3.1. Critical sets of the onsite energy f_B and their geometry

We review the polar coordinate system for $3\times3$ Q-tensors and the classification of the critical points of f_B, summarized in Lemma 3.1. We also review an alternative system of coordinates that is used to describe the neighbourhood of the critical sets of f_B. The construction uses the invariance of f_B under the adjoint action of the rotation group, see Definition 1.

For any $Q \in \hbox{Sym}^3_0 $ we have the spectral decomposition

(3.1)\begin{equation} Q = \lambda_1 {\hat n} \otimes {\hat n} + \lambda_2 {\hat m} \otimes {\hat m} + \lambda_3 {\hat p} \otimes {\hat p}, \end{equation}

with $\lambda_1 \geq \lambda_2 \geq \lambda_3$ the eigenvalues of Q, and ${\hat n}$, ${\hat m}$, ${\hat p} \in \mathbb{R}^3$ the corresponding orthonormal system of eigenvectors. In the case $\lambda_1 \gt \lambda_2 \gt \lambda_1$ the vectors $\hat n$, $\hat m$, $\hat p$ are determined uniquely up to signs.

Since Q is traceless, we have $\lambda_3 = - \lambda_1 - \lambda_2$ and we see that

(3.2)\begin{equation} Q = s ( {\hat n} \otimes {\hat n} - \frac{1}{3} I) + r ( {\hat m} \otimes {\hat m} - \frac{1}{3} I), \end{equation}

where

(3.3)\begin{equation} s = 2 \lambda_1 + \lambda_2, \quad r = 2 \lambda_2 + \lambda_1, \end{equation}

and I is the $3 \times 3$ identity matrix.

We examine the range of values the λ ₁, λ ₂, assuming $\lambda_1 \geq \lambda_2 \geq \lambda_3$. The case $\lambda_1 = 0 $ implies $\lambda_2 = \lambda_3 = 0$, and corresponds to Q = 0.

If $\lambda_1 \neq 0$ the trace condition implies $\lambda_1 \gt 0 $ and $\lambda_3 \lt 0$. Considering any $\lambda_1 \gt 0$, the values of λ ₂ belong to two intervals: region I where $\lambda_1 \geq \lambda_2 \geq 0 \gt \lambda_3 $, and region II where $\lambda_1 \gt 0 \gt \lambda_2 \geq \lambda_3 $.

In region I, we have $\lambda_2 \in [0, \lambda_1]$, and by (3.3), $r \in [s/2,s]$, with $s \in [2 \lambda_1, 3 \lambda_1]$.

In region II, we have $\lambda_2 \in [- \lambda_1/2, 0)$, and by (3.3), $r \in [0, s/2)$, $s \in [3/2 \lambda_1, 2 \lambda_1)$.

The boundaries of regions I, and II intersect at $\lambda_2 = 0$. The second (upper) boundary of interval I, $\lambda_2 = \lambda_1$, corresponds to $r = s = 3 \lambda_1$. In that case, by (3.2) we have

(3.4)\begin{equation} Q = - s ( {\hat p} \otimes {\hat p} - \frac{1}{3} I) = - 3 \lambda_1 ( {\hat p} \otimes {\hat p} - \frac{1}{3} I). \end{equation}

The second (lower) boundary of interval II, $\lambda_2 = - \frac{1}{2} \lambda_1 = \lambda_3$, corresponds to r = 0, $s = \frac{3}{2} \lambda_1 = - 3 \lambda_2$. Then

(3.5)\begin{equation} Q = s ( {\hat n} \otimes {\hat n} - \frac{1}{3} I) = - 3 \lambda_2 ( {\hat n} \otimes {\hat n} - \frac{1}{3} I). \end{equation}

Recall that Q is termed uniaxial if it has two equal nonzero eigenvalues, and isotropic if Q = 0. Otherwise Q is biaxial. The uniaxial Q-tensors correspond to the boundary of the union of regions I and II.

Lemma 3.1. Let $x_+$, $x_-$ denote the positive and negative roots of

(3.6)\begin{equation} - a^2 + 6 c^2 x^2 + b^2 x = 0, \end{equation}

respectively. Then the critical points of f_B consist of the sets defined by

(i) $\lambda_1 = \lambda_2 = 0$, where Q = 0,

(ii) $\lambda_1 \gt 0$ with $r = s = 3 \lambda_1 \gt 0$ (upper boundary of region I), where

(3.7)\begin{equation} Q = - 3 x_+ ( {\hat p} \otimes {\hat p} - \frac{1}{3} I), \quad {\hat p} \in S^2 \subset \mathbb{R}^3, \end{equation}

(iii) $\lambda_1 \gt 0$ with r = 0, $s = \frac{3}{2} \lambda_1$ (lower boundary of region II), where

(3.8)\begin{equation} Q = - 3 x_- ( {\hat n} \otimes {\hat n} - \frac{1}{3} I), \quad {\hat n} \in S^2 \subset \mathbb{R}^3. \end{equation}

The infimum of f_B is attained at the set (iii), denoted by $\Sigma^3_-$. The set (ii) is denoted by $\Sigma^3_+$ and is not a set of local extrema of f_B. The results are summarized by Proposition 2.1, see [Reference Majumdar20, Reference Majumdar and Zarnescu21, Reference Panayotaros26] for details.

In the polar representation the value of r quantifies the deviation from the set of uniaxial tensors $\Sigma^3_\pm$. However, the polar representation is singular at the critical points of f_B, and we need alternative systems of coordinates for the neighbourhood of the sets $\Sigma^3_\pm$. The system of coordinates we define below is motivated by the symmetries of f_B. We recall the following.

