Characterization of optimal carbon nanotubes under stretching and validation of the Cauchy-Born rule

Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two-and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy-Born rule in this setting.


Introduction
Nanostructured carbon has emerged over the last two decades as one of the most promising materials available to mankind. The discovery of fullerenes [48,49], followed by that of carbon nanotubes [41] and graphene [37,62], sparked an interest in low-dimensional materials. The fascinating electronic and mechanical properties of single-atom-thick surfaces and structures are believed to offer unprecedented opportunities for innovative applications, ranging from nextgeneration electronics to pharmacology, to batteries and solar cells [39,58,59]. New findings are emerging at an always increasing pace, cutting across materials science, physics, and chemistry, and extending from fundamental science to novel applications [23,61].
Carbon nanotubes are long, hollow structures exhibiting cylindrical symmetry [18]. Their walls consist of a single (or multiple) one-atom-thick layer of carbon atoms forming sp 2 covalent bonds [12] arranged in a hexagonal pattern. This molecular structure is responsible for amazing mechanical properties: Carbon nanotubes are presently among the strongest and stiffest known materials with a nominal Young's modulus [47,71] of 1 TPa and ideal strength greater than 100 MPa [3]. In addition, they are electrically and thermally conductive, chemically sensitive, transparent, and light weight [72]. Nanotubes can be visualized as the result of rolling up a patch of a regular hexagonal lattice. Depending on the different possible realizations of this rollingup, different topologies may arise, giving rise to zigzag, armchair, and chiral nanotubes. These topologies are believed to have a specific impact on the mechanical and electronic properties of the nanotube, which can range from highly conducting to semiconducting [9,10].
In contrast to the ever-growing material knowledge, the rigorous mathematical description of two-dimensional carbon systems is considerably less developed. Ab initio atomistic models are believed to accurately describe some features of the carbon nanotube geometry and mechanics [54,65,76]. These methods are nevertheless computational in nature and cannot handle a very large number of atoms due to the rapid increase in computational complexity. On the other hand, a number of continuum mechanics approaches have been proposed where carbon nanotubes are modeled as rods [63], shells [3,4,28,66], or solids [73]. These bring the advantage of possibly dealing with long structures, at the price however of a less accurate description of the detailed microscopic behavior.
The unique mechanical behavior of nanotubes under stretching is a crucial feature of these structures. As such, it has attracted attention from the theoretical [4,29,66,79], the computational [1,9,40,44], and the experimental side [17,47,74,77]. Still, a reliable description of nanotubes under stretching requires one to correctly resolve the atomic scale and, simultaneously, to rigorously deal with the whole structure. We hence resort to the classical frame of molecular mechanics [2,53,64] which identifies carbon nanotubes with point configurations {x 1 , . . . , x n } ∈ R 3n corresponding to their atomic positions. The atoms are interacting via a configurational energy E = E(x 1 , . . . , x n ) given in terms of classical potentials and taking into account both attractive-repulsive two-body interactions, minimized at a certain bond length, and three-body terms favoring specific angles between bonds [6,69,70]. The sp 2 -type covalent bonding implies that each atom has exactly three first neighbors and that bond angles of 2π/3 are energetically preferred [12]. The reader is referred to [16,20,32,57,68] for a collection of results on local and global minimizers in this setting and to [27,51] for additional results on carbon structures.
The focus of this paper is to show the local minimality of periodic configurations, both in the unstreched case and under the effect of small stretching. More specifically, we prove that, by applying a small stretching to a zigzag nanotube, the energy E is locally strictly minimized by a specific periodic configuration where all atoms see the same local configuration (Theorem 3.3). Local minimality is here checked with respect to all small perturbations in R 3n , namely not restricting a priori to periodic perturbations. On the contrary, periodicity is proved here to emerge as effect of the global variational nature of the problem.
The novelty of this result is threefold. At first, given the periodicity of the mentioned local minimizers, the actual configuration in R 3n can be determined by solving a simple minimization problem in R 2 , which consists in identifying the length of two specific bond lengths between neighboring atoms. This is indeed the standpoint of a number of contributions, see [1,8,30,31,43,44,46,50] among many others, where nevertheless periodicity is a priori assumed. In this regard, our result offers a justification for these lower-dimensional approaches. Our assumptions on E are kept fairly general in order to include the menagerie of different possible choices for energy terms which have been implemented in computational chemistry codes [7,11,38,60,75]. A by-product of our results is hence the cross-validation of these choices in view of their capability of describing carbon nanotube geometries.
Secondly, we rigorously check that, also in presence of small stretching, the geometrical model obtained via local minimization corresponds neither to the classical rolled-up model [18,19,45], where two out of three bond angles at each atom are 2π/3, nor to the polyhedral model [14,15,52], where all bond angles are equal. The optimal configuration lies between these two (Proposition 3.4), a fact which remarkably corresponds to measurements on very thin carbon nanotubes [80]. Moreover, in accordance with the results in [44], local minimizers are generically characterized by two different bond lengths.
Finally, our result proves the validity of the so-called Cauchy-Born rule for carbon nanotubes: By imposing a small tension, the periodicity cell deforms correspondingly and global periodicity is preserved. This fact rests at the basis of a possible elastic theory for carbon nanotubes. As a matter of fact, such periodicity is invariably assumed in a number of different contributions, see [4,29,40,79] among others, and then exploited in order to compute tensile strength as well as stretched geometries. Here again our results provide a theoretical justification of such approaches.
While the Cauchy-Born rule plays a pivotal role in mechanics [25,26,78], rigorous results are scarce. Among these we mention [36,13], which assess its validity within two-and d-dimensional cubic mass-spring systems, respectively. More general interactions are considered in [21,22], where the Cauchy-Born rule is investigated under a specific ellipticity condition applying to the triangular and hexagonal lattice, both in the static and the dynamic case. Our result is, to the best of our knowledge, the first one dealing with a three-dimensional structure which is not a subset of a Bravais lattice nor of a multilattice. Note though the Saint Venant principle in [24], which corresponds to the validity of an approximate version of the Cauchy-Born rule, up to a small error. However, the setting of [24] is quite different from the present one, where long-range purely two-body interactions are considered.
This work is the culmination of a series on the geometry and mechanics of nanotubes [55,56]. The theoretical outcomes of this paper have been predicted computationally in [55], where stability of periodic configurations have been investigated with Monte Carlo techniques, both for zigzag and armchair topologies under moderate displacements. A first step toward a rigorous analytical result has been obtained in [56] for both zigzag and armchair topologies under no stretching. In [56], stability is checked against a number of non-periodic perturbations fulfilling a specific structural constraint, which is related to the nonplanarity of the hexagonal cells induced by the local geometry of the nanotube. Here, we remove such constraint and consider all small perturbations, even in presence of stretching.
Indeed, removing the structural assumption and extending the result of [56] to the present fully general setting requires a remarkably deeper analysis. In a nutshell, one has to reduce to a cell problem and solve it. The actual realization of this program poses however substantial technical challenges and relies on a combination of perturbative and convexity techniques.
