Existence and Regularity of Spheres Minimising the Canham–Helfrich Energy

We prove the existence and regularity of minimisers for the Canham–Helfrich energy in the class of weak (possibly branched and bubbled) immersions of the 2-sphere. This solves (the spherical case) of the minimisation problem proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to proving the main results we establish the lower semicontinuity of the Canham–Helfrich energy under weak convergence of (possibly branched and bubbled) weak immersions.


Introduction
The basic structural and functional unit of all known living organisms is the cell. The interior material of a cell, the cytoplasm, is enclosed by biological membranes. Most of the cell membranes of living organisms are made of a lipid bilayer, which is a thin polar membrane consisting of two opposite oriented layers of lipid molecules.
In 1970, in order to explain the biconcave shape of red blood cells, Canham [7] proposed a bending energy density dependent on the squared mean curvature.
Three years later Helfrich proposed the following curvature elastic energy per unit area of a closed lipid bilayer [18,Equation (12)]: where H is the mean curvature, K is the Gauss curvature, c 0 is the so-called spontaneous curvature, and k c ,k c are the curvature elastic moduli. The values of the parameters can be measured experimentally (for example see [11,13] for c 0 , and [30] for k c ). The constantk c is not important for the purpose of this paper as by the Gauss-Bonnet Theorem, the integrated Gauss curvature is a topological constant. Lipid bilayers are very thin compared to their lateral dimensions, thus are usually modelled as surfaces. Suppose the surface and hence the membrane is represented by a smooth isometric embedding : S 2 → R 3 of the 2-sphere S 2 . We will be concerned with the following integrated version of (1.1): (1.2) where again, H is the mean curvature, c 0 is a constant, and μ is the Radon measure corresponding to the pull back of the Euclidean metric along . The integral in (1.2) is known as Canham-Helfrich energy. It is also referred to as Canham-Evans-Helfrich or just Helfrich energy. Its most important reduction is the Willmore energy, where c 0 = 0. Due to its simplicity and fundamental nature, the Willmore energy appears in many areas of science and technology, and has been studied a lot in the past. Its first appearances were found in the works of Poisson [31] in 1814 and Germain [14] in 1821. It was finally brought onto physical grounds by Kirchhoff [22] in 1850 as the free energy of an elastic membrane. In the early 20th century, Blaschke considered the Willmore energy in the context of differential geometry and proved its conformal invariance, see for instance [6]. The difference between Willmore energy and Canham-Helfrich energy comes from the constant c 0 , known as spontaneous curvature. According to Seifert [36], it is mainly caused by asymmetry between the two layers of the membrane. Geometrically, the asymmetric area difference between the two layers is given by the total mean curvature, that is the integrated mean curvature. This is due to the fact that the infinitesimal variation of the area, that is the area difference between two nearby surfaces, is the total mean curvature. Döbereiner et al. [12] observed that spontaneous curvature may also arise from differences in the chemical properties of the aqueous solution on the two sides of the lipid bilayer. Many approaches about how to derive the Canham-Helfrich energy density as the energy density of a lipid bilayer have appeared in the literature. We refer to Seifert [36] for more details.
Our goal is to minimise the Canham-Helfrich energy as well as to study the regularity of minimisers (and more generally of critical points). In the language of the calculus of variations we are concerned with Problem 1.2, as stated in Bernard et al. [5,Introduction,Problem (P1)]. Given a smooth embedding : S 2 → R 3 denote by Area = S 2 dμ the area of the surface (S 2 ) and by Vol the enclosed volume.
in order to solve Problem 1.2 by the so-called direct method of calculus of variations, lower semi continuity is required. According to Röger [27], it was an open question under which conditions/in which natural weak topology on the space of immersions one obtains lower semi continuity of the Canham-Helfrich energy. This is presumably the reason why Problem 1.2 was only partially solved for non-zero spontaneous curvature c 0 . The axisymmetric case was solved in 2013 by Choksi and Veneroni [9], who proved the existence of a minimiser of the Canham-Helfrich energy among a suitable class of axisymmetric (possibly singular) surfaces under fixed surface area and enclosed volume constraints. Five years later, Dalphin [10] showed existence of minimisers in a class of C 1,1 surfaces whose principal curvatures are bounded by a given constant 1/ε. Though, in his setting, it is still unclear how to get compactness and lower semi-continuity as ε tends to zero.
As already alluded to, we tackle Problem 1.2 by the direct method of calculus of variations. The issue is of course to find a suitable class F A 0 ,V 0 of admissible maps (endowed with a suitable topology) having area A 0 and enclosed volume V 0 such that the Canham-Helfrich energy is lower semi-continuous and has (pre-) compact sub-levels. A natural choice is the class of weak (Sobolev) immersions in W 2,2 (S 2 , R 3 ), already employed in the context of the Willmore energy for instance by Rivière [33] or Kuwert and Li [23]. We will use the space of bubble trees of weak possibly branched immersions. It will shortly become clear why we have to allow branched points and multiple bubbles.