Definition 1. Given $A \in SO(3)$, we define $\Psi_A $ on the set $\hbox{Mat}(3)$ of real $3 \times 3 $ matrices by

(3.9)\begin{equation} \Psi_A (Q) = A Q A^T = A Q A^{-1}. \end{equation}

Also define $\Psi:SO(3) \times \hbox{Mat}(3) \rightarrow \hbox{Mat}(3)$ by $\Psi(A,Q) = \Psi_A(Q)$.

The restrictions of the $\Psi_A$, $A \in SO(3)$, to $\hbox{Sym}^3_0$, and Ψ to $ SO(3) \times \hbox{Sym}^3_0$ are denoted by $\Psi_A$, Ψ, respectively.

By Lemma 3.2 the maps $\Psi_A$, $A \in SO(3)$, in $\hbox{Mat}(3)$, $ \hbox{Sym}^3_0 $ can be represented by matrix multiplication in $\mathbb{R}^9$, $\mathbb{R}^5$, respectively. The matrices depend on the choice of basis in $\hbox{Mat}(3)$, $\hbox{Sym}^3_0$, respectively.

Consider a basis $\{{\hat W}_j\}_{j = 1}^5$ in $\hbox{Sym}^3_0$, with $Q = \sum_{j=1}^5 w_j {\hat W}_j$, $\forall Q \in \hbox{Sym}^3_0$, and the isomorphism ${\overline{\beta}}^w(Q) = w := [w_1, \ldots, w_5]^T \in \mathbb{R}^5$. Then the matrix representation of $\Psi_A$, $A \in SO(3)$, in this basis is the unique matrix $T^w_A$ satisfying $T^w_A({\overline{\beta}}^w(Q)) = {\overline{\beta}}^w( \Psi_A(Q))$. $T^w_A$ has inverse $T^w_{A^{-1}}$, $\forall A \in SO(3)$. Also define $T^w:SO(3) \times \mathbb{R}^5 \rightarrow \mathbb{R}^5$ by $ T^w(A,q) = T^w_Aq$.

The application of the actions Ψ, T^w to the analysis of the critical sets of f_B is based on the following observations, see [Reference Panayotaros26] for more details.

The lemma is used to simplify computations in the neighbourhood of the sets ${\overline{\beta}}^w(\Sigma^3_\pm)$. In particular, we can compute quantities of interest related to the derivative of f_B at a single point and use the properties of T^w to extend the result to all points of ${\overline{\beta}}^w(\Sigma^3_\pm)$.

To implement this idea it is also convenient to define bases of $\hbox{Mat}(3)$ that lead to a representation of the $\Psi_A$, $A\in SO(3)$, and Ψ, restricted to $\hbox{Sym}^3_0$, by rotation matrices. This construction allows us to understand the local geometry of the critical sets of f_B by computations at a single point. This point is chosen below to further simplify the computations.

Consider first the standard basis of $\hbox{Mat}(3) $ of matrices ${\hat Q}_{i,j}$ with $i,j$th entry unity and all other entries zero. We let ${\hat Z}_k = {\hat Q}_{i,j}$, where the enumeration $k(i,j)$ is defined by

(3.10)\begin{equation} Q = \left[ \begin{array}{ccc} z_1 & z_2 & z_3 \\ z_4 & z_5 & z_6 \\ z_7 & z_8 & z_9 \end{array} \right], \quad Q = \sum_{k=1}^9 z_k {\hat Z}_k. \end{equation}

We define the inner product $\langle \cdot,\cdot \rangle$ in $\hbox{Mat(3)}$ by $\langle R,S\rangle = \hbox{tr} (R S^T) = \sum_{k = 1}^9 r_k s_k$, $ R = \sum_{k=1}^9 r_k {\hat Z}_k$, $ S = \sum_{k=1}^9 s_k {\hat Z}_k$. The corresponding norm is denoted by $||\cdot||$.

The isomorphism $\beta^z: \hbox{Mat}(3) \rightarrow \mathbb{R}^9$ defined by $\beta^z(Q) = z := [z_1, \ldots, z_9]^T$, $Q = \sum_{k=1}^9 z_k {\hat Z}_k$, is an isometry between $\hbox{Mat}(3) $ with $|| \cdot ||$, and $\mathbb{R}^9$ with the Euclidean norm. The map $\Psi_A$, $A \in SO(3)$, is represented by multiplication by the matrix $[\Psi_A]^z$, that is $\beta^z(\Psi_A(Q)) = [\Psi_A]^z \beta^z(Q)$, $\forall Q \in \hbox{Mat}(3)$. Then $[\Psi_A]^z$ is orthogonal in $\mathbb{R}^9$ with the Euclidean norm, $\forall A \in SO(3)$, see [Reference Panayotaros26].

Consider a new basis $\{{\hat Y}_j \}_{j = 1}^9$ of $\hbox{Mat}(3)$, let $Q = \sum_{k=1}^9 y_k {\hat Y}_k$, $\forall Q \in \hbox{Mat}(3)$, and define the isomorphism $\beta^y: \hbox{Mat}(3) \rightarrow \mathbb{R}^9$ by $\beta^y(Q) = y := [y_1,\ldots,y_9]^T$. Assume that (i): ${\hat Y}_i = \sum_{j = 1}^9 M_{i,j} {\hat Z}_j$, with M an orthogonal matrix with respect to the Euclidean inner product on $\mathbb{R}^9$, and (ii): $Q \in \hbox{Sym}^3_0$ if and only if $(\beta^y(Q))_i = 0$, for $i \in \cal I$, $\mathcal{I} \subset \{1, \ldots, 9\}$, $| \mathcal{I } | = 4$.