Whereas the proof in [56] was essentially based on the convexity of the energy given by the three bond angles at one atom, in the present context we have to reduce to a cell which includes eight atoms and is slightly nonplanar. The convexity of cell energies for various Bravais lattices has already been investigated in the literature [13,34,36,67], particularly for problems related to the validation of the Cauchy-Born rule. In our setting, however, we need to deal with an almost planar structure embedded in the three-dimensional space and therefore, to confirm convexity of the cell energy, a careful analysis in terms of the nonplanarity is necessary, see Section 7.2 and Theorem 7.6. In this context, an additional difficulty lies in the fact that the reference configuration of the cell is not a stress-free state.
The convexity is then crucially exploited in order to obtain a quantitative control of the energy defect in terms of the symmetry defect produced by symmetrizing a cell (Theorem 4.4). On the other hand, a second quantitative estimate provides a bound on the defect in the nonplanarity of the cell (called angle defect) with respect to the symmetry defect of the cell (Lemma 4.1). The detailed combination of these two estimates and a convexity and monotonicity argument (Proposition 4.3) proves that ground states necessarily have symmetric cells, from which our stability result follows (Theorem 3.3).
The validation of the Cauchy Born rule essentially relies on the application of a slicing technique which has also been used in [34] in a more general setting: One reduces the problem to a chain of cells along the diameter of the structure and shows that identical deformation of each cell is energetically favorable. In the present context, however, additional slicing arguments along the cross sections of the nanotube are necessary in order to identify correctly the nonplanarity of each hexagonal cell.
The paper is organized as follows. In Section 2 we introduce some notation and the mathematical setting. Section 3 collects our main results. In Section 4 we present the proof strategy, the essential auxiliary statements (Lemma 4.1 -Theorem 4.4), and the proof of Theorem 3.3. The proofs of the various necessary ingredients are postponed to Sections 5-7.

Carbon-nanotube geometry
The aim of this section is to introduce some notation and the nanotube configurational energy. Let us start by introducing the mathematical setting as well as some preliminary observations.
As mentioned above, carbon nanotubes (nanotubes, in the following) are modeled by configurations of atoms, i.e., collections of points in R 3 representing the atomic sites. Nanotubes are very long structures, measuring up to 10 7 times their diameter. As such, we shall not be concerned with describing the fine nanotube geometry close to their ends. We thus restrict our attention to periodic configurations, i.e., configurations that are invariant with respect to a translation of a certain period in the direction of the nanotube axis. Without loss of generality we consider only nanotubes with axis in the e 1 := (1, 0, 0) direction. Therefore, a nanotube is identified with a configuration where L > 0 is the period of C and C n := {x 1 , . . . , x n } is a collection of n points x i ∈ R 3 such that x i · e 1 ∈ [0, L). In the following, we will refer to C n as the n-cell of C, and since C is characterized by its n-cell C n and its period L, we will systematically identify the periodic configuration C with the couple (C n , L), i.e., C = (C n , L).
2.1. Configurational energy. We now introduce the configurational energy E of a nanotube C, and we detail the hypotheses on E that we assume throughout the paper. We aim here at minimal assumptions in order to include in the analysis most of the many different possible choices for energy terms that have been successfully implemented in computational chemistry codes [7,11,38,60,75].
The energy E is given by the sum of two contributions, respectively accounting for two-body and three-body interactions among particles that are respectively modelled by the potentials v 2 and v 3 , see (1).
We assume that the two-body potential v 2 : (0, ∞) → [−1, ∞) is smooth and attains its minimum value only at 1 with v 2 (1) = −1 and v 2 (1) > 0. Moreover, we ask v 2 to be shortranged, that is to vanish shortly after 1. For the sake of definiteness, let us define v 2 (r) = 0 for r ≥ 1.1. These assumptions reflect the nature of covalent atomic bonding in carbon favoring a specific interatomic distance, here normalized to 1.
We say that two particles x, y ∈ C are bonded if |x − y| < 1.1, and we refer to the graph formed by all the bonds as the bond graph of C. Taking into account periodicity, this amounts to considering two particles x i and x j of the n-cell C n of C to be bonded if for every x i , x j ∈ C n . Let us denote by N the set of all couples of indices corresponding to bonded particles, i.e., The three-body potential v 3 : [0, 2π] → [0, ∞) is assumed to be smooth and symmetric around π, namely v 3 (α) = v 3 (2π−α). Moreover, we suppose that the minimum value 0 is attained only at 2π/3 and 4π/3 with v 3 (2π/3) > 0. Let T be the index set of the triples corresponding to first-neighboring particles, i.e., For all triples (i, j, k) ∈ T we denote by α ijk ∈ [0, π] the bond angle formed by the vectors x i −x j and x k − x j . The assumptions on v 3 reflect the basic geometry of carbon bonding in a nanotube: Each atom presents three sp 2 -hybridized orbitals, which tend to form 2π/3 angles.
The configurational energy E of a nanotube C = (C n , L) is now defined by where the factors 1/2 are included to avoid double-counting the interactions among same atoms. Let us mention that the smoothness assumptions on v 2 and v 3 are for the sake of maximizing simplicity rather than generality and could be weakened. Observe that our assumptions are generally satisfied by classical interaction potentials for carbon (see [69,70]). Since the energy E is clearly rotationally and translationally invariant, in the following we will tacitly assume that all statements are to be considered up to isometries. We say that a nanotube C = (C n , L) is stable if (C n , L) is a strict local minimizer of the interaction energy E.

2.2.
Geometry of zigzag nanotubes. We now introduce a specific two-parameter family of nanotubes which will play a crucial role in the following. This is the family of so-called zigzag nanotubes having the minimal period µ > 0. The term zigzag refers to a specific topology of nanotubes, which can be visualized as the result of a rolling-up of a graphene sheet along a specific lattice direction, see Figure 1. The resulting three-dimensional structure is depicted zigzag armchair Figure 1. Rolling-up a graphene sheet to a zigzag nanotube: the vector illustrates the identification of the two dashed vertical lines. The term zigzag refers to the orientation of this vector with respect to bonds. Different vectors correspond indeed to different nanotube topologies. The dotted line indicates the identification direction for armchair nanotubes.
in Figure 2. Note that our preference for the zigzag topology is solely motivated by the sake of definiteness. The other classical choice, namely the so-called armchair topology, could be considered as well. The reader is referred to [56] for some results on unstretched armchair geometries. e 1 Figure 2. Zigzag nanotube.
We let ∈ N, > 3, and define the family F (µ) as the collection of all configurations that, up to isometries, coincide with k(λ 1 + σ) + j(2σ + 2λ 1 ) + l(2σ + λ 1 ), ρ cos π(2i + k) , ρ sin π(2i + k) for some choice of λ 1 ∈ (0, µ/2), λ 2 ∈ (0, µ/2), σ ∈ (0, µ/2), and ρ ∈ 0, µ 4 sin(π/(2 )) such that Of course, the configurations in F (µ) are periodic with minimal period µ. The parameter ρ indicates the diameter of the tube and λ 1 , λ 2 are the two possibly different lengths of the covalent bonds in each hexagon of the tube, where the bonds of length λ 1 are oriented in the e 1 direction (see Figure 4). These configurations are objective [42]: They are obtained as orbits of two points under the action of a prescribed isometry group. The latter group is generated by a translation and by a translation combined with a rotation about the e 1 -axis. Notice that our definition slightly differs from the one adopted in [55,56] in the sense that for fixed i, k the points identified by the quadruples (i, j, k, l) for j ∈ Z, l ∈ {0, 1} lie on a line parallel to e 1 (see Figure 3).