In what follows, we denote by · (resp. ×) the Euclidean scalar (resp. vector) product on R 3 . Definition 1.3. A map : S 2 → R 3 is called weak (possibly branched) immersion with finite total curvature if ∈ W 1,∞ (S 2 , R 3 ) and the following holds: 1. There exists C > 1 such that, for almost every p ∈ S 2 , (1.4) where the norms are taken with respect to the standard metric on S 2 and with respect to the Euclidean metric of R 3 , and where d × d is the tensor given in local coordinates on S 2 by 2. There exist a positive integer N and finitely many points b 1 , . . . , b N ∈ S 2 such that log |d | ∈ L ∞ loc (S 2 \{b 1 , . . . , b N }); 3. The Gauss map n, defined by in any local chart x of S 2 , satisfies The space of weak (possibly branched) immersions with finite total curvature is denoted by F.
Since by assumption is a Lipschitz map, it induces an L ∞ -metric g given by for elements X, Y of the tangent bundle T S 2 . In the usual way (see for instance [16, 1.2]), the L ∞ -metric g induces a Radon measure μ on S 2 which is mutually absolutely continuous to the 2-dimensional Hausdorff measure on S 2 . Using Müller-Svěrák theory of weak isothermic charts [29] and Hélein's moving frame technique [17] one can prove the following proposition (see for instance [34] where x is a local arbitrary conformal chart on S 2 for the standard metric. Moreover Remark 1.5. In view of Proposition 1.4, a careful reader could wonder why we do not work with conformal W 2,2 weak, possibly branched, immersions only and why we do not impose for the membership in F, to be conformal from the beginning. The reason why it is technically convenient not to impose conformality from the beginning is to allow for general perturbations in the variational problem, which do not have to respect infinitesimally the conformal condition.
The reason that we chose the class F as above is the following theorem of the first author and Rivière [28, Theorem 1.5] (see also [8] The theorem already gives (pre-)compactness, a notion of convergence, and lower semi-continuity (actually, continuity) of the third summand in (1.2) of the Canham-Helfrich energy, that is of the area functional. Indeed, T := ( f ∞ , ξ 1 ∞ , . . . , ξ N ∞ ) forms a bubble tree, see Definition 3.2. In particular, the limit T is not in the class F anymore. At a rough informal level, a non-expert reader can think of a bubble tree T := ( f , ξ 1 , . . . , ξ N ) as a "pearl necklace" where each "pearl" corresponds to the image of a possibly branched weak immersion ξ i (S 2 ) and f is a Lipschitz map from S 2 to R 3 "parametrising" the whole pearl necklace, in To get a better understanding of why we obtain a bubble tree in the limit, we will look at an example of Problem 1.2. Let Then, the infimum in Problem 1.2 is achieved by the bubble tree T = ( f , Id S 2 , Id S 2 ) of twice the unit sphere. Indeed, H c 0 ( T ) = 0 and H c 0 ( ) 0 for any other smooth immersion of S 2 into R 3 , so T achieves the infimum. A minimising sequence k (S 2 ) of smoothly embedded spheres converging to such a bubble tree can be achieved by gluing (1 + 1/k)S 2 to (1 − 1/k)S 2 via a small catenoidal neck of size 2/k. Notice also that if satisfies H c 0 ( ) = 0, then the image [S 2 ] is the unit sphere by a classical theorem of Hopf [19].
Getting a bubble tree in the limit is in accordance with the earlier result on existence of minimisers by Choksi and Veneroni [9] in the axisymmetric case: indeed the minimiser in [9, Theorem 1] is made by a finite union of axisymmetric surfaces.
Moreover, the bubbling phenomenon is also known as budding transition in biology and has been recorded with video microscopy, see Seifert [36] or Seifert, Berndl, and Lipowsky [37].
In Chapter 3 we sharpen Theorem 1.6 in a way that we get lower semi-continuity for the Canham-Helfrich functional. This can be seen as a possible answer to the aforementioned open question raised by Röger [27]. In Chapter 4 we compute the Euler-Lagrange equation for the Canham-Helfrich energy in divergence form. Moreover, we prove that all the weak branched conformal immersions of a minimising bubble tree (actually more generally for a critical bubble tree) are smooth away from their branch points. Our proof is based on the regularity theory for Willmore surfaces developed by Rivière [32]. This relies on conservation laws discovered by Rivière [32] in the context of the Willmore energy and adjusted by Bernard [4] for the Canham-Helfrich energy. We get the following final result.
Moreover, for each i ∈ {1, . . . , N } there exist a non-negative integer N i and finitely many points Proof. Let 1 , 2 , . . . be a minimising sequence of (1.6). There holds where H k is the mean curvature corresponding to k , see (2.3). By the Gauss-Bonnet theorem (see (2.6) for the precise statement in case of weak branched immersions and (2.8) for the estimate below), which means the first inequality of (3.17) is satisfied. Moreover, (2.10) implies the second inequality of (3.17). Hence, we can apply Theorem 3. A more general form of the Canham-Helfrich energy is given by for ∈ F, where the parameter α 0 is referred to as tensile stress, and ρ 0 as osmotic pressure. We get the following solution of Problem (P2) from the introduction in [5]: Moreover, if the inequality is strict, then there exist 0 ∈ F, a positive integer N 0 , which proves the first statement. Now assume 1 , 2 , . . . is a sequence in F such that As α > 0, we have sup k Area k < ∞ and thus, using (1.7), also sup k A simple contradiction argument using (2.10) now leads to Therefore, analogously to the proof of Theorem 1.7, we can apply Theorem 3.3 to obtain an integer N and 1 , . Obviously, and since there are no constraints, we simply get N = 1. Letting 0 := 1 , we infer from (1.7) that in the case that |c 0 | √ α, The conclusion follows from Theorem 4.3, and (2.12).