Denote the matrix representation of $\Psi_A$ in the basis $\{{\hat Y}_j \}_{j = 1}^9$ by $[\Psi_A]^y$. Let the matrix T_A that represents the restriction of $\Psi_A$ to $\hbox{Sym}^3_0$ in the basis $\{{\hat Y}_j \}_{j \in \mathcal{J}}$, $\mathcal{J} = \{1, \ldots,9\}\setminus\mathcal{I}$. We can show the following, see [Reference Panayotaros26].

An example of a basis $\{{\hat Y}_j \}_{j = 1}^9$ satisfying (i), (ii) is defined by

(3.11)\begin{equation} y_{2} = \frac{1}{\sqrt{2}} ( z_2 + z_4), \quad y_{4} = \frac{1}{\sqrt{2}} ( z_2 - z_4), \end{equation}

(3.12)\begin{equation} y_{3} = \frac{1}{\sqrt{2}} ( z_3 + z_7), \quad y_{7} = \frac{1}{\sqrt{2}} ( z_3 - z_7), \end{equation}

(3.13)\begin{equation} y_{6} = \frac{1}{\sqrt{2}} ( z_6 + z_8), \quad y_{8} = \frac{1}{\sqrt{2}} ( z_6 - z_8), \end{equation}

(3.14)\begin{equation} y_{9} = \frac{1}{\sqrt{3}}( z_1 + z_5 + z_9), \quad y_{1} = \frac{1}{\sqrt{2}}( z_1 - z_5 ), \quad y_{5} = \frac{1}{\sqrt{6}} ( z_1 + z_5 - 2 z_9 ). \end{equation}

Thus $Q \in \hbox{Sym}^3_0$ is equivalent to

\begin{equation*} y_4 = y_7 = y_8 = y_9 = 0, \end{equation*}

i.e. $\mathcal{J} = \{1, 2, 3, 5, 6 \}$.

The onsite energy f_B of (2.2) in terms of y_j, $j \in \cal J$, is then computed from

(3.15)\begin{equation} \hbox{tr}(Q^2) = y_1^2 + y_2^2 + y_3^2 + y_5^2 + y_6^2, \end{equation}

(3.16)\begin{equation} \hbox{tr}(Q^3) = \frac{3}{\sqrt{6}} y_2^2 y_5 + \frac{3}{\sqrt{2}} y_2 y_3 y_6 + \frac{3}{2\sqrt{2}} (y_3^2 y_1 - y_6^2 y_1 ) - \frac{3}{2\sqrt{6}} (y_3^2 y_5 + y_6^2 y_5) - \frac{1}{\sqrt{6}} y^3_5 + \frac{3}{\sqrt{6}} y_1^2 y_5. \end{equation}

The critical sets $\Sigma^\pm$ of f_B can be also parametrized by polar coordinates. Using

(3.17)\begin{equation} {\hat n}(\theta,\phi) = [\sin \theta \cos \phi, \sin \theta \sin \phi,\cos \theta]^T, \quad (\theta, \phi) \in (0,\pi)\times [0,2\pi), \end{equation}

for unit vectors in $\mathbb{R}^3$, $\hat n \in S^2\setminus\{{\hat e}_1,-{\hat e}_1\} $, the corresponding Q-tensors are

(3.18)\begin{equation} Q_\pm({\hat n}(\theta,\phi)) = - 3 x_\pm \left[ \begin{array}{ccc} \sin^2 \theta \cos^2 \phi - \frac{1}{3} & \frac{1}{2} \sin^2 \theta \sin 2 \phi & \frac{1}{2} \sin 2 \theta \cos \phi \\ \frac{1}{2} \sin^2 \theta \sin 2 \phi & \sin^2 \theta \sin^2\phi - \frac{1}{3} & \frac{1}{2} \sin 2\theta \sin \phi \\ \frac{1}{2} \sin 2 \theta \cos \phi & \frac{1}{2} \sin 2\theta \sin \phi & \cos^2 \theta - \frac{1}{3} \end{array} \right], \end{equation}

see (2.8), (2.9). By (3.10) and (3.11)-(3.14) their coordinates in the variables y_j, $j \in \cal J$, are

\begin{equation*} \nonumber y_1^\pm = - \frac{3 x_\pm}{\sqrt{2}} \sin^2 \theta (\cos^2 \phi - \sin^2 \phi), \quad y_2^\pm = - \frac{3 x_\pm}{\sqrt{2}}\sin^2 \theta \sin 2 \phi, \quad y_3^\pm = - \frac{3 x_\pm}{\sqrt{2}}\sin 2 \theta \cos \phi, \end{equation*}

(3.19)\begin{equation} y_5^\pm = - \frac{3 x_\pm}{\sqrt{6}}(\sin^2 \theta - 2 \cos^2 \theta), \quad y_6^\pm = - \frac{3 x_\pm}{\sqrt{2}}\sin 2 \theta \sin \phi. \end{equation}

These expressions can be used to produce local coordinates for $\mathbb{R}\mathrm{P}^2$ and to describe its embedding to $\hbox{Sym}^3_0$, details are given below.

In the remaining part of this subsection, we construct coordinates for a tubular neighbourhood of the critical sets $\Sigma^3_\pm$ of f_B. The results and notation are used in the proofs of subsection 3.2. The construction uses the spectral decomposition of $\nabla^2 f_B$.