For fixed µ > 0, F (µ) is a two-parameter smooth family of configurations since each configuration in F (µ) is uniquely determined by λ 1 and λ 2 by taking relation (3) into account. Later we will consider different values for the minimal period µ in order to model nanotubes under stretching.
We state the following basic geometric properties of configurations in F (µ) (see Figure 3). The analogous properties in the case λ 1 = λ 2 = 1 have already been discussed in [55].  (b) Atoms in F are arranged in planar sections, perpendicular to e 1 , obtained by fixing j, k, and l in (2). Each of the sections contains exactly atoms, arranged at the vertices of a regular -gon. For each section, the two closest sections are at distance σ and λ 1 , respectively. (c) The configuration F is invariant under a rotation of 2π/ around e 1 , under the translation µe 1 , and under a transformation consisting of a rotation of π/ around e 1 and a translation along the vector (λ 1 + σ)e 1 (see Figure 4). (d) Let i ∈ {1, . . . , }, j ∈ Z and k, l ∈ {0, 1}: the quadruple (i, j, k, l) identifies points of F, denoted by x j,l i,k , where (0, j, k, l) is identified with ( , j, k, l). Given x j,0 i,0 ∈ F, the two points x j−1,1 i,1 and x j,0 i−1,1 is λ 2 and the distance from x j+1,0 Figure 3 for the analogous notation of x j,0 i,1 and x j,1 i,1 .
Notice that for fixed λ 1 and λ 2 the other parameters range between two degenerate cases: ρ = 0 (the cylinder is reduced to its axis) and σ = 0 (sections collide). We shall however impose further restrictions, for each atom should have three bonds. In particular, the only three bonds per atom should be the ones identified by point (d) of Proposition 2.1. By recalling that two particles are bonded if their distance is less than the reference value 1.1, since the distance between two consecutive sections is either λ 1 or σ, we require λ 1 > 0.9 and σ > 0.2. Additionally, we require λ 1 , λ 2 < 1.1, which also implies σ < 1.1 by (3). On the other hand, on each section, the edge of the regular -gon should be greater than 1.1. Such length is given by 2ρ sin γ , where γ is the internal angle of a regular 2 -gon, i.e., Therefore, we need to impose ρ > ρ − := 0.55/ sin γ . With these restrictions we have the following Proposition 2.2 (Parametrization of the family). Let F ∈ F (µ) with ρ > ρ − , σ > 0.2 and λ 1 , λ 2 ∈ (0.9, 1.1). Then, all atoms in F have exactly three (first-nearest) neighbors, two at distance λ 2 and one at distance λ 1 , where the bond corresponding to the latter neighbor is parallel to e 1 . Among the corresponding three bond angles, which are smaller than π, two have amplitude α (the ones involving atoms in three different sections), and the third has amplitude β (see Figure 4), where α ∈ (π/2, π) is obtained from and β ∈ (π/2, π) is given by The proof for the case λ 1 = λ 2 = 1 was detailed in [55]. The extension to our setting is a straightforward adaption and is therefore omitted. As already mentioned, the collection F (µ) is a two-parameter family where all its configurations are uniquely determined by the specification of λ 1 and λ 2 . The corresponding element will be denoted by F λ1,λ2,µ . Restricting the minimal period µ to the interval (2.6, 3.1) we observe by (3) and an elementary computation that the constraints λ 1 , λ 2 ∈ (0.9, 1.1) and > 3 automatically imply 0.2 < σ < 0.65 and ρ > ρ − . Therefore, the assumptions of Proposition 2.2 hold.

Main results
In this section we collect our main results. The corresponding proofs will then be presented in Sections 4-7.
For a fixed integer > 3, let us consider a configuration F in the family F (µ). As F is periodic, it can be identified with the couple (F n , L), where F n is the corresponding n-cell (n = 4m for some m ∈ N), and L = L µ m := mµ is the period parameter, corresponding to the cell length (notice that for m = 1 we get the minimal period of the configuration). In view of (1) and the properties stated in Proposition 2.2, the energy can be written as 3.1. Unstrechted nanotubes. A first natural problem to be considered is the energy minimization restricted to the families F (µ), with the values of µ in the reference interval µ ∈ (2.6, 3.1). Let us denote by F λ1,λ2,µ an element of F (µ) with bond lengths λ 1 , λ 2 . If we minimize among nanotubes F λ1,λ2,µ with respect to µ ∈ (2.6, 3.1) and λ 1 , λ 2 in a neighborhood of 1, we reduce to the case λ 1 = λ 2 = 1. Indeed, we can replace λ 1 , λ 2 by 1, leave α unchanged, and choose µ according to (3) and (5) such that the energy (8) decreases. We notice that {F 1,1,µ | µ ∈ (2.6, 3.1)} is a one-parameter family. It follows from Proposition 2.2 and (3) that this family can also be parametrized in terms of the bond angle α introduced in Proposition 2.2 using the relation µ = 2(1 − cos α). We indicate these configurations by G α .
Let us report the idea of the proof. Exploiting the monotonicity properties of v 3 and β (the latter being decreasing as a function of α), one derives that the minimum is attained for α in a small left neighborhood I of 2π/3, e.g., I := (2π/3 − σ, 2π/3] for some small σ > 0. Using in addition the convexity of v 3 and the concavity of β, it follows that α → E(F) = −3n/2 + n 2v 3 (α) + v 3 (β(α, γ )) is strictly convex in I, which implies the assertion.
The result in particular shows that neither the polyhedral nor the rolled-up configuration is a local minimizer of the energy E. The corresponding minimal period of the nanotube is given by cf. (3) and (5), and we notice G α us = F 1,1,µ us . Nanotubes with µ = µ us will be referred to as unstretched nanotubes. The aim of [55,56] was to prove that G α us is a local minimizer. This has been illustrated numerically in [55] and checked analytically in [56], for a restricted class of perturbations. Our stability result Theorem 3.3 below delivers an analytical proof of stability with respect to all small perturbations. As such, it generalizes and improves known results, even in the unstreched case.
3.2. Nanotubes under stretching. Let us now move forward to the case of stretched nanotubes. This corresponds to choosing µ = µ us . Indeed, we impose a tensile or compressive stress on the nanotube by simply modifying its minimal period. Given the role of periodicity in the definition of the energy E, see (1), this has the net effect of stretching/compressing the structure. Note that this action on the structure is very general. In particular, it includes, without reducing to, imposed Dirichlet boundary conditions, where only the first coordinate of the boundary atoms is prescribed. For fixed µ ∈ (2.6, 3.1) we consider the minimization problem We obtain the following existence result.
Theorem 3.2 (Existence and uniqueness of minimizer: General case). There exist 0 ∈ N and, for each ≥ 0 , an open interval M only depending on v 2 , v 3 , and , with µ us ∈ M , such that for all µ ∈ M there is a unique pair of bondlengths (λ µ 1 , λ µ 2 ) such that F λ µ In the following the minimizer is denoted by F * µ . Note that we have F * µ us = G α us by Proposition 3.1.
