Connected surfaces with boundary minimizing the Willmore energy

For a given family of smooth closed curves $\gamma^1,...,\gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary $\gamma=\gamma^1\cup...\cup\gamma^\alpha$. This is realized by minimizing the Willmore energy $\mathcal{W}$ on a suitable class of competitors. While the direct minimization of the Area functional may lead to limits that are disconnected, we prove that, if the infimum of the problem is $<4\pi$, there exists a connected compact minimizer of $\mathcal{W}$ in the class of integer rectifiable curvature varifolds with the assigned boundary conditions. This is done by proving that varifold convergence of bounded varifolds with boundary with uniformly bounded Willmore energy implies the convergence of their supports in Hausdorff distance. Hence, in the cases in which a small perturbation of the boundary conditions causes the non-existence of Area-minimizing connected surfaces, our minimization process models the existence of optimal elastic connected compact generalized surfaces with such boundary data. We also study the asymptotic regime in which the diameter of the optimal connected surfaces is arbitrarily large. Under suitable boundedness assumptions, we show that rescalings of such surfaces converge to round spheres. The study of both the perturbative and the asymptotic regime is motivated by the remarkable case of elastic surfaces connecting two parallel circles located at any possible distance one from the other. The main tool we use is the monotonicity formula for curvature varifolds, that we extend to varifolds with boundary, together with its consequences on the structure of varifolds with bounded Willmore energy.

Let ϕ : Σ → R 3 be an immersion of a 2-dimensional manifold Σ with boundary ∂Σ in the Euclidean space R 3 . We say that an immersion is smooth if it is of class C 2 . In such a case we define the second fundamental form of ϕ in local coordinates as II ij (p) = (∂ ij ϕ(p)) ⊥ , Date: November 15, 2019. 1 for any p ∈ Σ \ ∂Σ, where (·) ⊥ denotes the orthogonal projection onto (dϕ(T p Σ)) ⊥ . Denoting by g ij = ∂ i ϕ, ∂ j ϕ the induced metric tensor on Σ and by g ij the components of its inverse, we define the mean curvature vector by H(p) = 1 2 g ij (p)II ij (p), for any p ∈ Σ \ ∂Σ, where sum over repeated indices is understood. The normalization of H is such that the mean curvature vector of the unit sphere points inside the ball and it has norm equal to one. Denoting by µ ϕ the volume measure on Σ, we define the Willmore energy of ϕ by For an immersion ϕ : Σ → R 3 we will denote by co ϕ : ∂Σ → R 3 the conormal field, i.e. the unit vector field along ∂Σ belonging to dϕ(T Σ) ∩ (dϕ| ∂Σ (T ∂Σ)) ⊥ and pointing outside of ϕ(Σ).
The study of variational problems involving the Willmore energy has begun with the works of T. Willmore ([32], [33]), in which he proved that round spheres minimize W among every possible immersed compact surface without boundary. The Willmore energy of a sphere is 4π. In [32] the author proposed his celebrated conjecture, claiming that the infimum of W among immersed smooth tori was 2π 2 . Such conjecture (eventually proved in [18]) motivated the variational study of W in the setting of smooth surfaces without boundary. In such setting many fundamental results have been achieved, and some of them (in particular [31], [14], and [26]) developed a very useful variational approach, that today goes under the name of Simon's ambient approach. Such method relies on the measure theoretic notion of varifold as a generalization of the concept of immersed submanifold. We remark that, more recently, an alternative and very powerful variational method based on a weak notion of immersions has been developed in [23], [24], and [25]. Following Simon's approach, the concept of curvature varifold with boundary ( [17], [13]), considered as a good generalization of smooth immersed surfaces, will be fundamental in this work. Such notion is recalled in Appendix A. We will always consider integer rectifiable curvature varifolds with boundary, that we will usually call simply varifolds. Roughly speaking a rectifiable varifold is identified by a couple v(M, θ V ), where M ⊂ R 3 is 2-rectifiable and θ V : M → N ≥1 is locally H 2 -integrable on M , and we think at it as a 2-dimensional object in R 3 whose points p come with a weight θ V (p). We recall here that a 2-dimensional varifold V = v(M, θ V ) has weight measure µ V = θ V H 2 ¬ M , that is a Radon measure on R 3 ; moreover it has (generalized) mean curvature vector H ∈ L 1 loc (µ V ; R 3 ) and generalized boundary where σ V is a Radon R 3 -valued measure on R 3 of the form σ V = ν V σ, with |ν V | = 1 σ-ae and σ is singular with respect to µ V ; also div T M X(p) = tr(P ⊤ • ∇X(p)) where P ⊤ is the matrix corresponding to the projection onto T p M , that is defined H 2 -ae on M . By analogy with the case of sooth surfaces, we define the Willmore energy of a varifold V = v(M, θ V ) by setting if V has generalized mean curvature H, and W(V ) = +∞ otherwise. A rectifiable varifold V = v(M, θ V ) defines a Radon measure on G 2 (R 3 ) := R 3 × G 2,3 , where G 2,3 is the Grassmannian of 2-subspaces of R 3 , identified with the metric space of matrices corresponding to the orthogonal projection on such subspaces. More precisely for any f ∈ C 0 c (G 2 (R 3 )) we define In this way a good notion of convergence in the sense of varifolds is defined, i.e. we say that a sequence V n = v(M n , θ Vn ) converges to V = v(M, θ V ) as varifolds if for any f ∈ C 0 c (G 2 (R 3 )).
More recently, varifolds with boundary and Simon's method have been used also in the study of variational problems in the presence of boundary conditions. A seminal work is [26], in which the author constructs branched surfaces with boundary that are critical points of the Willmore energy with imposed clamped boundary conditions, i.e. with fixed boundary curve and conormal field. Another remarkable work is [10], in which an analogous result is achieved in the minimization of the Helfich energy. We also mention [22], in which the minimization problem of the Willmore energy of surfaces with boundary with fixed topology is considered, and the only constraint is the boundary curve, while the conormal is free, yielding the so-called natural Navier boundary condition.