Let $ Q \in \hbox{Sym}^3_0$, then $ Q = \sum_{k=1}^5 w_j {\hat W}_j$, with

(3.20)\begin{equation} {\hat W}_1 = {\hat Y}_1, \quad {\hat W}_2 = {\hat Y}_2, \quad {\hat W}_3 = {\hat Y}_3, \quad {\hat W}_4 = {\hat Y}_5, \quad {\hat W}_5 = {\hat Y}_6. \end{equation}

Define ${\overline{\beta}}^w: \hbox{Sym}^3_0 \rightarrow \mathbb{R}^5$ by ${\overline{\beta}}^w(Q) = w :=[w_1, \ldots, w_5]^T$. The maps $\Psi_A$ are represented by multiplication by the matrix $T^w_A$, and $(T^w_A)_{i,j} = \langle [\Psi_A]^y {\hat W}_j , {\hat W}_i \rangle $, i, $j = 1, \ldots, 5$, i.e. $T^w_A$ coincides with T_A, assuming the enumeration of $\cal J$ in (3.20). The trace inner product $\langle \cdot, \cdot \rangle $ in $\hbox{Sym}^3_0$ is given by $\langle w, w' \rangle = \sum_{i=1}^5 w_i w'_i = \sum_{j \in \mathcal{J}}^5 y_j y'_j $. Lemma 3.4 implies that multiplication by $T^w_A$ is an isometry, $\forall A \in SO(3)$.

The application of the rotations $\Psi_A$, and their representation by orthogonal matrices to the analysis of the critical points of f_B is a consequence of the following observations.

Let $\mathcal{L} = f_B \circ {\overline \gamma}^w: \mathbb{R}^5 \rightarrow \mathbb{R}$, ${\overline \gamma}^w = ({\overline{\beta}}^w)^{-1}$, and $h:\mathbb{R}^5 \rightarrow \mathbb{R}^5$ defined by $h(q) = T^w_Aq$, $A \in SO(3)$. We have that q a critical point of $\cal L$ implies that h(q) is also a critical point of $\cal L$, moreover the Hessians of at q, h(q) are related by

(3.21)\begin{equation} (\nabla^2 \mathcal{L})(q) = [Dh(q)]^T [(\nabla^2 \mathcal{L} )(h(q))] [Dh(q)]. \end{equation}

Thus to check that $\Sigma^3_\pm$ are critical sets, it suffices to check that Df_B vanishes at any point of $\Sigma^3_\pm$. Also, since the Hessians of $\cal L$ at Q, $Q'= \Psi_A(Q) \in \Sigma^3_\pm$, $A \in SO(3)$, are related by (3.21) the spectrum of the Hessian is then same at all points of the two nontrivial critical sets of $\cal L$. Also, the fact that $T^w_A$ is an isometry implies that if v an eigenvector of $(\nabla^2 \mathcal{L})(q) $ then $Dh(q) v$ an eigenvector of $(\nabla^2 \mathcal{L})(h(q)) $, corresponding to the same eigenvalue. The computation of the spectral data of $\nabla^2 \mathcal{L}$ is summarized by the following.

The proof follows from computations of the Q-tensors (3.7), (3.8) obtained from ${\hat p} = {\hat n} = {\hat e}_1 := [1,0,0]^T $ respectively, see [Reference Panayotaros26]. By (3.17), (3.11)-(3.14) these critical points are

(3.22)\begin{equation} y_1 = - \frac{3}{\sqrt{2}} x_\pm, \quad y_5 = - \frac{3}{\sqrt{6}} x_\pm, \quad y_2 = y_3 = y_6 = 0. \end{equation}

The Hessian of $\mathcal{L} = f_B \circ {\overline \gamma}^w$ at this point is simplified and its eigenvalues and eigenvectors are computed explicitly.

We use the information of Lemma 3.5 to describe the sets $\Sigma^3_\pm $ as the image of embeddings of the projective plane to $\hbox{Sym}^3_0$, and to construct a normal sets of coordinates in the neighbourhood of the $\Sigma^3_\pm$.

The projective plane $\mathbb{R}\mathrm{P}^2$ is the set of points of the 2-sphere S ² with antipodal points identified, i.e. pairs $\{n,-n\}$, $n \in S^2$, with $\{n,-n\} = \{n',-n'\}$, n, $n' \in S^2$, if and only if $n = n'$ or −nʹ. Equivalently, $\mathbb{R}\mathrm{P}^2$ is the set of projectors $ {\hat \nu} \otimes {\hat \nu}$, ${\hat \nu} \in S^2$, see e.g. [Reference Lee17], p.18, for the topology in each description.

Consider the maps $Q_\pm:S^2 \rightarrow \hbox{Sym}_0^3 \subset \hbox{Mat}(3)$ defined in (2.8), (2.9). We have $Q_\pm(-{\hat \nu}) = Q_\pm({\hat \nu})$. Define ${\tilde Q}_\pm:\mathbb{R}\mathrm{P}^2 \rightarrow \hbox{Sym}_0^3$ by ${\tilde Q}_\pm(\{{\hat n}, -{\hat n} \}) := Q_\pm({\hat n})$, and let

(3.23)\begin{equation} {\Gamma}_\pm := {\overline{\beta}}^y \circ Q_\pm:S^2 \rightarrow \mathbb{R}^5, \quad {\tilde \Gamma}_\pm := {\overline{\beta}}^y \circ {\tilde Q}_\pm:\mathbb{R}\mathrm{P}^2 \rightarrow \mathbb{R}^5. \end{equation}

Then $\Sigma^3_\pm = {\tilde Q}_\pm(\mathbb{R}\mathrm{P}^2)$.

To study these maps locally we define a chart in a neighbourhood $\boldsymbol{\cal{U}}$ of $ {\hat e}_1 \in S^2$ and its antipodal points, and compose with rotations to cover $\mathbb{R}\mathrm{P}^2$. We further use this idea to construct tangential and normal coordinates for points in the neighbourhood of the critical sets $\Sigma^3_\pm \subset \hbox{Sym}^3_0$: we first construct normal coordinates for points in $\boldsymbol{\cal{U}}$ and then apply the matrices representing rotations to translate the normal basis vectors to all other critical points. Furthermore, our choice of tangential and normal coordinates for the neighbourhood of $\boldsymbol{\cal{U}}$ makes the Hessian of f_B diagonal at all points of the critical sets. This simplifies the computations of the next section.