Our aim is to investigate the local stability of F * µ . To this end, we consider general small perturbationsF of F * µ with the same bond graph, i.e., each atom keeps three and only three bonds, and we can identify the three neighboring atoms of the perturbed configurations with the ones for the configuration F (7) for m ∈ N with n = 4m . We define small perturbations The parameter η > 0 will always be chosen sufficiently small such that the topology of the bond graph remains invariant. η will in general also depend on . Moreover, we recall E(F) = E(F n , L µ m ). We obtain our main result, concerning local stability under small stretching.

Theorem 3.3 (Local stability of minimizers).
There exist 0 ∈ N and for each ≥ 0 some µ crit > µ us and η > 0 only depending on v 2 , v 3 , and such that for all ≥ 0 and for all The theorem asserts that, under prescribed and small stretchings (i.e., the value of L µ m is prescribed), there exists a periodic strict-local minimizer F * µ that belongs to the family F (µ). In other words, given µ > µ us , the µ-periodic configuration F * µ is a local minimizer among configurations subject to the same macroscopic stretching, i.e., the atoms follow the macroscopic deformation. This can be seen as a validation of the Cauchy-Born rule in this specific setting. Especially, the result justifies the reduction of the 3n-dimensional minimization problem min{E(F)| F ∈ P η (µ)} to the two-dimensional problem (10).
In the following statement we collect the main properties of the local minimizer.
Notice that Theorem 3.3 provides a stability result only for the case of expansion µ ≥ µ us and for values µ near µ us . The situation for compression is more subtle from an analytical point of view and our proof techniques do not apply in this case. However, we expect stability of nanotubes also for small compression and refer to [55] for some numerical results in this direction. Let us complete the picture in the tension regime by discussing briefly the fact that for larger stretching cleavage along a section is energetically favored. More precisely, we have the following result.
Notice that the configuration H µ corresponds to a brittle nanotube cleaved along a crosssection. The energy is given by E(H µ ) = E(F 1,1,µ us ) + 4 since in the configuration H µ there are 4 less active bonds per n-cell than in F 1,1,µ us . Moreover, H µ is a stable configuration in the sense of Theorem 3.3 for all µ ≥ µ us , which can be seen by applying Theorem 3.3 separately on the two parts of H µ , consisting of the points x j,l i,k with j < m/2 and j ≥ m/2, respectively. As mentioned, nanotubes are long structures. In particular, m should be expected to be many orders of magnitude larger than . The case of large m is hence a sensible one and for m large enough we have µ frac ,m < µ crit , with µ crit from Theorem 3.3. Hence, by combining Theorem 3.3 with Theorem 3.5, for all µ ≥ µ us we obtain a stability result for an elastically stretched or cleaved nanotube, respectively.
The proof of Theorem 3.5 is elementary and relies on the fact that the difference of the energy associated to F * µ and H µ can be expressed by for µ in a small neighborhood around µ us , where we used Property 1 in Proposition 3.4 and n = 4m . We close the section by noting that the scaling of µ frac ,m − µ us in m is typical for atomistic systems with pairwise interactions of Lennard-Jones type and has also been obtained in related models, cf. [5,33,34].

Existence and stability: Proof of Theorem 3.2 and Theorem 3.3
In this section we consider small perturbationsF of configurations in F (µ) with the same bond graph, as defined in (11). The atomic positions ofF will be indicated by x j,l i,k and are labeled as for a configuration F (µ), cf. Proposition 2.1(d). We first introduce some further notation needed for the proof of our main result. In particular, we introduce a cell energy corresponding to the energy contribution of a specific basic cell.
Centers and dual centers. We introduce the cell centers and the dual cell centers Note that for a configuration in F(µ) for fixed j the 2 points z i,j,0 and z dual i,j−1,1 for i = 1, . . . , lie in a plane perpendicular to e 1 . Likewise, z i,j,1 and z dual i,j,0 for i = 1, . . . , lie in a plane perpendicular to e 1 .
Cell energy. The main strategy of our proof will be to reduce the investigation of (10) to a cell problem. In order to correctly capture the contribution of all bond lengths and angles to the energy, it is not enough to consider a hexagon as a basic cell, but two additional atoms have to be taken into account.
x 1 x 2 x 3 x 4 x 6 x 5 Let be given a center z i,j,k and number the atoms of the corresponding hexagon by i,k and the remaining clockwisely by x 3 , x 4 , x 5 , x 6 as indicated in Figure 5, such that x 3 is consecutive to x 1 , see also (54) below. Additionally, the atoms bonded to x 1 and x 2 , respectively, which are not contained in the hexagon, are denoted by x 7 and x 8 . Note that . . , 6 we define the bondlengths b i as indicated in Figure 6 By ϕ i we denote the interior angle of the hexagon at x i . By ϕ 7 , ϕ 8 we denote the remaining two angles at x 1 and by ϕ 9 , ϕ 10 we denote the remaining two angles at x 2 , see again Figure 6. Figure 6. Notation for the bond lengths and angles in the basic cell.
We define the cell energy by Notice that the cell energy is a function depending on the bond lengths and angles in the cell. However, as we identify each cell with its center z i,j,k , for simplicity we use the notation E cell = E cell (z i,j,k ). Furthermore, also for notational convenience we do not put indices i, j, k on bond lengths and angles. To derive convexity properties of E cell it is convenient to take also the contribution of the angles ϕ 7 , . . . , ϕ 10 into account. Observe that Indeed, each bond not (approximately) parallel to e 1 is contained exactly in two cells. Each bond (approximately) parallel to e 1 is contained in four cells, twice in form of a bond in a hexagon, once as a bond left of a hexagon and once as a bond right of a hexagon. Moreover, angles with index {1, 2} are contained exactly in one cell and angles with index {3, . . . , 10} are contained in exactly two cells.

Symmetrization of cells.
A basic cell is a configuration of eight points of R 3 . By x kink ∈ R 3×8 we denote the unstretched kink configuration: a basic cell as found in the unstretched configuration G α us from Section 3, see (54) below for the exact definition. Notice that the coordinates given in (54) correspond to a convenient choice of a new reference orthonormal system in R 3 . Indeed, consider a cell of the nanotube G α us , where the eight points are ordered from x 1 to x 8 according to the convention of the previous subsection (see Figure 5), in particular the points x 3 , x 4 , x 5 , x 6 are numbered clockwisely with respect to an observer lying in the interior of the tube. We fix a new reference coordinate system as follows: we let the center of the cell be the origin, e 1 (axis direction) be the direction of x 2 − x 1 , e 2 the direction of x 3 − x 6 , and e 3 = e 1 ∧ e 2 . Sometimes we will write R 2 × {0} for the plane generated by e 1 , e 2 . If x ∈ R 3×8 denotes a generic cell, possibly after a rigid motion we may always assume that, with respect to the new reference system, the second and third components of (x 1 + x 7 )/2, (x 2 + x 8 )/2 are zero and the points x 4 , x 5 lie in a plane parallel to R 2 × {0}.