1.2. Elastic surfaces with boundary. If γ = γ 1 ∪ ... ∪ γ α is a finite disjoint union of smooth closed compact embedded curves, a classical formulation of the Plateau's problem with datum γ may be to solve the minimization problem that is one wants to look for the surface of least area having the given boundary. From a physical point of view, solutions of the Plateau's problem are good models of soap elastic films having the given boundary ( [19]). Critical points of the Plateau's problem are called minimal surfaces and they are characterized by having zero mean curvature (this is true also in the non-smooth context of varifolds in the appropriate sense, see [30]). In particular, minimal surfaces or varifolds with vanishing mean curvature have zero Willmore energy. However, as we are going to discuss, the Plateau's problem, and more generally the minimization of the Area functional, may be incompatible with some constraints, such as a connectedness constraint.
In this paper we want to study the minimization of the Willmore energy of varifolds V with given boundary conditions, i.e. both conditions of clamped or natural type on the generalized boundary σ V , adding the constraint that the support of the varifold must connect the assigned curves γ 1 , ..., γ α . Hence the minimization problems we will study have the form for some assigned vector valued Radon measure σ 0 , or for some assigned positive Radon measure µ with suppµ = γ. Let us introduce a remarkable particular case that motivates our study. Let C = [0, 1] 2 / ∼ be a cylinder. Let R ≥ 1 and h > 0. We define that is a disjoint union of two parallel circles of possibly different radii. We consider the class of immersions . In particular for any h > h 0 there are no minimal surfaces (and thus no solutions of the Plateau's problem) connecting the two components of Γ R,h , even in a perturbative setting h ≃ h 0 + ε. This rigidity in the behavior of minimal surfaces suggests that in some cases an energy different from the Area functional may be a good model for connected soap films, like for describing the optimal elastic surface connecting Γ R,h in the perturbative case h ≃ h 0 + ε. Since surfaces with zero Willmore energy recover critical points of the Plateau's problem, we expect the minimization of W to be a good process for describing optimal elastic surfaces under constraints, like connectedness ones, that do not match with the Area functional. Also, from the modeling point of view, we remark the importance of Willmore-type energies, like the Helfrich energy, in the physical study of biological membranes ( [11], [29]), and in the theory of elasticity in engineering (see [12] and references therein).
We have to mention some remarkable results about critical points of the Willmore energy (called Willmore surfaces) with boundary. Apart from the above cited [26], Willmore surfaces with a boundary also of the form Γ R,h have been studied together with the rotational symmetry of the surface in [4], [6], [7], [8], and [9]; a new result about symmetry breaking is [16]. Also, interesting results about Willmore surfaces in a free boundary setting is contained in [1]. A relation between Willmore surfaces and minimal surfaces is investigated in [5].
1.3. Main results. Let us collect here the main results of the paper. If γ = γ 1 ∪ ... ∪ γ α is a disjoint union of smooth embedded compact 1-dimensional manifolds, we give a sufficient condition guaranteeing existence in minimization problems of the form (2) or (3). We obtain the following two Existence Theorems.
If inf P < 4π, then P has minimizers.
If inf P < 4π, then P has minimizers.
Both Existence Theorems are obtained by applying a direct method in the context of varifolds. In both cases the connectedness constraint passes to the limit by means of the following theorem, that relates varifolds convergence with convergence in Hausdorff distance of the supports of the varifolds. Suppose that suppσ Vn = γ 1 n ∪ ...∪ γ α n where the γ i n 's are disjoint compact embedded 1-dimensional manifolds, γ 1 , ...,γ β with β ≤ α are disjoint compact embedded 1-dimensional manifolds, and assume that γ i n →γ i in d H for i = 1, ..., β and that H 1 (γ i n ) → 0 for i = β + 1, ..., α.
The paper is organized as follows. In Section 2 we recall the monotonicity formula for curvature varifolds with boundary and its consequences on the structure of varifolds with bounded Willmore energy. Such properties are proved in Appendix B. In Section 3 we prove some properties of the Hausdorff distance and we prove Theorem 3.4. Section 4 is devoted to the proof of the Existence Theorems 4.1 and 4.2; we also describe remarkable cases in which such theorems apply, such as in the above discussed perturbative setting. Theorem 3.4 and the monotonicity formula give us results also about the asymptotic behavior of connected varifolds with suitable boundedness assumptions; more precisely we prove that rescalings of a sequence of varifolds V n with diam(suppV n ) → ∞ converge to a sphere both as varifolds and in Hausdorff distance (Corollary 5.2). Finally in Section 6 we apply all the previous results to the motivating case of varifolds with boundary conditions on curves of the type of Γ R,h . We prove that for any R and h the minimization problem of type Q has minimizers and their rescalings asymptotically approach a sphere (Corollary 6.2). Appendix A recalls the definitions about curvature varifolds with boundary and a useful compactness theorem.
1.4. Notation. We adopt the following notation.
• The symbol B r (p) denotes the open ball of radius r and center p in R 3 .
• The symbol ·, · denotes the Euclidean inner product.
• The symbol H k denotes the k-dimensional Hausdorff measure in R 3 .
• The symbol d H denotes the Hausdorff distance.
• If ϕ : Σ → R 3 is a smooth immersion of a 2-dimensional manifold with boundary, then in local coordinates we denote by II ij the second fundamental form, by H the mean curvature vector, by g ij the metric tensor, by g ij its inverse, by µ ϕ the volume measure on Σ induced by ϕ, and by co ϕ the conormal field.
hence v ⊥ is defined H 2 -ae on M and it implicitly depends on the point p ∈ M .
M is the weight measure. If they exist, the generalized mean curvature and boundary are usually denoted by H (or H V ) and σ V .
• The symbol C denotes a fixed cylinder, i.e. C = [0, 1] 2 / ∼ . • For given R ≥ 1 and h > 0, the symbol Γ R,h denotes an embedded 1-dimensional manifold of the form that is a disjoint union of two parallel circles of possibly different radii. Observe that the distance between the two circles is equal to 2h. • For a given boundary datum Γ R,h as above, we define the class F R,h := ϕ : C → R 3 | ϕ smooth immersion, ϕ| ∂C : ∂C → Γ R,h smooth embedding .