Let $\boldsymbol{\cal{U}}$ be the set of unit vectors

(3.24)\begin{equation} {\hat n}(\theta,\phi) = [\sin \theta \cos \phi, \sin \theta \sin \phi,\cos \theta]^T \in \mathbb{R}^3, \quad (\theta, \phi) \in B, \end{equation}

where

(3.25)\begin{equation} (\theta, \phi) \in B = (\pi/2 - \Delta, \pi/2 + \Delta) \times (- \Delta,\Delta). \end{equation}

Choose Δ small enough so that $\boldsymbol{\cal{U}}$ does not contain antipodal points. The set $\boldsymbol{\cal{U}}$ is an open neighbourhood of ${\hat e}_1 = [1, 0 , 0]^T \in S^2$. The inverse $\psi_1 := {\hat n}^{-1}$ of $ {\hat n}:B \rightarrow \mathcal{U} $ is a coordinate chart for $\mathcal{U} \in S^2$. Given a pair of antipodal points $\{m, -m\}$, with m or $-m \in \mathcal{ U}$, we identify the unique point that belongs to $\boldsymbol{\cal{U}}$ and map it to B via ψ ₁. This defines a function (chart) from $ \tilde{\mathcal{U} } = \{\mathcal{U},-\mathcal{U}\}$ that we denote by ${\tilde \psi}_1 $.

We generate charts around other points of S ² and $\mathbb{R}\mathrm{P}^2$ using rotations of $\boldsymbol{\cal{U}}$, $\tilde{\mathcal{U}}$.

Letting $\hat \nu \in S^2$, we let $H^\pm(\nu)$ denote the open hemisphere of unit vectors $\hat \mu $ satisfying ${\hat \mu} \cdot {\hat \nu} \gt rless 0$. Consider a collection $\{A_p\}_{p \in \mathcal{K}}$ of matrices $A_p \in SO(3)$, $\forall p \in \mathcal{K}$, where $\mathcal{K} \subset \mathbb{N}$ is an index set with the properties $1 \in \mathcal{K}$, $A_1 = I$, the identity matrix, and that the sets $ \{A_p \mathcal{U}, - A_p \mathcal{U} \}$, $p \in \mathcal{K}$, form a cover of of $\mathbb{R}\mathrm{P}^2$. The set $\cal K$ can be assumed finite by compactness. We use the notation $\mathcal{U}_p = A_p \mathcal{U}$.

Given $p \in \mathcal{K}$, and the rotation matrix A_p, define $ {\hat n}_p: B \rightarrow \mathcal{U}_p$ by ${\hat n}_p = A_p {\hat n} $, and let $\psi_p = {\hat n}^{-1}_p:\mathcal{U}_p \rightarrow B$. ψ_p is a coordinate chart on $\mathcal{U}_p \subset S^2$. The ${\hat n}_p$ are real analytic and injective in B, therefore the ψ_p are also real analytic in $\mathcal{U}_p$, $\forall p \in \mathcal{K}$. Given the set $\tilde{\mathcal{U}}_p = \{\mathcal{U}_p, - \mathcal{U}_p\}$ define the chart ${\tilde \psi}_p:\tilde{\mathcal{U}}_p \rightarrow B$ by letting ${\tilde \psi}_p(\{m,-m\}) = \psi_p(m)$ if $m \in H^+({A}_p {\hat e}_1)$, and ${\tilde \psi}_p(\{m,-m\}) = \psi_p(-m)$ if $-m \in H^+({A}_p {\hat e}_1)$. Given $m \in \tilde{\mathcal{U}}_p$, $p \in \mathcal{K}$, we use the notation $(\theta_p, \phi_p) = (\theta_p(m), \phi_p(m)) = ({\tilde \psi}_{p,1}(m),{\tilde \psi}_{p,2}(m)) = ({\tilde \psi}_{p,1},{\tilde \psi}_{p,2})$ for the two components of the chart ${\tilde \psi}_p$, i.e. we use the same symbol for the charts and the variables. For p = 1 we have $(\theta_1, \phi_1) = (\theta,\phi)$.

To see the transition functions consider $\tilde{\mathcal{U}}_p $, $\tilde{\mathcal{U}}_p' \subset \mathbb{R}\mathrm{P}^2$, p, $p' \in \mathcal{K}$ with nonempty intersection. Then either $\mathcal{U}_p \cap \mathcal{U}_{p'} \neq \emptyset$ or $\mathcal{U}_p \cap (-\mathcal{ U}_{p'}) \neq \emptyset$ (and not both). Let $\mathcal{V}_{p'}$ be the element of the pair $\mathcal{U}_{p'}$, $-\mathcal{U}_{p'}$ that intersects $\mathcal{U}_p$. Then $\mathcal{V}_p = A'_{p'} \mathcal{U}$, for some $A'_{p'} \in SO(3)$, and

(3.26)\begin{equation} {\hat n}(\theta_{p'},\phi_{p'}) = (A'_{p'})^{-1} A_{p} {\hat n}(\theta_{p},\phi_{p}). \end{equation}

The transition function ${\tilde \psi}_{p'} \circ {\tilde \psi}^{-1}_p: {\tilde \psi}_p( \tilde{\mathcal{U}}_p \cap \tilde{\mathcal{U}}_{p'} ) \rightarrow {\tilde \psi}_{p'}(\tilde{\mathcal{U}}_p \cap \tilde{\mathcal{U}}_{p'}) $ is the map from $ (\theta_{p},\phi_{p}) \in {\tilde \psi}_{p}(\tilde{\mathcal{U}}_p \cap \tilde{\mathcal{U}}_{p'}) \subset B$ to the unique $(\theta_{p'},\phi_{p'}) \in {\tilde \psi}_{p'}(\tilde{\mathcal{U}}_p \cap \tilde{\mathcal{U}}_{p'}) \subset B$ satisfying (3.26). Real analyticity of the transition function follows from the fact that the right hand side of (3.26), and ${\hat n}^{-1}$ are real analytic.