A key step in our analysis will be to show that the minimization of the cell energy (13) can be reduced to a special situation with high symmetry. To this end, we introduce the symmetrization of a cell. For y = (y 1 , y 2 , y 3 ) ∈ R 3 we let r 1 (y) := (−y 1 , y 2 , y 3 ) and r 2 (y) := (y 1 , −y 2 , y 3 ). For the generic cell x = (x 1 , . . . , x 8 ) ∈ R 3×8 we define the reflections S 1 interchanges the pair of points (x 3 , x 6 ) and (x 4 , x 5 ), and changes the sign of the second components of all points. On the other hand, S 2 interchanges the pair of points (x 1 , x 2 ), (x 3 , x 4 ), (x 5 , x 6 ), and (x 7 , x 8 ), and changes the sign of the first components of all points.
We let If x is seen as a perturbation of x kink , x S1 (resp. x S2 ) is the reflected perturbation with respect to the plane generated by e 1 , e 3 (resp. e 2 , e 3 ). The symmetry of the configurations implies therefore We define the symmetrized perturbations We also introduce the symmetry defect Notice that for notational simplicity in (18) we do not put indices i, j, k on x, x , and S(x). A property that we remark is that for a basic cell x with center z i,j,k the quantity |z dual i,j,k − z dual i,j−1,k | does not change when passing to S(x) since the second and third component of are assumed to be zero. Below we will see that the difference of the cell energy of S(x) and x can be controlled in terms of ∆(z i,j,k ) due to strict convexity of the energy.
Angles between planes. In the following we denote the plane through three points p 1 , p 2 , and p 3 by {p 1 p 2 p 3 }, i.e., Furthermore, for each y = x j,l i,k we denote by y 1 , y 2 , y 3 the three atoms that are bonded with y, where the three points are numbered such that y 3 − y is (approximately) parallel to the axis direction e 1 . Let θ = θ(x) ≤ π denote the angle between the planes defined by {y 3 yy 1 } and {y 3 yy 2 }. More precisely, let n 13 , n 23 denote unit normal vectors to the planes {y 3 yy 1 } and {y 3 yy 2 }, respectively. Then we have θ(y) = max π − arccos(n 13 · n 23 ), arccos(n 13 · n 23 ) (19) as represented in Figure 7. With these preparations we will now define angles corresponding to centers and dual centers. Let z i,j,k = 1 2 (x j,0 i,k + x j,1 i,k ) be a center of a given hexagon. As before we denote the points of the hexagon by x 1 , . . . , x 6 . By θ l (z i,j,k ) we denote the angle between the planes The angle between the planes {y 3 yy 1 } and {y 3 yy 2 } is denoted by θ(y).
In Section 5 we prove the following lemma which provides a linear control for the oscillation of plane angles of a perturbed configurationF with respect to those of a configuration in F (µ) in terms of the symmetry defect from (18).

Lemma 4.1 (Symmetry defect controls angle defect).
There is a universal constant c > 0 such that for η > 0 small enough for allF ∈ P η (µ) with ∆(z i,j,k ) ≤ η for all centers z i,j,k we have Note that the sum on the left equals exactly 4m(2 − 2)π ifF ∈ F (µ).
Reduced energy. A key step in our analysis will be to show that the minimization of the cell energy (13) can be reduced to a special situation with high symmetry. As represented in Figure  8, this corresponds to the conditions with the angles introduced in (19). The notationμ is reminiscent of the fact that we have indeed µ = µ for a basic cell of a nanotube in F (µ). Under (20), arguing along the lines of Proposition 2.2, we obtain β = β(α 1 , γ 1 ) = 2 arcsin sin α 1 sin By elementary trigonometry, cf. Figure 8, we also get We now introduce the symmetric energy by Notice In general, we show that, up to a small perturbation, the symmetric energy E sym µ,γ1,γ2 delivers a lower bound for E cell for cells satysfying (20).
Since E sym µ,γ1,γ2 is symmetric in (α 1 , γ 1 ) and (α 2 , γ 2 ), we observe that E red is symmetric in γ 1 and γ 2 , i.e., E red (µ, γ 1 , γ 2 ) = E red (µ, γ 2 , γ 1 ). The following result, which is proved in Section 6, collects the fundamental properties of E red .  We denote the unique minimzer again by F * µ and recall the definition of small perturbations P η (µ) in (11). Based on the properties of the reduced energy E red , we are able to show that, up to a linear perturbation in terms of the symmetry defect ∆ defined in (18), E red bounds the cell energy E cell from below. More precisely, we have the following.
Theorem 4.4 (Energy defect controls symmetry defect). There exist C > 0 and 0 ∈ N only depending on v 2 and v 3 , and for each ≥ 0 there are η > 0 and an open interval M containing µ us such that for all µ ∈ M ,F ∈ P η (µ), and centers z i,j,k we have and alsoμ (z i,j,k ).
Possibly passing to a smaller η , we get |z dual i,j,k − z dual i,j−1,k | ∈ M andθ(z i,j,k ) ∈ G for all i, j, k. By Theorem 4.4 we have for each cell if 0 is chosen sufficiently large. Then, taking the sum over all cells and using Property 2. of Proposition 4.3, we get by (14) Possibly passing to a smaller η , we can assume that ∆(z i,j,k ) ≤ η for all centers with η from Lemma 4.1. Then using Lemma 4.1 and recalling (25) we find where in the last step we have used the fact that γ = π(1 − 1/ ), see (4). This together with Property 3 of Proposition 4.3 yields for some C > 0 only depending on v 3 . Recalling the constraint in definition (11), we get for fixed i and k that and therefore, by taking the sum over all i and k, we getμ ≥ µ ≥ µ us . Then we derive by Property 4 and 5 of Proposition 4.3 for 0 sufficiently large and a possibly smaller constant C > 0. Note that in this step of the proof we have fundamentally used that µ ≥ µ us , i.e., the nanotube is stretched, so that a monotonicity argument can be applied. It remains to confirm the strict inequality E(F) > E(F * µ ). If ∆(z i,j,k ) > 0 for some center z i,j,k , this follows directly from the previous estimate. Otherwise, asF is a nontrivial perturbation, one of the angles in (25) or one of the lengths |z dual i,j,k − z dual i,j−1,k | does not coincide with the corresponding mean value and then at least one of the inequalities (26)-(27) is strict due to the strict convexity and monotonicity of the mappings considered in Proposition 4.3.

Symmetry defect controls angle defect: Proof of Lemma 4.1
This short section is devoted to the proof of Lemma 4.1. Recall the definition of the centers in (12), the angles (19), and the symmetry defect (18).