Monotonicity formula and its consequences
Here we recall the fundamental monotonicity formula for curvature varifolds with boundary, together with some immediate consequences on surfaces and on the structure of varifolds with finite Willmore energy. This classical formula is completely analogous to its version without boundary ( [31], [14]), hence the technicality behind the results we are going to state is developed in Appendix B.
Let 0 < σ < ρ and p 0 ∈ R 3 . If V is an integer rectifiable curvature varifold with boundary with bounded Willmore energy (here the support of V is not necessarily bounded), with µ V the induced measure in R 3 , and generalized boundary σ V , it holds that In particular the function ρ → A(ρ) is non-decreasing.
When more than a varifold is involved, we will usually denote by A V (·) the monotone quantity associated to V for chosen p 0 ∈ R 3 .
It is useful to remember that Let us list some immediate consequences on surfaces with boundary.
Lemma 2.1. Let Σ ⊂ R 3 be a compact connected immersed surface with boundary. Then In particular

Moreover calling d H the Hausdorff distance (see Section 3) and writing
Proof. It suffices to prove (10). Since Σ is smooth we have that Since Σ is smooth, by (6) we have that while by compactness it holds that and we get (10).
More importantly, the monotonicity formula implies fundamental structural properties on varifolds with bounded Willmore energy. First we remark such results in the case of varifolds without boundary, as proved in [14].
be an integer rectifiable varifold with σ V = 0 and finite Willmore energy. Then at any point p 0 ∈ R 3 there exists the limit and θ V is upper semicontinuous on R 3 (see (A.7) and (A.9) in [14]). In particular Recall that if suppV is also compact and non-empty, then W(V ) ≥ 4π ((A. 19) in [14]) and θ V is uniformly bounded on R 3 by a constant depending only on W(V ) ((A.16) in [14]).
In complete analogy with Remark 2.2 we prove in Appendix B (see Proposition B.1) that if V is a 2dimensional integer rectifiable curvature varifold with boundary, denoting by S a compact 1-dimensional embedded manifold containing the support suppσ V with |σ V |(S) < +∞ and assuming that Whenever a varifold v(M, θ V ) satisfies the above assumptions, we will always assume that

Convergence in the Hausdorff distance
The convergence of sets with respect to the Hausdorff distance will play an important role in our study. For every sets X, Y ⊂ R 3 we define the Hausdorff distance d H between X and Y by (14) d We say that a sequence of sets X n converges to a set X in d H if lim n d H (X n , X) = 0. Now we prove some useful properties of the Hausdorff distance.
ii) If X n is connected for any sufficiently large n and X is bounded, then X is connected as well.
Proof. i) Just note that if X ⊂ N ε 2 (X n ), then X ⊂ N ε (X n ). ii) By i) we can assume without loss of generality that X is closed, and thus compact. Suppose by contradiction that there exist two closed sets A, B ⊂ X such that A ∩ B = ∅, A = ∅, B = ∅, and A ∪ B = X. Since X is compact, A and B are compact as well, and thus d(A, B) := inf x∈A,y∈B |x − y| = ε > 0. By assumption, for any n ≥ n( ε 4 ) we have that X n ⊂ N ε The sets N ε 4 (A) ∩ X n and N ε 4 (B) ∩ X n are disjoint and definitively non-empty, and open in X n . This implies that X n is not connected for n large enough, that gives a contradiction.
Lemma 3.2. Suppose X n is a sequence of uniformly bounded closed sets in R 3 and let X ⊂ R 3 be closed. Then X n → X in d H if and only if the following two properties hold: a) for any subsequence of points y n k ∈ X n k such that y n k − → k y, we have that y ∈ X, b) for any x ∈ X there exists a sequence y n ∈ X n converging to x.
Proof. Suppose first that d H (X n , X) → 0. If there exists a converging subsequence y n k ∈ X n k with limit y / ∈ X, then d(y n k , X) ≥ ε 0 > 0, and thus X n k ⊂ N ε 0 2 (X) for k large, that is impossible; so we have proved a). Now let x ∈ X be fixed. Consider a strictly decreasing sequence ε m ց 0 . For any ε m > 0 let n εm be such that X ⊂ N εm (X n ) for any n ≥ n εm . This means that B εm (x) ∩ X n = ∅ for any n ≥ n εm and any m ∈ N. We can define the sequence The sequence ε mn converges to 0 as n → ∞, otherwise there exists η > 0 such that X n ∩ B η (x) = ∅ for any n large, but this contradicts the convergence in d H . Hence x n → x and we have proved b). Suppose now that a) and b) hold. If there is ε 0 > 0 such that X n ⊂ N ε 0 (X) for n large, then a subsequence x n k converges to a point y such that d(y, X) ≥ ε 0 > 0, that is impossible. If there is ε 0 > 0 such that X ⊂ N ε 0 (X n ) for n large, then there is a sequence z n ∈ X such that d(z n , X n ) ≥ ε 0 > 0. By b) we have that X is bounded, then a subsequence z n k converges to z ∈ X, and d(z, X n k ) ≥ ε 0 2 definitely in k. But then z is not the limit of any sequence x n k ∈ X n k . However z is the limit of a sequencex n ∈ X n by b), and thus it is the limit of the subsequencex n k , and this gives a contradiction.
Proof. Both X and Y are bounded. We can apply Lemma 3.2, that immediately implies that X ⊂ Y and Y ⊂ X using the characterization of convergence in d H given by points a) and b).
The above properties allow us to relate the convergence in the sense of varifolds to the convergence of their supports in Hausdorff distance.