Remark 3.6. By (3.24) we have $ {\hat n}(\theta, \phi) = A_\phi A_\theta {\hat e}_1$, and $\mathcal{U} $ is the set of $A {\hat e}_1$, $ A = A_\phi A_\theta$, where $(\theta, \phi) \in B$,

(3.27)\begin{equation} A_\phi = \left[ \begin{array}{ccc} \cos \phi & - \sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{array} \right], \quad A_\theta = \left[ \begin{array}{ccc} \sin \theta & 0 & - \cos \theta \\ 0 & 1 & 0 \\ \cos \theta & 0 & \sin \theta \end{array} \right]. \end{equation}

Part of the statement of Proposition 2.1 is that the ${\tilde \Gamma}_\pm$ are embeddings, see [Reference Panayotaros26]. We can then define tubular neighbourhoods of their image ${\overline{\beta}}^y (\Sigma_\pm^3)$ using the atlas $\{\tilde{\mathcal{U}}_p, {\tilde \psi}_p \}_{p \in \mathcal{K}} $ for the component along the ${\overline\beta}^y (\Sigma_\pm^3)$, and coordinates along normal directions, see [Reference Lee17], ch. 3, [Reference Burns and Gidea6], section 6.2.

In particular, given any $p \in \mathcal{K}$, and the open set $\tilde{\mathcal{U}}_p \in \mathbb{R}\mathrm{P}^2$, there exists an open neighbourhood of ${\tilde \Gamma}_\pm(\tilde{\mathcal{U}}_p)$ in $\mathbb{R}^5$ that consists of points

(3.28)\begin{equation} y^\pm({\tilde \psi}_p,r_p) = {\tilde \Gamma}_\pm(\theta_p,\phi_p) + \sum_{j = 1}^3 r_{p,j} {\hat N}^\pm_{p,j}(\theta_p,\phi_p), \quad {\tilde \psi}_p = (\theta_p,\phi_p) \in B, \quad r \in I_\delta^3, \end{equation}

$I_\delta = (-\delta,\delta)$, δ > 0, ${\tilde \psi}_p = (\theta_p,\phi_p)$, $r_p = (r_{p,1}, r_{p,2}, r_{p,3})$, B as in (3.25), (3.24), where the ${\hat N}_{p,j}^\pm:\tilde{\mathcal{U}}_p \rightarrow S^5$, $j = 1, 2, 3$ are smooth unit vector fields that are normal to $\tilde \Gamma_\pm( \tilde{\mathcal{U}}_p)$ with respect to the Euclidean inner product in $\mathbb{R}^5$.

The local normal vector fields ${\hat N}^\pm_{p,j}$ of (3.28) are not unique. We choose normal vectors that are eigenvectors of $\nabla^2 f_B$, see Lemma 3.9 below. This choice simplifies the proof of Lemma 3.11 in the next section, see [Reference Panayotaros26] for details.

Define first local normal vector fields ${N}^\pm_{1,j}:{\mathcal{U}} \rightarrow S^5$, j = 1, 2, 3: let ${N}^\pm_1 (\pi/2,0) = ||\Gamma_\pm( {\hat e}_1)||^{-1} \Gamma_\pm( {\hat e}_1)$, and choose two unit vectors ${N}^\pm_j (\pi/2,0) $, j = 2, 3, so that ${N}^\pm_j (\pi/2,0) $, j = 1, 2, 3 form a orthonormal basis of the orthogonal complement of the tangent space of ${\Gamma}_\pm( \mathcal{U}_p)$ at at $\Gamma_\pm({\hat e}_1)$. (The choice of the ${N}^\pm_j (\pi/2,0) $, j = 2, 3 is not unique.) Also let

(3.29)\begin{equation} {N}^\pm_{1,j}(\theta,\phi) = T_{A_\theta A_\phi} {N}^\pm_j (\pi/2,0), \quad j = 1,2,3, \quad (\theta,\phi) \in B. \end{equation}

We then define the ${\hat N}^\pm_{p,j}:\tilde{\mathcal{U}}_p \rightarrow S^5$ by

(3.30)\begin{equation} {\hat N}^\pm_{p,j} (\{\psi_p, - \psi_p \}) = T_{A_p} {N}^\pm_{1,j}({\psi}_1), \quad \quad {\tilde \psi}_1 \in B, \quad j = 1, 2, 3, \quad p \in \mathcal{K}. \end{equation}

By Lemma 3.5, and Remark 3.6 the ${\hat N}^\pm_j(p,m)$ are real analytic vector fields and form an orthonormal set of eigenvectors of $\nabla^2 f_B$ at $m \in \tilde{\mathcal{U}}_p$ corresponding to $\lambda^\pm_j$, $j = 1, 2, 3$, respectively, for all $m \in \tilde{\mathcal{U}}_p$, $p \in \mathcal{K}$.

Recall that the matrices T_A, $A \in SO(3)$, represent rotations in the basis y and leave the image of $ {\overline{\beta}}^y (\hbox{Sym}^3_0) $ invariant. The following show that the particular choice of normal coordinates for $\boldsymbol{\cal{U}}$, and its extension to the whole critical set by rotations makes the Hessian of f_B diagonal everywhere in the critical set.