Proof of Lemma 4.1. LetF be a small perturbation of F ∈ F (µ), with ∆(z i,j,k ) ≤ η for all centers z i,j,k . Due to the symmetry of the problem it suffices to show For brevity we write θ i = θ l (z i+1 2 ,j,0 ) for i = 1, 3, . . . , 2 − 1 and θ i = θ l (z dual i 2 ,j−1,1 ) for i = 2, 4, . . . , 2 . (Note that for convenience we do not include the index j in the notation.) Let n i , n i+1 be unit normal vectors as introduced before (19) such that n i · n i+1 is near 1 and the smallest angle between them, which we denote by (n i , n i+1 ), is given by (n i , n i+1 ) = π − θ i for i = 1, 3, . . . , 2 − 1. For a suitable ordering of n i and n i+1 we then also obtain (n i , n i+1 ) = π−θ i for i = 2, 4, . . . , 2 . Fix a center x 0 ∈ R 3 and let P be the 2 -gon with vertices v i := x 0 +n i , i = 1, . . . , 2 . Denote the interior angles accordingly by ϕ i . Note that each edge of P forms a triangle with x 0 with angles π − θ i , ψ 1 i , and ψ 2 i , where ψ 1 i is the angle at the vertex v i and ψ 2 i is the angle at v i+1 . The key ingredient in the proof is now the observation that there exists a universal c > 0 such that for i = 1, 3 . . . , 2 − 1, where it is understood that ψ 2 0 = ψ 2 2 and z 0,j,0 = z ,j,0 . We defer the derivation of this property to the end of the proof. Notice that θ i = ψ 1 i + ψ 2 i for i = 1, . . . , 2 and that 2 i=1 ϕ i ≤ (2 − 2)π since P is a 2 -gon. We now obtain by (28) The assertion then follows by taking the sum over all j = 1, . . . , m. It remains to confirm (28). Fix i = 1, 3, . . . , 2 − 1 and let N i+1 be the plane containing the points v i , v i+1 , and v i+2 . By d i+1 we denote the distance of x 0 from N i+1 and by n i+1 the orthogonal projection of the vector n i+1 onto N i+1 . Note that d i+1 ≤ δ for δ small, depending only on the choice of η, and that |n i+1 0, which holds after possibly changing the signs of the vectors. Using that (v i+2 − v i+1 ) · (n i+1 − n i+1 ) = 0 and recalling that d i+1 is small, we calculate by a Taylor expansion ). Since ϕ i+1 =ψ 1 i+1 +ψ 2 i , to conclude (28a), it therefore remains to show for a universal constant c > 0. To see this, we first note that we have d i+1 = 0 whenever ∆(z i+1 2 ,j,0 ) + ∆(z i+3 2 ,j,0 ) = 0. Indeed, if ∆(z i+1 2 ,j,0 ) + ∆(z i+3 2 ,j,0 ) = 0, the high symmetry of the atoms in the cells with centers z i+1 2 ,j,0 and z i+3 2 ,j,0 (cf. (18)) implies that the three normal vectors n i , n i+1 , and n i+2 are coplanar. Thus, x 0 is contained in N i+1 and therefore d i+1 = 0.
We close this section with the proof of Proposition 3.4.

Energy defect controls symmetry defect: Proof of Theorem 4.4
This section is devoted to the proof of Theorem 4.4. The fact that the minimum of the cell energy is attained for a special configuration with high symmetry (see (20)) essentially relies on convexity properties of the cell energy E cell defined in (13). Throughout the section we consider a cell consisting of eight points x = (x 1 , . . . , x 8 ) ∈ R 3×8 as defined before (13), see Figure 5. Likewise, the bond lengths are again denoted by b 1 , . . . , b 8 and the angles by ϕ 1 , . . . , ϕ 10 , see Figure 6. With a slight abuse of notation we denote the cell energy for a given configuration x by E cell (x). 7.1. Relation between atomic positions, bonds, and angles. We will investigate the convexity properties of E cell near the planar reference configuration x 0 = (x 0 1 , . . . , x 0 8 ) ∈ R 3×8 defined by where γ = π(1 − 1/ ) and σ us = − cos α us with α us as given by Proposition 3.1 (cf. also (5)).
Note that x kink represents the mutual position of atoms in a cell for the unstretched nanotube G α us found in Proposition 3.1. For later use we note that by Lemma 6.1 and a Taylor expansion we find for some universal C > 0 large enough. In order to discuss the convexity properties of E cell we need to introduce a specific basis of R 3×8 , i.e., the space of cell configurations. This will consist of three collections of vectors, denoted by V degen , V good , and V bad , where the sets are defined as follows: We introduce the translations and infinitesimal rotations V trans = (e 1 , . . . , e 1 ), (e 2 , . . . , e 2 ), (e 3 , . . . , e 3 ) ⊂ R 3×8 and set V degen = V trans ∪ V rot . The family V good contains the 13 vectors The first 6 vectors keep the angles fixed and modify only the bond lengths, see Figure 9. The vectors u 8 , . . . , u 11 keep the bond lengths fixed to first order and change the angles, see Figure  10. Eventually, the remaining vectors u 12 and u 13 modify both angles and bonds as in Figure  11. By V bad we denote the collection of the vectors It is elementary to check that the vectors V degen ∪ V good ∪ V bad are linearly independent and thus form a basis of R 3×8 . Note that the vectors in V good are perpendicular to the vectors in V bad . Clearly, the cell energy is strictly convex as a function of the bond lengths and angles by the assumptions on the potentials v 2 and v 3 . Our goal is to show that the same property holds if   Then the cell energy reads as with the factors κ b . . = κ a 10 = 1/2. Before analyzing the mapping T , we need to introduce some more notation for the sum of angles ϕ i . From here on, we denote by e 1 , . . . , e 10 the canonical basis of R 10 , and we let a 1 := e 1 + . . . + e 6 , a 2 := e 1 + e 7 + e 8 , a 3 := e 2 + e 9 + e 10 be vectors in R 10 . Elementary geometry yields T a (x 0 )·a 1 = 4π and T a (x 0 )·a j = 2π for j = 2, 3 as well as T a (x) · a 1 ≤ 4π and T a (x) · a j ≤ 2π for j = 2, 3 for each x ∈ R 3×8 . Indeed, the sum of the interior angles in a hexagon is always smaller or equal to 4π and exactly 4π if the hexagon is planar. Likewise one argues for a triple junction.
Lemma 7.1 (Properties of T ). The mapping T is smooth in a neighborhood of x 0 . There is a constant c kink > 0 such that Proof. First, to see Property 1, we note that span(V degen ∪ V bad ) is a subset of Ker(DT (x 0 )) since each vector in V degen ∪ V bad does not change bond lengths and angles to first order. On the other hand, each vector in V good changes bond lengths or angles to first order and is therefore not contained in the kernel of DT (x 0 ). Indeed, the first six vectors of V good are directions of perturbations that do not change angles to first order, but bond lengths. Vectors u 7 , . . . , u 11 are perturbations that do not change bond lengths in first order, but angles. Vectors u 12 and u 13 are in-plane displacements of a single atom and change both bond lengths and angles to first order. More precisely, for the changes of bond lengths we get (We prefer not to give details of the computation, but rather refer the reader to Figures 9-11 where the situation of the different directions is indicated.). It is elementary to check that the vectors DT (x 0 )u i , i = 1, . . . , 13, are linearly independent which concludes the proof of Property 1 by dimension counting.
Since dim(Ker(DT (x 0 ))) = 11 and in V good only the first six vectors do not change angles to first order, Property 2 holds. Property 3 follows from the fact that the mapping t → T a (x 0 + tv) · a j has a local maximum at t = 0 for j = 1, 2, 3 and for all v ∈ R 3×8 as noticed before the statement of the lemma.