Proof. Let us first observe that by the uniform boundedness of M n , we get that γ i n converges to some compact set X i in d H up to subsequence for any i = β + 1, ..., α. Each X i is connected by Lemma 3.1, then by Golab Theorem we know that H 1 (X i ) ≤ lim inf n H 1 (γ i n ) = 0, hence X i = {p i } for any i = β + 1, ..., α for some points p β+1 , ..., p α . Call X = {p β+1 , ..., p α }. By assumption we know that µ Vn ⋆ ⇀ µ V as measures on R 3 , also M n and M can be taken to be closed. Moreover suppσ V ⊂ X ∪γ 1 ∪ ... ∪γ β . In fact V n are definitely varifolds without generalized boundary on any open set of the form N ε (X ∪γ 1 ∪ ... ∪γ β ) and they converge as varifolds to V on such an open set with equibounded Willmore energy. We want to prove that the sets M n and M ∪ X ∪γ 1 ∪ ... ∪γ β satisfy points a) and b) of Lemma 3.2 and that X ⊂ M .
then by assumption and Lemma 3.2 there is a sequence of points in suppσ Vn converging to x. So let x ∈ M \ (γ 1 ∪ ... ∪γ β ∪ X). We know that there exists the limit lim ρց0 There exists a sequence ρ m ց 0 such that lim n µ Vn (B ρm (x)) = µ V (B ρm (x)) for any m. Hence M n ∩ B ρm (x) = ∅ for any m definitely in n. Arguing as in Lemma 3.2 we find a sequence x n ∈ M n converging to x, and thus the property b) of Lemma 3.2 is achieved.
we want to check property a) of Lemma 3.2 for such sets. Once this convergence is established, we get that M n → M ∪ X ∪γ 1 ∪ ... ∪γ β in d H and we can show that the whole thesis follows. In fact we have that for any ε > 0 for any η > 0 it holds that for any n ≥ n ε,η . In particular for any n ≥ n ε,η . Setting ε = η we see that for any η > 0 it holds that for any n ≥ n 2η,η . Hence M n → M ∪ X ∪γ 1 ∪ ... ∪γ β in d H . Therefore M ∪ X ∪γ 1 ∪ ... ∪γ β is closed and connected. Moreover we get that X ⊂ M , in fact for any p i ∈ X for any K ∈ N ≥1 by connectedness of M n we find some subsequence y n k ∈ M n ∩ ∂B 1 Since M is closed, passing to the limit K → ∞ we see that p i ∈ M . In particular M n → M ∪γ 1 ∪ ... ∪γ β in d H and the proof is completed.
So we are left to prove that M n \ A ε converges to M ∪ X ∪γ 1 ∪ ... ∪γ β \ A ε = M \ A ε in d H for any fixed ε > 0. Consider any converging sequence y n k ∈ M n k \ A ε . For simplicity, let us denote y n such sequence. Suppose by contradiction that y n → y but y ∈ M ∪ A ε . Since M is closed, there exist ζ > 0 such that B ζ (y) ∩ M = ∅ for n large. Since M n is connected and M = ∅ we can write that ∂B ζ (y) ∩ M n = ∅ for any σ ∈ ( ζ 4 , ζ 2 ) for n large enough. Since y n ∈ A ε , up to choosing a smaller ζ we can assume that B ζ (y) does not intersect suppσ Vn for n large. Fix N ∈ N with N ≥ 2 and consider points The open balls are pairwise disjoint. Passing to the limit σ ց 0, setting ρ = ζ 8N , and using Young's inequality in Equation (6) evaluated on the varifold V n at the point p 0 = z n,k we get that for any n large and any k = 1, ..., N − 1. Since (15) and passing to the limit n → ∞ we get that Since N can be chosen arbitrarily big from the beginning, we get a contradiction with the uniform bound on the Willmore energy of the V n 's. Hence we have proved that M n → M ∪γ 1 ∪ ... ∪γ β in d H . By Lemma 3.1 we get that M ∪γ 1 ∪ ... ∪γ β is connected. that M is closed. Suppose that suppσ Vn is as in Theorem 3.4. If a subsequence y n k ∈ M n k converges to y, then y ∈ M ∪γ 1 ∪ ... ∪γ β .
Observe that the supports M n , M are not necessarily bounded here.
Remark 3.6. The connectedness assumption in Theorem 3.4 is essential. Consider in fact the following example: let M n = ∂B 1 (0)∪∂B 1 n (0) and θ Vn (p) = 1 for any p ∈ M n . Hence the varifolds v(M n , θ Vn ) converge to v(∂B 1 (0), 1) as varifolds and they have uniformly bounded energy equal to 8π, but clearly M n does not converge to ∂B 1 (0) in d H .
If inf P < 4π, then P has minimizers.
Proof. Let V n = v(M n , θ Vn ) be a minimizing sequence for the problem P. Call I = inf P < 4π, and suppose without loss of generality that W(V n ) < 4π for any n. For any p 0 ∈ M n \ γ passing to the limits σ → 0 and ρ → ∞ in the monotonicity formula (6) we get Hence the sequence M n is uniformly bounded in R 3 . Integrating the tangential divergence of the field X(p) = χ(p) (p) where χ(p) = 1 for any p ∈ B R 0 (0) ⊃ M n for any n we get that for any n, and then µ Vn is uniformly bounded. By the classical compactness theorem for rectifiable varifolds ( [30]) we have that V n → V = v(M, θ V ) in the sense of varifolds (up to subsequence), and M is compact. By an argument analogous to the proof of Theorem 3.4 we can show that V = 0. Suppose in fact that V = 0. Since α ≥ 2 and the curves γ 1 , ..., γ α are disjoint and embedded, there exist a embedded torus φ : Since M n is connected and uniformly bounded, there is a sequence of points y n ∈ M n ∩ φ(S 1 × S 1 ) with a converging subsequence y n k → y. Observe that there is ∆ > 0 such that d(y n , γ) ≥ ∆. (z n,j ) ≤ µ V (B 3 4 ∆ (y)) = 0, summing over j = 1, ..., N and passing to the limit in n we get that gives a contradiction. Hence Theorem 3.4 implies that suppV ∪γ = M ∪γ is connected. Since W(V ) ≤ I by lower semicontinuity, we are left to show that σ V = σ 0 .