Notation
We adopt the conventions of [34]. To avoid indices and to get clearly arranged equations, we will employ the following suggestive notation. For R 3 valued maps e and f defined on the unit disk D 2 , we write where · denotes the Euclidean inner product and × denotes the usual vector product on R 3 . Similarly, for λ : Moreover, for a vector field The m-dimensional Lebesgue measure is denoted by L m .

Weak (Possibly Branched) Conformal Immersions
We adapt the notion of weak immersions which was independently formalized by Rivière [33] and Kuwert and Li [23].
Let ( , c 0 ) be a smooth closed Riemann surface (in the rest of the paper we will take ( , c 0 ) to be the 2-sphere endowed with the standard round metric). Without loss of generality we can assume that ( , c 0 ) is endowed with a metric g c 0 of constant curvature and area 4π (see for instance [20]). For the definition of the Sobolev spaces W k, p ( , R 3 ) on see for instance Hebey [16]. A map : → R 3 is called a weak branched conformal immersion with finite total curvature if and only if there exists a positive integer N , finitely many points almost everywhere for any conformal chart x of , , and its Gauss map n defined by in any local positive chart x of satisfies The space of weak branched conformal immersions with finite total curvature is denoted by F or just F in case = S 2 . We define the L ∞ -metric g pointwise for almost every p ∈ by for elements X, Y of the tangent space T p . In the usual way, the L ∞ -metric g induces a Radon measure μ g on . The conformality condition (2.1) implies that g = e 2λ g c 0 for some λ ∈ L ∞ loc ( \{b 1 , . . . , b N }) called conformal factor. Moreover, we define the second fundamental form I pointwise for almost every p ∈ by Then ∈ W 2,2 ( ) and the conformal factor λ is an element of L 1 ( ). Moreover, for each singular point b j , j = 1, . . . , N , there exists a strictly positive integer n j ∈ N such that the following holds: such that is a conformal immersion of a neighbourhood of b j . • The conformal factor λ satisfies the following singular Liouville equation in distributional sense: where δ b j is the Dirac delta centred at b j , K is the Gaussian curvature of , and K 0 ∈ R is the (constant) curvature of ( , g c 0 ).
By integrating the singular Liouville equation (2.5), we obtain the Gauss-Bonnet Theorem for weak branched immersions: where χ( ) is the Euler Characteristic of . Note in particular that, once the topology of is fixed, the number of branch points counted with multiplicity is bounded by the Willmore energy: Moreover, the Willmore energy controls the L 2 norm squared of the second fundamental form: (2.8)

Simon's Monotonicity Formula and Li-Yau Inequality for Weak Branched
Immersions Let ∈ F be any weak branched conformal immersion with finite total curvature and branch points {b 1 , . . . , b N }. In the usual way (by splitting the vector field in its tangential and normal parts and using integration by parts), one shows that A simple cut-off argument together with (2.4) shows that the first variation formula (2.9) is true for all X ∈ W 1,2 ( , R 3 ). In what follows we will gather a couple of facts that are well known for weak unbranched immersions and, due to (2.9), are also valid for weak branched conformal immersions with finite total curvature. Firstly, letting X ( p) := ( p) − (a 0 ) for p ∈ and some fixed a 0 ∈ , one has div X = 2 and hence, see where the weak mean curvature is almost everywhere given by Moreover, if ∈ F with H 2 dμ < 8π , then is an embedding (compare also with Proposition 2.1).