To see Property 4, we first consider the special case v ∈ V bad . In this situation the property follows from an elementary computation, which we detail only in the case v = (e 3 |0| . . . |0). In this case, after some calculations, we obtain (T a (x 0 + tv)) i = arccos(−1/2 + 3t 2 /2) + O(t 3 ) ≤ 2π/3−ct 2 for some c > 0 for i = 1, 7, 8, i.e., for the angles at the triple junction at point x 1 . Using also Property 1, this indeed implies (v T D 2 T a (x 0 )v) · a 2 ≤ −c, i.e., by a perturbation out of the plane the sum of the angles is reduced to second order. For the other triple junction and the interior angles of the hexagon we argue analogously. This shows the property for perturbations in the directions V bad . Likewise, we proceed for directions in span(V bad ).
Now consider the general case v = v trans + v rot + v bad ∈ span(V degen ∪ V bad ) for v trans ∈ span(V trans ), v rot ∈ span(V rot ), and v bad ∈ span(V bad ).
First, since T (x + w) = T (x) for all x ∈ R 3×8 and all w ∈ V trans , we get DT (x)w = 0 and w T D 2 T (x)w = 0 for all w ∈ span(V trans ), w ∈ R 3×8 , and x ∈ R 3×8 . Consequently, we deduce Moreover, let A ∈ R 3×3 skew be such that v rot = Ax 0 and observe that there is a rotation R t ∈ SO(3) such that x 0 t := R t (x 0 + tv rot ) is contained in the plane R 2 × {0} and one has |R t − (I − tA)| = O(|tA| 2 ), cf. [35, (3.20)]. (Here I ∈ R 3×3 denotes the identity matrix.) Consequently, we get |x 0 − x 0 t | = O(|tA| 2 ). This implies for some w ∈ R 3×8 with |w| ≤ c|A| 2 and the property that the third component of each vector in w is zero. A Taylor expansion and Property 1 of the lemma then yield As the sum of the angles in the hexagon and at the triple junctions remains invariant under perturbation w, we then deduce The desired result now follows from the fact that 3 j=1 v T bad D 2 T a (x 0 )v bad · a j ≤ −c|v bad | 2 has already been established in the first part of the proof, where we also note that |v bad | ≥ c|v − v degen | with v degen being the orthogonal projection of v onto span(V degen ).
For later purpose we also introduce the mappingẼ : [0, 2π] 10 × [0, +∞) 8 → R defined bỹ Lemma 7.2 (Properties ofẼ). The mappingẼ is smooth and there are constants 0 < c E,1 < c E,2 and 0 ∈ N depending only on v 2 and v 3 such that for ≥ 0 1. (DẼ(T (x kink ))) i = 0 for i = 11, . . . , 18, Proof. Property 1 follows from the fact that T b (x kink ) = (1, . . . , 1) ∈ R 8 and v 2 (1) = 0. To see Property 2, we apply Lemma 6.1 to find (T a (x kink )) i ∈ (2π/3 − c 2 −2 , 2π/3 − c 1 −2 ) for i = 1, . . . , 10 and the fact that v 3 ∈ C 2 with v 3 (2π/3) = 0, v 3 (2π/3) > 0. Likewise, Property 3 follows from v 2 (1) > 0 and v 3 (2π/3) > 0, respectively. 7.2. Convexity of the cell energy. The following theorem gives a first property of the Hessian of E cell at the kink configuration x kink . Theorem 7.3 (Convexity of E cell in good directions). Let 0 < r < 1. Then there exist 0 ∈ N and a constant c > 0 depending only on v 2 , v 3 , and r such that for ≥ 0 and each v ∈ R 3×8 with |v · w| ≤ r|w||v| for all w ∈ span(V degen ∪ V bad ) Proof. First, by the regularity of the mapping T , Property 1 in Lemma 7.1, and the fact that x kink → x 0 for → ∞, we find 0 ∈ N sufficiently large such that for ≥ 0 the kernel of DT (x kink ) has dimension at most 11. Then we find universal constants 0 < c 1 < c 2 such that for all ≥ 0 , possibly for a larger 0 , we have For the second property we used (55). Let be given v ∈ R 3×8 with |v · w| ≤ r|w||v| for all w ∈ span(V degen ∪ V bad ). The vector can be written as We further observe that by Lemma 7.2, Property 1 and 2, there is a constant c 3 only depending on c E,2 such that Then collecting (57)-(59) and using Property 3 of Lemma 7.2 we derive For 0 large enough (depending also on r) this implies the assertion of the lemma for ≥ 0 .
To investigate the convexity properties in the directions V bad , we need some further preparations. Recall the reflections introduced in (15). The following lemma is a consequence of Theorem 7.3 and shows that variations in the directions V good decrease the energy only to higher order. Lemma 7.4 (Energy decrease in good directions). There exist 0 ∈ N and a constant C > 0 depending only on v 2 and v 3 such that for ≥ 0 and each v ∈ span(V good ) Proof. Let v ∈ span(V good ) be given and define a perturbation of v by for some universal s > 0 to be specified below. (Note that the direction v − v increases the third components of the points x 3 , . . . , x 6 of the basic cell). By Property 1 and 2 of Lemma 7.2 and the fact that |v − v | ≤ 4s|v| −1 it clearly suffices to show To this end, we will show thatẼ for all t > 0 small. Then (61) follows by taking the limit t → 0. Consider x = x kink + tv for t > 0 small. Possibly after applying a rigid motion we can assume that the second and third components of (x 1 + x 7 )/2 and (x 2 + x 8 )/2 are zero, the points x 1 , x 2 , x 7 , x 8 lie in the plane R 2 × {0} and that the points x 3 , x 4 , x 5 , x 6 lie in a plane parallel to R 2 × {0}. (Recall that v induces an in-plane perturbation, i.e., the third component of each vector in v is zero.) We replace x by a symmetrized version as follows.
Define x S1 by (16) and note that E cell (x S1 ) = E cell (x). Moreover, it is elementary to see that the third component of each vector in w 1 := x S1 − x is zero. Consequently, w 1 is perpendicular to V bad , V trans , and the rotations v 2 , v 3 . Clearly, as the reflection S 1 leaves the points (x 1 + x 7 )/2 and (x 2 + x 8 )/2 unchanged, we also have that w 1 is not parallel to the rotation v 1 . Consequently, by Theorem 7.3 and a continuity argument with t small enough, the mapping t → E cell (x + t w 1 ) is convex on [0, 1]. This implies for x = 1 2 (x + x S1 ) (see (17a)) that E cell (x ) ≤ 1 2 (E cell (x) + E cell (x S1 )) = E cell (x). Likewise, we consider x S2 := x kink + S 2 (x − x kink ) and note that E cell (x S2 ) = E cell (x ). Similarly as before, the vector w 2 := x S2 −x is perpendicular to the vectors V bad and not parallel to V degen . Using Theorem 7.3 we get E cell (S(x)) ≤ E cell (x ) ≤ E cell (x) for S(x) = 1 2 (x + x S2 ) (see (17b)).