Since γ is smooth we can write that as p → q 0 with p ∈ γ for some constant C γ depending on the curvature of γ. Let 0 < σ < s with s = s(γ) such that (17) holds for p ∈ γ ∩ B s (q) for any q ∈ γ. For any q 0 ∈ γ the monotonicity formula (6) at q 0 on V n gives In particular (18) µ Vn (B σ (q)) ≤ C(I, γ, σ 0 )σ 2 for any q 0 ∈ γ, any σ ∈ (0, s), and any n. Consider now any X ∈ C 0 c (B r (q 0 )) for fixed q 0 ∈ γ and r ∈ (0, s). By varifold convergence we have that where we wrote σ V = ν V |σ V |. Now let m ∈ N be large and consider the cut off function Moreover, there exists a constant C(γ) such that B r (q 0 ) ∩ N 1 Hence setting X = Λ m Y in (19) and letting m → ∞ we obtain for any Y ∈ C 0 c (B r (q 0 )). Since q 0 ∈ γ is arbitrary we conclude that σ V = σ 0 , and thus V is a minimizer. Theorem 4.2. Let γ = γ 1 ∪...∪γ α be a disjoint union of smooth embedded compact 1-dimensional manifolds with α ∈ N ≥2 . Let m : γ → N ≥1 by H 1 -measurable with m ∈ L ∞ (H 1 ¬ γ). Let Q be the minimization problem If inf P < 4π, then P has minimizers.
Proof. We adopt the same notation used in the proof of Theorem 4.1. In this case the generalized boundaries of the minimizing sequence V n = v(M n , θ Vn ) are denoted by σ Vn = ν Vn |σ Vn |, and |σ Vn | ≤ mH 1 ¬ γ. The very same strategy used in Theorem 4.1 shows that V n converges up to subsequence in the sense of varifolds to a limit V = v(M, θ V ) = 0 with M ∪ γ compact and connected by Theorem 3.4 and Remark 3.7, and W(V ) ≤ inf Q. Hence, to see that V is a minimizer, we are left to show that |σ V | ≤ mH 1 ¬ γ. Calling µ := mH 1 ¬ γ, we find as in Theorem 4.1 that there exist constants C = C(inf Q, γ, µ) and s = s(γ) such that µ Vn (B σ (q)) ≤ Cσ 2 , for any q ∈ γ, any σ ∈ (0, s), and any n large. For any X ∈ C 0 c (B r (q 0 )) for fixed q 0 ∈ γ and r ∈ (0, s) the convergence of the first variation of varifolds reads (23) for Y ∈ C 0 c (B r (q 0 )) and Λ m as in (20). Estimating as in (21) and taking the limit m → ∞ we obtain Remark 4.3. Assuming in the above existence theorems that the connected components of the boundary datum are at least two (i.e. α ≥ 2) is technical, but it is also essential in order to obtain a non-trivial minimization problem, i.e. a problem that does not necessarily reduces to a Plateau's one. In fact if we consider a single closed embedded smooth oriented curve γ, Lemma 34.1 in [30] guarantees the existence of a minimizing integer rectifiable current T = τ (M, θ, ξ) with compact support and with boundary γ. Hence by Lemma 33.2 in [30] the integer rectifiable varifold V = v(M, θ) is stationary and suppσ V ⊂ γ. Then we can take M = suppT , that is compact. Since ∂T = γ and T is minimizing, the set M ∪ γ is connected and W(V ) is trivially zero.
The Existence Theorems 4.1 and 4.2 can be applied in different perturbative regimes, as discussed in the following corollaries and remarks.

Remark 4.6.
Many examples in which the Existence Theorems 4.1 and 4.2 and Corollary 4.4 apply are given by defining the following boundary data. We can consider any compact smooth surface S without boundary such that W(S) < 8π. Then the monotonicity formula (see also [14] and [15]) implies that S is embedded. We remark that there exist examples of such surfaces having any given genus ( [31] and [3]). Considering any suitable plane π that intersects S in finitely many disjoint compact embedded curves γ 1 , ..., γ α , we get that one halfspace determined by π contains a piece Σ of S with W(Σ) < 4π and ∂Σ = γ 1 ∪ ... ∪ γ α . Calling co Σ the conormal field of Σ we get that problems suppV ∪ ∂Σ compact, connected , and suitably small perturbations P ε , Q ε of them have minimizers.
Remark 4.7. Suppose that γ = γ 1 ∪ ... ∪ γ α is a disjoint union of compact smooth embedded 1-dimensional manifolds and that γ is contained in some sphere S 2 R (c). Up to translation let c = 0. If there is a point N ∈ S 2 R (0) such that for any i the image π N (γ i ) via the stereographic projection π N : S 2 has minimizers. In fact under such assumption there exists a connected submanifold Σ of S 2 R (0) with ∂Σ = γ, thus W(Σ) < 4π and Theorem 4.2 applies.
Suppose that h 0 > 0 is the critical value for which a connected minimal surface Σ with ∂Σ = Γ R,h exists if and only if h ≤ h 0 . Let Σ 0 be a minimal surface with ∂Σ 0 = Γ R,h 0 . Applying Corollary 4.5 we get that for ε > 0 sufficiently small the minimization problem has minimizers. Let us anticipate that in the case of boundary data of the form Γ R,h we will see in Corollary 6.2 that actually existence of minimizers for the problem Q ε is guaranteed for any ε > 0.

Asymptotic regime: limits of rescalings
As we recalled in Remark 2.2, it is proved in [14] that the infimum of the Willmore energy on closed surfaces coincide with the infimum taken over non-zero compact varifolds without boundary. First we prove that such infima are both achieved by spheres. This result is certainly expected by experts in the field, but up to the knowledge of the authors it has not been proved yet without appealing to highly non-trivial regularity theorems.
