) < ε, we thus find that for measurable functions a i and b on N ∩ A with where c is a non-negative, bounded measurable function defined on A ∩ N . Overall (5.8) can be rewritten as where we have set for brevity d = J N ψ u / 1 + |D * u| 2 , so that d − 1 L ∞ (A∩N ) ≤ C(N ) ε. We finally formulate (5.11) as an elliptic PDE on a domain of R n . To this end, up to decrease the size of A, we can introduce coordinates on A ∩ N by means of an embedding F : U ⊂ R n → R n+1 of an open set U with smooth boundary in the unit ball of R n with A ∩ N = F (U ) and A ∩ bd (N ) = F (bd (U )). We set σ i = (∂F/∂x i ) • F −1 so that is also a frame of T p N for each p ∈ A ∩ N , and we have for the symmetric, bounded and uniformly elliptic tensor field Notice that the ellipticity of Λ relies on the facts that {v i (x)} n i=1 is a basis of T x R n , F is an embedding, and {τ i (p)} n i=1 is a basis of T p N . Thus we can understand (5.11) as wherec is non-negative and bounded, and provided q > n/2 and assuming that the right-hand side is finite. Changing variables one more time, and exploiting a covering argument, we thus find (5.13) We now assume the strict stability of N , and use the formulas above in order to obtain a sharp quantitative estimates for Lipschitz graphs which only involves a very weak notion of deficit.
Theorem 5.1 (Weak-deficit estimate on Lipschitz graphs). Let N be a smooth compact orientable n-dimensional surface in R n+1 with boundary, and let u : N → R be a Lipschitz function with u = 0 on ∂N . Consider the almost-mean curvature deficit If H N ≡ 0 and N is strictly stable, in the sense that, for some λ > 0, and if H Ψu ∈ L q (Ψ u ) for some q > n/2, then u C 0 (N ) ≤ C(N, q) H Ψu L 2 (Ψu) + H Ψu L q (Ψu) . (5.18) Proof of Theorem 5.1. We first notice that if H Ψu ∈ L 2 (Ψ u ), then Indeed, by (5.4), by Hölder inequality and by the Poincaré inequality on N , if ϕ ∈ C 1 c (N ) and π N is the normal projection over N , then so that (5.19) immediately follows. In particular, (5.17) is a consequence of (5.15), which we now prove. For the sake of clarity we shall first prove the theorem in the flat case when κ i ≡ 0 for all i = 1, . . . , n, and thus N is an open bounded set with smooth boundary in some n-plane of R n+1 . Clearly we have 0 ≤ a ≤ 1 2 δ(u) ∇u L 2 (N ) .
Having proved (5.24), we now complete the proof as follows. By (5.20),