Remark 3.7. Let $\Omega^\pm_p = ({\tilde \psi}_p,r^\pm_{p}) \in B \times I_\delta^3$, where B, I_δ are defined in (3.25)-(3.24), and (3.28) respectively. Then (3.28)-(3.29) imply

(3.31)\begin{equation} D_{\Omega_p} y^\pm {(\tilde \psi}_p,0) = [ t^\pm_{\theta_p}, t^\pm_{\phi_p}, {\hat N}^\pm_{p,1}, {\hat N}^\pm_{p,2}, {\hat N}^\pm_{p,3}]({\tilde \psi}_p,0), \end{equation}

where $ t^\pm_{\theta_p}({\tilde \psi}_p,0)$, $t^\pm_{\phi_p}({\tilde \psi}_p,0)$ are tangent to the critical set at $({\tilde \psi}_p,0)$, and the ${\hat N}^\pm_j({\tilde \psi}_p,0)$ are normal to the critical set at $({\tilde \psi}_p,0)$. Thus $ D_{\Omega_p} y^\pm {(\tilde \psi}_p,0) $ is invertible and it follows that the map defined by (3.28) is an analytic diffeomorphism from $B \times I_\delta^3$ to an open subset of $\mathbb{R}^5$ that intersects ${\tilde \Gamma}(\tilde{\mathcal{U}}_p)$.

Lemma 3.9. Let $m \in \tilde{\mathcal{U}}_p \subset \mathbb{R}\mathrm{P}^2$, $p \in \mathcal{K}$. Define the coordinates $\Omega^\pm_p = ({\tilde \psi}_p,r^\pm_{p}) \in B \times I_\delta^3$, by (3.28) in the neighbuorhoods $y^\pm(B \times I_\delta^3)$ of ${\tilde \Gamma}_\pm(m)$, and let ${\tilde f}_B(\Omega_p^\pm) = f_B(y^\pm(\Omega_p^\pm))$. Then, (i) $\nabla^2_{\Omega_p^\pm} {\tilde f_B}{(\tilde \psi}_p,0)$ is diagonal with

(3.32)\begin{equation} \frac{\partial^2 {\tilde f}_B}{\partial {\tilde \psi}^2_{p,j}}({\tilde \psi}_p,0) = 0, \quad j = 1, 2; \quad \frac{\partial^2 {\tilde f}_B}{\partial r^{\pm 2}_{p,j}}({\tilde \psi}_p,0) = \lambda^\pm_j, \quad j = 1, 2, 3, \end{equation}

$\forall {\tilde \psi}_p \in B$. Also, (ii) $ \frac{\partial {\tilde f}_B}{\partial {\tilde \psi}_{p,j}}(\Omega_p^\pm ) = 0$, $j = 1, 2$, $\forall \Omega_p^\pm \in B \times I_\delta^3$.

3.2. Reduced equation and continuation of solutions

In this section, we outline the proofs of Propositions 2.3, 2.4, and 2.8. The main step is Lemma 3.11 below on the equivalence of the equation for critical points of $\overline{\mathcal{F}}$ for small $|L|$ to a reduced equation for the “angular variables” parametrizing the L = 0 critical set. In the case $U_- = \overline{G}$ the bifurcation equation is a discrete version of the equation for Oseen–Frank equilibria.

To set up Lemma 3.11, we parametrize a neighbourhood of the critical sets $\mathcal{C}_0(\overline{\mathcal{F}_4},U_+,U_-)$ of $\overline{\mathcal{F}}_4$ in $X({\overline{G}})$. We let $U = U_+ \cup U_-$, and $U^c = G\setminus U$. U is assumed nonempty.

The set $X({\overline{G}})$ is a product of the linear spaces $\hbox{Sym}^3_0$, and we use the coordinates y_j (and w_j) of the previous section at each site: $Q(k) \in \hbox{Sym}_0^3$, $k \in U \cup U^c$, is written as $ Q(k) = \sum_{k \in U \cup U^c} \sum_{j \in \mathcal{J}} y_{k,j} {\hat Y}_j = \sum_{k \in U\cup U^c} \sum_{i = 1}^5 w_{k,i} {\hat W}_i $, see (3.20). This defines an isomorphism (or chart) $\overline{\mathcal{B}}^w: (\hbox{Sym}_0^3)^{|U|+|U^c| } \rightarrow \mathbb{R}^{5( |U| +|U^c|)}$, where $(\overline{\mathcal{B}}^w(Q))(k) = w(k) := [w_{k,1}, \ldots, w_{k,5}]^T$. (An alternative notation is $\overline{\mathcal{B}}^y$.) The Riemannian metric on $\mathbb{R}^{5( |U| +|U^c|)}$ is the one induced by the standard Euclidean inner product, as in subsection 3.2. This metric is invariant under multiplication by the $5 \times 5$ matrices $T_{A_k}$, $A_k \in SO(3)$, applied (independently) at each site $k \in U \cup U^c$.

Also, the $\mathcal{C}_0(\overline{\mathcal{F}_4},U_+,U_-)$ of $\overline{\mathcal{F}}_4$ in $X({\overline{G}})$ are products of projective planes, and their neighbourhood can be described using products of the coordinate systems of the previous section at the sites of U. The tangential directions at each site of U parametrize the critical sets.