Passing to the limits σ → 0 and ρ → ∞ in the monotonicity formula for varifolds we get that for any p 0 ∈ R 3 . Hence θ V (p 0 ) = 1 for any p 0 ∈ M , and also for H 2 -ae p ∈ M and for every p 0 ∈ M . Fix δ > 0 small and two points p 1 , p 2 ∈ M with p 2 ∈ B 2δ (p 1 ). For H 2 -ae p ∈ M we can write Since M is bounded, we get that H ∈ L ∞ (µ V ). Therefore, since θ V = 1 on M , by the Allard Regularity Theorem ( [30]) we get that M is a closed surface of class C 1,α for any α ∈ (0, 1). Since M is closed, it is also compact, and thus it is connected, for otherwise W(V ) ≥ 8π. Let p ∈ M be any fixed point such that (28) holds, and call ν p the unit vector such that ν ⊥ p = T p M . Up to translation let p = 0. Consider the axis generated by ν 0 and any point p 0 ∈ M \ {0}. We can write p 0 = q + w with q = αν 0 and w, ν 0 = 0. Writing analogously (q + w ′ ) ∈ M \ {0} another point with the same component on the axis generated by ν 0 , (28) implies that Hence, whenever q = 0, we have that |w| = |w ′ |; that is points in M of the form αν 0 + w with α = 0 and w ∈ ν ⊥ 0 lie on a circle. It follows that M is invariant under rotations about the axis {tν 0 | t ∈ R}. This argument works at H 2 -almost any point of M . Therefore we have that for any p ∈ M , the set M is invariant under rotations about the axis p + {tν p | t ∈ R}. Still assuming 0 ∈ M , up to rotation suppose that ν 0 = (0, 0, 1). Let a ∈ M be such that ν a = (1, 0, 0). There exists a point b ∈ M such that b = tν 0 = (0, 0, t) for some t ∈ R \ {0}. We can write 0 = q + w and b = q + w ′ for the same q ∈ a + {tν a | t ∈ R} and some w, w ′ ∈ ν ⊥ a . Since |w| = |w ′ |, it follows that q = 0, otherwise b = 0. Since q = 0, the rotation of the origin about the axis a + {tν a | t ∈ R} implies that M contains a circle C of radius r > 0 passing through the origin, and the plane containing C is orthogonal to ν ⊥ 0 . Since M is of class C 1 , the circle C has to be tangent at 0 to the subspace ν ⊥ 0 . Thus by invariance with respect to the rotation about the axis {tν 0 | t ∈ R}, we have that M contains the sphere with positive radius given by the rotation of C about {tν 0 | t ∈ R}. Since the Willmore energy of a sphere is 4π, it follows that M coincide with such sphere. Now we can prove the above mentioned result on the asymptotic behavior of connected varifolds.
Corollary 5.2. Let V n = v(M n , θ Vn ) be a sequence of integer rectifiable curvature varifolds with boundary satisfying the hypotheses of Theorem A.2. Suppose that M n is compact and connected for any n.
and suppσ Vn is a disjoint union of uniformly finitely many compact embedded 1-dimensional manifolds, then the sequenceṼ ,θ n whereθ n (x) = θ Vn (diam(suppV n ) x), converges up to subsequence and translation to the varifold where S is a sphere of diameter 1, in the sense of varifolds and in Hausdorff distance.
Proof. Up to translation let us assume that 0 ∈ suppV n . Then suppṼ n is uniformly bounded with diam(suppṼ n ) = 1. We have that and thus Theorem A.2 implies thatṼ n converges to a limit varifold V (up to subsequence). Also σṼ diam(suppVn) = 0; hence V has compact support and no generalized boundary. Let us say that suppσṼ n is the disjoint union of the smooth closed curves γ 1 n , ..., γ α n . By the uniform boundedness of suppṼ n , we get that γ i n converges to some compact set X i in d H up to subsequence. Each X i is connected by Lemma 3.1, then by Golab Theorem we know that H 1 (X i ) ≤ lim inf n H 1 (γ i n ) = 0, hence X i = {p i } for any i for some points p 1 , ..., p α , and we can assume that p i = 0 for any i = 1, ..., α. Using ideas from the proof of Theorem 3.4, we can show that V = 0. In fact suppose by contradiction that V = 0. Fix N ∈ N with N ≥ 4. By connectedness of M n , since diam(suppṼ n ) → 1, and the boundary curves converge to a discrete sets, for j = 1, ..., N there are points z n,j ∈ ∂B j 2N (0) ∩ suppṼ n for n large. We can also choose N so that d(z n,j , suppσṼ for any n and j = 1, ..., N . Since V = 0 we have that lim sup n µṼ n (B 1 4N (z n,j )) ≤ lim sup n µṼ n (B 2 (0)) = 0.
Hence summing on j = 1, ..., N in (29) and passing to the limit n → ∞ we get that gives a contradiction. Therefore we can apply Theorem 3.4 to conclude that suppṼ n converges to M in d H . Finally, since V is a compact varifold without generalized boundary and 4π ≤ W(V ) ≤ lim inf n W(V n ) = 4π, by Proposition 5.1 we conclude that V is a round sphere of multiplicity 1. By Lemma 3.2 the diameter of M is the limit lim n diam(suppṼ n ) = 1.

The double circle boundary
In this section we want to discuss how the Existence Theorems 4.1 and 4.2 and the asymptotic behavior described in Corollary 5.2 relate with the remarkable case that motivates our study, namely the immersions in the class F R,h .
First, the monotonicity formula provides the following estimates on immersions ϕ ∈ F R,h .
Lemma 6.1. Fix R ≥ 1 and h > 0. It holds that: i) ii) Proof. i) We can consider as competitor in F R,h the truncated sphere is the point on the z-axis located at the same distance from the two connected components of Γ R,h . The surface Σ is contained in another truncated sphere Σ ′ having the same center and radius and symmetric with respect to the plane {z = 1−R 2 4h }. The boundary of Σ ′ is the disjoint union of two circles of radius 1. We have ii) Let ϕ ∈ F R,h and Σ = ϕ(C ). By connectedness there is a point p ∈ Σ \ ∂Σ lying in the plane z = 0. Hence d H (Σ, ∂Σ) ≥ h, and by (12) we have h and the thesis follows by using i) by letting h → ∞.