Canham-Helfrich Energy
Given real numbers c 0 ∈ R and α, ρ 0 as well as a weak branched conformal immersion with finite total curvature : → R 3 , we define the Canham-Helfrich energy H c 0 α,ρ ( ) in its most general form by Note that, in case : → R 3 is a smooth (actually Lipschitz is enough) embedding, by the Divergence Theorem the last integral equals the volume enclosed by ( ). The parameter α is referred to as tensile stress, ρ as osmotic pressure. Compare this definition for instance with [4, Equation (3.6)] or [5].

Existence of Minimisers
In this chapter we will prove the compactness of sequences with uniformly bounded Willmore energy and area as well as lower semi-continuity of the Canham-Helfrich energy under this convergence, see Theorem 3.3. The proof of Theorem 3.3 will build on top of [28] and the next Lemma 3.1 which establishes the convergence of the constraints and the lower semi-continuity of the Willmore energy away from the branch points (Lemma 3.1 should be compared with [34, Lemma 5.2]). Then, there exists a sequence of positive numbers s 1 , s 2 , . . . converging to zero such that Proof. Suppose U is an open subset of S 2 \{b 1 , . . . , b N }, K is a compact subset of U , and x : U → R 2 is a conformal chart for S 2 . Denote by the conformal factors. Notice that the volume element corresponding to ξ k is given by e 2λ k . In a first step we will show that for any 1 p < ∞, as well as A simple argument by contradiction shows that it is enough to prove the statement after passing to a subsequence of k. Since the ξ k 's and ξ ∞ are conformal and x is a conformal chart, we can write the mean curvature vector as where is the flat Laplacian with respect to x. By Hypothesis (3.3), we have that , which implies (3.10). By the Rellich-Kondrachov Compactness Theorem, after passing to a subsequence, there holds for any 1 p < ∞. Therefore, using Hypothesis (3.2) and passing to a further subsequence, it follows that

It follows that
, which implies (3.11) by lower semi-continuity of the L 2 -norm under weak convergence. Similarly, from Hypothesis (3.2) and the strong convergence (3.12), we infer (3.8).
Again by the strong convergence (3.12) and Hypothesis (3.2), we can extract a subsequence such that by dominated convergence, Using this and the fact that by the Rellich-Kondrachov Compactness Theorem for any 1 p < ∞, one verifies (3.9). Next, let r k be any sequence of positive numbers converging to zero and abbreviate First, notice that for any Borel function f on which is a consequence of the dominated convergence theorem and the fact that finite sets have μ ∞ measure zero. Let n 0 = 1. For each positive integer j, we use (3.8) to inductively choose n j > n j−1 such that Moreover, define l k = j for all integers k with n j−1 < k n j and define s k = r l k . Then, we have that s k → 0 as k → ∞ as well as (3.14) which in particular remains valid for s k replaced by any t k s k . Hence, by (3.13) we can deduce (3.4). Using the convergence on compact sets (3.9)-(3.11), Equations (3.5)-(3.7) follow similarly. It only remains to show that Equation (3.5) is still valid after replacing s k by any sequence t k s k converging to zero. Hence, we only have to show that This follows because, by Hölder's inequality, The first factor on the right hand side is bounded by (3.1). To see that the second factor goes to zero as k tends to infinity, we apply (3.14) and the fact that Next we will define the notion of a bubble tree. The idea is that the different bubbles can be parametrised by decomposing a single 2-sphere. The bubbles can then be attached to each other by a Lipschitz map, see (3.15) and (3.16). which extends to a Lipschitz map Moreover, for all i ∈ {1, . . . , N }, 15) and for all j ∈ {1, . . . , N i } there exists p i, j ∈ R 3 such that The next theorem establishes the weak closure of bubble trees, as well as the convergence of the constraints in the Helfrich problem and the lower semi-continuity of the Willmore energy. The proof builds on [28].