By this symmetrization procedure we get that the eight points S(x) are contained in two kinked planes (similarly as x kink ). We denote the incidence angle of the two planes by γ ≤ π and note that γ ≤ γ if the constant s > 0 in (60) is chosen sufficiently large. The bond lengths For the angles ϕ 1 = ϕ 2 and ϕ 3 = . . . = ϕ 10 holds.
Now taking γ ≤ γ into account and recalling that α us is optimal angle from Proposition 3.1, we find where the last step follows from (54). This shows (62) and concludes the proof.
The next lemma shows that a perturbation of the angles, which does not change the sum of the angles, essentially does not decrease the energy to first order.
Lemma 7.5. There exist 0 ∈ N and a constant C > 0 depending only on v 2 and v 3 such that for ≥ 0 and each w = (w 1 , . . . , w 10 ) ∈ R 10 with w · a j = 0 for j = 1, 2, 3 we have Proof. From Property 2 of Lemma 7.1 we have that the image of the affine mapping DT a (x 0 ) has dimension 7. Moreover, we have (DT a (x 0 )v) · a j = 0 for j = 1, 2, 3 and all v ∈ R 3×8 . Indeed, write v = v good + v bad with v good ∈ span(V good ) and v bad ∈ span(V degen ∪ V bad ). Note that DT a (x 0 )v = DT a (x 0 )v good by Property 1 of Lemma 7.1. For each t ∈ R the eight points x 0 + tv good are contained in the plane R 2 × {0}. This implies T a (x 0 + tv good ) · a j ∈ {2π, 4π} for all t ∈ R and j = 1, 2, 3, which gives (DT a (x 0 )v good ) · a j = 0 for j = 1, 2, 3, as desired.
The dimension of the image of DT a (x 0 ) together with the fact that w · a j = 0 for j = 1, 2, 3 show that there exists a vector v ∈ span(V good ) such that DT a (x 0 )v = w. By applying Lemma 7.4 we get where C is the constant from Lemma 7.4. By a continuity argument and (55) we get |DT (x kink )− DT (x 0 )| ≤ c −1 . This together with Property 2 of Lemma 7.2 shows for C = C(C , c E,2 , c). The fact that DT a (x 0 )v = w, |v | ≤ c|w| for a constant c > 0 (depending on DT a (x 0 )) and Property 1 of Lemma 7.2 conclude the proof.
We now improve Theorem 7.3 and prove convexity of E cell at the kink configuration x kink .
Theorem 7.6 (Convexity of E cell ). Let 0 < r < 1. Then there exist 0 ∈ N and a constant c > 0 depending only on v 2 , v 3 , and r such that for ≥ 0 and each v ∈ R 3×8 with |v · w| ≤ r|w||v| for all w ∈ span(V degen ) Proof. As in the proof of Theorem 7.3 we consider the mapping f v as defined before (58). The goal is to show f v (0) ≥ c|v| 2 −2 . We write v = v degen + v bad + v good with three orthogonal vectors, where v degen + v bad ∈ span(V degen ∪ V bad ), v degen ∈ span(V degen ), v bad ∈ span(V degen ) ⊥ , and v good ∈ span(V degen ∪ V bad ) ⊥ . By assumption we obtain after a short calculation |v good | 2 + |v bad | 2 ≥ (1 − r 2 )|v| 2 .

7.3.
Proof of Theorem 4.4. As a last preparation for the proof of Theorem 4.4, we need to investigate how the angles between planes behave under reflection of a configuration (see (15)- (17)). Let a center z i,j,k be given and, as before, denote by x ∈ R 3×8 the atoms of the corresponding cell. We introduce the angles between the planes as in Section 4. By θ l (x) we denote the angle between the planes {x 1 x 3 x 4 } and {x 1 x 6 x 5 }. By θ r (x) we denote the angle between the planes {x 3 x 4 x 2 } and {x 2 x 5 x 6 }. Moreover, we let θ dual l (x) = θ(x 1 ) and θ dual r (x) = θ(x 2 ) with θ(x i ), i = 1, 2, as defined in (19). Recall also the definition of ∆(z i,j,k ) in (18). Lemma 7.7 (Symmetry defect controls angle defect). There exist a universal constant C > 0 and 0 ∈ N, and for each ≥ 0 there exists η > 0 such that for allF ∈ P η (µ), µ ∈ (2.6, 3.1), and all centers z i,j,k we have θ l (S(x)) + θ r (S(x)) ≤ θ l (x) + θ r (x) + C∆(z i,j,k ), θ dual l (S(x)) + θ dual r (S(x)) ≤ θ dual l (x) + θ dual r (x) + C∆(z i,j,k ), where x ∈ R 3×8 denotes the position of the atoms in the cell with center z i,j,k and S(x) as in (17b).
We postpone the proof of this lemma to the end of the section and now continue with the proof of Theorem 4.4.
Proof of Theorem 4.4. LetF ∈ P η (µ) be a given configuration, where η is specified below, and let x ∈ R 3×8 be the points of one cell as introduced in Section 4. As usual, possibly after a rigid motion we can assume that the second and third components of (x 1 +x 7 )/2, (x 2 +x 8 )/2 are zero and the points x 4 , x 5 lie in a plane parallel to R 2 × {0}. We now perform a symmetrization argument as in the proof of Lemma 7.4.
We define x S1 by (16). Clearly the vector w 1 := x S1 −x is perpendicular to V trans . Moreover, we have |w 1 · v i | ≤ r|w 1 ||v i | for i = 1, 2, 3 for a universal constant r ∈ (0, 1). In particular, r is independent of the perturbation x. Indeed, for v 1 and v 2 this follows from the fact that the points (x 1 + x 7 )/2 and (x 2 + x 8 )/2 are left unchanged. For v 3 it follows from the assumption that the points x 4 , x 5 lie in a plane parallel to R 2 × {0}.
Likewise, we consider x S2 := x kink + S 2 (x − x kink ) and, similarly as before, the vector w 2 := x S2 − x is perpendicular to V trans and satisfies |w 2 · v i | ≤ r|w 2 ||v i | for i = 1, 2, 3 for a universal constant r ∈ (0, 1). Indeed, for v 1 and v 2 this follows as before and for v 3 it suffices to note that also for the configuration x = (x 1 , . . . , x 8 ) the points x 4 , x 5 lie in a plane parallel to R 2 × {0}. Using again Theorem 7.6 we get E cell (S(x)) + c −2 |w 2 | 2 ≤ E cell (x ) with S(x) from (17b). Possibly passing to a smaller constant c > 0 (not relabeled) and using (18), we observe E cell (S(x)) + c −2 ∆(z i,j,k ) ≤ E cell (x).
Finally, we give the proof of Lemma 7.7.
Proof of Lemma 7.7. The proof is mainly based on a careful Taylor expansion for the angles under the symmetrization of the atomic positions in the cell, which is induced by the reflections s k j (x ) this implies |s k j (x ) · e 1 | = n k (x ) × n k j (x ) · e 1 = |n k (x ) · e 3 ||n k j (x ) · e 2 | ≤ C sin( π − ϕ 2 )|w 2 | + C|w 2 | 2 .