We already discussed in Remark 4.8 the existence of minimization problems arising by perturbations of minimal catenoids in some F R,h . By Lemma 6.1 we can complete the picture about existence of optimal connected elastic surfaces with boundary Γ R,h for any R ≥ 1 and h > 0, as well as the asymptotic behavior of almost optimal surfaces having such boundaries.
Corollary 6.2. Fix R ≥ 1 and h > 0. 1) Then the minimization problem Then (up to subsequence) S k converges in Hausdorff distance to a sphere S of diameter 1, and the varifolds corresponding to S k converge to V = v(S, 1) in the sense of varifolds.
Proof. 1) The result follows by point i) in Lemma 6.1 by applying Corollary 4.4. 2) Identifying S k with the varifold it defines, we estimate the total variation of the boundary measure by . Moreover, by the Gauss-Bonnet Theorem the L 2 -norm of the second fundamental form of S k is uniformly bounded. Hence Corollary 5.2 applies and the thesis follows.
Using the notation of point 2) in Corollary 6.2, we remark that even if we know that the rescalings S k converge to a sphere in d H and as varifolds, it remains open the question whether at a scale of order h the sequence Σ k approximate a big sphere. More precisely it seems a delicate issue to understand if diamΣ k ∼ 2h k as k → ∞. We conclude with the following partial result: the monotonicity formula gives us some evidence in the case we assume that diamΣ k h k → ∞.
then M is a plane containing the z-axis and θ Z ≡ 1.
Proof. We identify M k with the varifold it defines. First we can establish the convergence up to subsequence in the sense of varifolds by using Theorem A.2. In fact we have that H 1 (∂M k ) → 0,´M k |II M k | 2 is scaling invariant and thus finite. Moreover, since d(0, ∂M k ) ≥ 1, by monotonicity (6) we get that where A M k (·) is the monotone quantity centered at 0 evaluated on M k , and therefore µ M k (B σ (0)) ≤ C(σ) for any σ ≥ 1. Hence the hypotheses of Theorem A.2 are satisfied and we call Z = v(M, θ Z ) the limit varifold of M k . Observe that σ Z = 0 and W(Z) < +∞. From now on assume that diamΣ k /h k → ∞. Arguing as in the proof of Corollary 5.2 we can prove that Z = 0. In fact suppose by contradiction that Z = 0. Fix N ∈ N with N ≥ 4. By connectedness of M k , for j = 1, ..., N there are points z k,j ∈ ∂B j N (0, 0, 1) ∩ M k and z k,j ∈ ∂M k for k large. The open balls are pairwise disjoint. Hence the monotonicity formula (6) applied on M k at points z k,j with σ → 0 and ρ = 1 2N gives for any k and j = 1, ..., N . Since Z = 0 we have that lim sup (z k,j )) ≤ lim sup k µ M k (B 2 (0, 0, 1)) = 0.
Hence, summing on j = 1, ..., N in (32) and passing to the limit k → ∞ we get that gives a contradiction. Also the support of Z is unbounded. In fact suppose by contradiction that suppZ ⊂⊂ B R (0), and thus M is closed by Proposition B.1. Since M k is connected, there exists q ′ k ∈ M k ∩ ∂B 2R (0) definitely in k for R sufficiently big. Up to subsequence q ′ k → q ′ . By Remark 3.5 we get that q ′ ∈ suppZ, that contradicts the absurd hypothesis. Since M is unbounded, by Corollary B.2 (or equivalently (A.22) in [14]) we know that By construction lim kˆB σ (0)∩∂M k p |p| 2 , co M k dH 1 (p) = 0, hence passing to the limit k → ∞ in the monotonicity formula (6) evaluated on M k we get that for ae σ > 0. By monotonicity On the other hand, by (A.14) in [14] we can write that Hence Z is stationary, lim ρ→∞ µ Z (Bρ(q)) ρ 2 = π, and M is closed. If p 0 is any point in M , the monotonicity formula for Z centered at p 0 reads In particular θ Z (p 0 ) = 1, and thus we can apply Allard Regularity Theorem at p 0 . Thus we get that M is of class C ∞ around p 0 (and analogously everywhere), and thus there exists the limit lim σ→0ˆB ρ(p0)\Bσ (p 0 ) Passing to the limits ρ → ∞ and σ ց 0 in (33), we get that Therefore |(p − p 0 ) ⊥ | = 0 for any p ∈ M , where we recall that (·) ⊥ is the orthogonal projection on T p M ⊥ . Since this is true for any p 0 ∈ M , we derive that M is a plane. Finally Remark 3.5 implies that M contains the vertical axis {(0, 0, t) | t ∈ R}.
Appendix A. Curvature varifolds with boundary In this appendix we recall the definitions and the results about curvature varifolds with boundary that we need throughout the whole work. This section is based on [17] (see also [30], [13]).
Let Ω ⊂ R k be an open set, and let 1 < n ≤ k. We identify a n-dimensional vector subspace P of R k with the k × k-matrix {P ij } associated to the orthogonal projection over the subspace P . Hence the Grassmannian G n,k of n-spaces in R k is endowed with the Frobenius metric of the corresponding projection matrices. Moreover given a subset A ⊂ R k , we define G n (A) = A × G n,k , endowed with the product topology. A general n-varifold V in an open set Ω ⊂ R k is a non-negative Radon measure on G n (Ω). The varifold convergence is the weak* convergence of Radon measures on G n (Ω), defined by duality with C 0 c (G n (Ω)) functions.
We denote by π : G n (Ω) → Ω the natural projection, and by µ V = π ♯ (V ) the push forward of a varifold V onto Ω. The measure µ V is called induced (weight) measure in Ω. Given a couple (M, θ) where M ⊂ Ω is countably n-rectifiable and θ : M → N ≥1 is H n -measurable, the symbol v(M, θ) defines the (integer) rectifiable varifold given by where T x M is the generalized tangent space of M at x (which exists H n -ae since M is rectifiable). The function θ is called density or multiplicity of v(M, θ). Note that µ V = θH n ¬ M in such a case.