sequence of bubble trees of weak immersions and
Then, there exists a subsequence of T k which we again denote by T k such that N k = N for some positive integer N and there exists a sequence of diffeomorphisms k of S 2 such that is a bubble tree of weak immersions and Proof. We first consider the special case where N k = 1 for all positive integers k.
and J i, j := {i ∈ J i, j : ∀k ∈ N : i :

and the necks
Finally, for any μ k integrable Borel function ϕ on S 2 , we get (3.20) We notice that by the strong convergence (1.5), for some finite number C > 0. Hence, by Hölder's inequality, By (3.17) and (3.19), the right hand side of each line goes to zero as k tends to infinity. That means the last term of Equation (3.20) goes to zero as k tends to infinity when ϕ is replaced by H k as well as when ϕ is replaced by n k · k . Therefore, using (3.5) and (3.6), we can conclude the convergence of the integrated mean curvature and the convergence of the volume from (3.20). Similarly, we can conclude the lower semi-continuity of the Willmore energy W from (3.20) by replacing ϕ with H 2 k , using super linearity of the limit inferior and by ignoring the non-negative second term in (3.20). Now, the general case follows analogously to the proof of [28, Theorem 7.2].

Regularity of Minimisers
Throughout this section, denotes a smooth, oriented, and closed 2-dimensional manifold. Moreover, c 0 , α, and ρ are the parameters of the Canham-Helfrich energy, that is c 0 ∈ R and α, ρ 0, see (2.14). A (possibly branched) weak immersion In what follows, we will first compute the Canham-Helfrich equation in divergence form, see Lemma 4.1. Then, we will prove that a weak immersion satisfying the Canham-Helfrich equation is smooth away from its branch points, see Theorem 4.3. The proof is based on the regularity theory for weak Willmore immersions developed by Rivière [32,34]. An important step in Riviere's regularity theory is the discovery of hidden conservation laws for weak Willmore immersions. In the framework of Canham-Helfich immersions, the corresponding hidden conservation laws were discovered by Bernard [4].
corresponds to the first variation of the Willmore energy.
Proof. After composing with a conformal chart away from the branch points, we may assume that is a map D 2 → R 3 . Let ω ∈ C ∞ c (D, R 3 ) and define t := + t ω for t ∈ R. The conformal factor λ is given by 2e 2λ = |∇ | 2 and the metric coefficients (g t ) i j by (g t ) i j = ∂ i t · ∂ j t . Standard computations (see for instance [34, (7.8)-(7.10)]) give Therefore, using Using that ω has compact support in D 2 , and using the symmetry of the second fundamental form, that is ∂ i n · ∂ j = ∂ j n · ∂ i , we compute further that From [34,Corollary 7.3] we know that (4.7) see also [32]. Moreover (see for instance [4,Chapter 3.3]), Putting (4.6) -(4.9) into (4.1) yields (4.2).

Remark 4.2.
Usually in the literature (see for instance [4,Chapter 3.3]) one finds the expression of the first variation for H dμ g written as It is not hard to check the equivalence of (4.10) with (4.6) proved above. The advantage of the expression (4.6) is twofold: first it invokes less regularity of the immersion map , second it is already in divergence form. Both advantages will be useful in establishing the regularity of weak Canham-Helfrich immersions: indeed, (4.10) would correspond to an L 1 term in the Euler-Lagrange equation (which is typically a problematic right hand side for elliptic regularity theory) while (4.6) corresponds to the divergence of an L 2 term (which is a much better right hand side in elliptic regularity).

Theorem 4.3. (Smoothness of weak Canham-Helfrich immersions) Suppose ∈ F is a weak Canham-Helfrich immersion. Then is a C ∞ immersion away from the branch points.
Proof. After composing with a conformal chart of away from the branch points onto the unit disk D 2 , we may assume that is a map D 2 → R 3 without branch points and satisfies the Canham-Helfrich equation (4.2). It is enough to show that ∈ C ∞ (B 1/2 (0)). The proof splits into three parts.
Therefore, from the system of conservation laws (4.12)-(4.15) we get step by step for all 1 q < ∞ ∈ W Define  In particular, we deduce that inf Let c 0 ∈ (−ε 4.4 (A 0 , V 0 ), ε 4.4 (A 0 , V 0 )) and let k be a minimizing sequence of inf ∈F A 0 ,V 0 S 2 (H − c 0 ) 2 dμ . For k large enough, it holds that Combining (4.25)-(4.27), we get where in the last identity we plugged in the definition of ε 4.4 (A 0 , V 0 ) as in (4.24).