From now on we will always understand that a varifold V is an integer rectifiable one.
We say that a function H ∈ L 1 loc (µ V ; R k ) is the generalized mean curvature of V = v(M, θ) and σ V Radon R k -valued measure on Ω is its generalized boundary if for any X ∈ C 1 c (Ω; R k ), where div T M X(p) is the H n -ae defined tangential divergence of X on the tangent space of M . Recall that σ V has the form σ V = ν V σ, where |ν V | = 1 σ-ae and σ is singular with respect to µ V .
If V has generalized mean curvature H, the Willmore energy of V is defined to be The operator X → δV (X) :=´div T M X dµ V is called first variation of V . Observe that for any X ∈ C 1 c (Ω; R k ), the function ϕ(x, P ) := div P (X)(x) = tr(P ∇X(x)) is continuous on G n (Ω). Hence, if V n → V in the sense of varifolds, then δV n (X) → δV (X).
By analogy with integration formulas classically known in the context of submanifolds, we say that a varifold V = v(M, θ) is a curvature n-varifold with boundary in Ω if there exist functions A ijk ∈ L 1 loc (V ) and a Radon R k -valued measure ∂V on G n (Ω) such that Gn(Ω) P ij ∂ x j ϕ(x, P ) + A ijk (x, P )∂ P jk ϕ(x, P ) dV (x, P ) = = nˆG n(Ω) ϕ(x, P )A jij (x, P ) dV (x, P ) +ˆG n(Ω) ϕ(x, P ) d∂V i (x, P ), for any i = 1, ..., k for any ϕ ∈ C 1 c (G n (Ω)). The rough idea is that the term on the left is the integral of a tangential divergence, while on the right we have integration against a mean curvature plus a boundary term. The measure ∂V is called boundary measure of V .
Theorem A.1 ( [17]). Let V = v(M, θ) be a curvature varifold with boundary on Ω. Then the following hold true. i) A ijk = A ikj , A ijj = 0, and A ijk = P jr A irk + P rk A ijr = P jr A ikr + P kr A ijr . ii) P il ∂V l (x, P ) = ∂V i (x, P ) as measures on G n (Ω). iii) P il A ljk = A ijk . iv) H i (x, P ) := 1 n A jij (x, P ) satisfies that P il H l (x, P ) = 0 for V -ae (x, P ) ∈ G n (Ω). v) V has generalized mean curvature H with components H i (x, T x M ) and generalized boundary σ V = π ♯ (∂V ).
We call the functions II k ij (x) := P il A jkl components of the generalized second fundamental form of a curvature varifold V . Observe that II k jj = P jl A jlk = A jjk − P kl A jjl = A jkj − P kl A jlj = nH k − nP kl H l = nH k , and A ijk = II k ij + II j ki .
In conclusion we state the compactness theorem that we use in this work.
i.e. the multiplicity function θ V is upper semicontinuous on R 3 \ S. Since θ V is integer valued, the set {p ∈ R 3 \ S | θ v (p) ≥ 1 2 } is closed in R 3 \ S. Therefore we can take the closed set M = {p ∈ R 3 \ S | θ v (p) ≥ 1 2 } ∪ S as the support of V . A particular case of our analysis can be summarized in the following statement.
Proposition B.1. Let V be a 2-dimensional integer rectifiable curvature varifold with boundary. Denote by σ V the generalized boundary and by S a compact set containing the support suppσ V . Assume that and S is a compact 1-dimensional manifold with H 1 (S) < +∞. Then the limit lim ρց0 µ V (B ρ (p)) ρ 2 exists at any point p ∈ R 3 \ S, the multiplicity function θ V (p) = lim ρց0 µ V (Bρ(p)) ρ 2 is upper semicontinuous on R 3 \ S and bounded by a constant C(d(p, S), |σ V |(S), K, W(V )) depending only on the distance d(p, S), |σ V |(S), K and W(V ). Moreover V = v(M, θ V ) where M = {p ∈ R 3 \ S | θ v (p) ≥ 1 2 } ∪ S is closed. Also, we can derive the following consequence.
Corollary B.2. Let V = v(M, θ V ) be a 2-dimensional integer rectifiable curvature varifold with boundary with W(V ) < +∞. Denote by σ V the generalized boundary and by S a compact set containing the support suppσ V . Assume that S is a compact 1-dimensional manifold with H 1 (S) < +∞. Then where M essentially unbounded means that for every R > 0 there is B r (x) ⊂ R 3 \ B R (0) such that µ V (B r (x)) > 0. Moreover, in any of the above cases the limit lim ρ→∞ µ V (Bρ(0)) ρ 2 ≥ π exists.
Proof. Suppose that M is essentially unbounded. We can assume that lim sup ρ→∞ µ V (Bρ(0)) ρ 2 ≤ K < +∞. Then Hence, assuming without loss of generality that 0 ∈ S, the monotone quantity A(ρ) evaluated on V with base point 0 gives and thus ∃ lim ρ→∞ µ V (Bρ(0)) ρ 2 ≤ K < +∞. Also the assumptions of Proposition B.1 are satisfied and we can assume that M is closed.
We can prove that M has at least one unbounded connected component. In fact any compact connected component N of M defines a varifold v(N, θ V | N ) with generalized mean curvature; now if S ∩ N = ∅ then W(N ) ≥ 4π, and thus there are finitely many compact connected components without boundary, if instead S ∩ N = ∅, S ⊂ B R 0 (0) by compactness, and ∃ p 0 ∈ N \ B r (0) for r > R 0 but N is compact, then the monotonicity formula applied on v(N, θ V | N ) at point p 0 gives (52) π ≤ lim σ→0 for any Y ∈ C 0 c (R 3 ). Finally the monotonicity formula applied on W gives lim n µ V (R n (0)) R 2 n ≥ lim inf n µ Vn (B 1 (0)) ≥ µ W (B 1 (0)) ≥ lim σ→0 A W (σ) ≥ π.