The proof follows by defining a map $\boldsymbol{\tilde{\Gamma}}:(\mathbb{R}\mathrm{P}^2)^{|U|} \rightarrow X({\overline G})$ that is an embedding and an immersion, see [Reference Panayotaros26]. Consider first the case where $U_+$, $U_-$, U^c are nonempty. We let

(3.33)\begin{equation} \boldsymbol{\tilde{\Gamma}}:(\mathbb{R}\mathrm{P}^2)^{|U_+|} \times (\mathbb{R}\mathrm{P}^2)^{|U_-|} \rightarrow \mathbb{R}^{5|U_+|} \times \mathbb{R}^{5|U_-|} \times \mathbb{R}^{5|U^c|}, \end{equation}

by

(3.34)\begin{equation} \boldsymbol{\tilde{\Gamma}}_k(m_+,m_-) = \begin{cases} {\tilde \Gamma}_-(m_{+,k}), & k \in U_+, \\ {\tilde \Gamma}_+(m_{-,k}), & k \in U_-, \\ 0, & k \in U^c, \end{cases} \end{equation}

where $m_\pm = \{m_{\pm,k}\}_{k \in U_\pm}$, and ${\tilde \Gamma_\pm}$ is as in (3.23). In the case where one of $U_+$, $U_-$, U^c is empty we delete the corresponding line from (3.34).

Assume that $U_+$, $U_-$, U^c are nonempty. We then define a tubular neighbourhood $\cal T$ of the sets $ \boldsymbol{\tilde{\Gamma}}(\mathcal{C}_0(\overline{\mathcal{F}}_4,U_+,U_-))$ in $\mathbb{R}^{5|G|}$ as the union over $\textbf{p} \in \mathcal{K}^{|U|}$ of the images of the maps $\textbf{y}_\textbf{p}: \tilde{\mathcal{U}}_\textbf{p} \times I_\delta^{3|U|} \times I_\delta^{5 |U^c|} \rightarrow \mathbb{R}^{5(|U|+|U^c|)}$ with components

(3.35)\begin{equation} \textbf{y}_{k,\textbf{p}}({\tilde \psi}_\textbf{p},r_\textbf{p},q_\textbf{p}) = \begin{cases} {\tilde \Gamma}_+({\tilde \psi}_{p_k}) + \sum_{i = 1}^3 r_{p_k, i} {\hat N}^+_{p_k, i}({\tilde \psi}_{p_k}), & k \in U_+, \\ {\tilde \Gamma}_-({\tilde \psi}_{p_k}) + \sum_{i = 1}^3 r_{p_k, i} {\hat N}^-_{p_k, i}({\tilde \psi}_{p_k}), & k \in U_-, \\ q_{p_k}, & k \in U^c, \end{cases} \end{equation}

with δ > 0, $ r_{p_k,i} \in I_\delta$, for all $k \in U$, $p_k \in \mathcal{K}$, $i = 1, 2, 3$, and $q_{p_k} \in I_\delta^5$, for all $k \in U^c$. The ${\hat N}^\pm_{p_k,i}({\tilde \psi}_{p_j})$, ${\tilde \psi}_{p_k} \in \tilde{\mathcal{U}}_{p_k}$, $p_k \in \mathcal{K}$, $k \in U$, are the normal vectors defined in the previous section. They are therefore eigenvectors of the Hessian of f_B with respect to the variables w(k) at $ {\tilde f}_\pm({\tilde \psi}_{p_k})$, for all $k \in U$. By Remark 3.8 the transition function between $r_{p_k}$, $r_{p'_k}$ is the identity, for any p_k, $p'_k \in \mathcal{K}$ with $\tilde{\mathcal{U}_{p_k}} \cap \tilde{\mathcal{U}}_{p'_k} \neq \emptyset$. Also $q_{p_k} = \{q_{p_k,i}\}_{i=1}^5 = \{w_{k,i}\}_{i=1}^5 $, i.e. “cartesian coordinates” w_i, see (3.20), at each site of U^c. Transition functions for these components are the identity.

By Remark 3.7, expressions (3.35) with δ > 0 sufficiently small define implicitly local coordinates for $\cal T$, in particular, $\cal T$ is a union of neighbourhoods of $\boldsymbol{\tilde{\Gamma}}(\tilde{\mathcal{U}}_\textbf{p})$, where we have the “polar-cartesian” coordinates

(3.36)\begin{equation} {\tilde \psi}_{p_k} \in \tilde{\mathcal{U}}_{p_k}, \quad r_{p_k} \in I_\delta^3, \quad k \in U, \quad p_k \in \mathcal{K}, \end{equation}

(3.37)\begin{equation} q_{p_k} \in I_\delta^5, \quad k \in U^c. \end{equation}

In the case where one of U_j is empty, we delete the first ( $j = +$) or second ( $j=-$) line of (3.35), and (3.36). In the case where U^c is empty we delete the last line of (3.35), and (3.37).

We consider critical points of $\overline{\mathcal{F}}$ in the neighbourhood $\cal T $ of the L = 0 nontrivial critical sets.

Consider first the case where U^c is nonempty. We let

(3.38)\begin{align} g_{\textbf{p},i}(k) & = \frac{\partial \overline{\mathcal{F}} }{\partial {{\tilde \psi}_{{p}_k,i}} }, \quad i = 1,2, \quad k \in U, \end{align}

(3.39)\begin{align} h_{\textbf{p},i}(k) & = \frac{\partial \overline{\mathcal{F}}}{\partial r_{{p}_{k},i}} , \quad i = 1,2,3, \quad k \in U, \end{align}

(3.40)\begin{align} w_{\textbf{p},i}(k) & = \frac{\partial \overline{\mathcal{F}} }{\partial q_{{p}_{k}, i} }, \quad \quad i = 1,\ldots,5, \quad k \in U^c. \end{align}

These are the components of the derivative $D \overline{\mathcal{F}}$ in local coordinates $ {\tilde \psi}_\textbf{p}$, $r_\textbf{p}$, $q_\textbf{p}$, $\textbf{p} \in \mathcal{K}^{|U|}$.

Denoting the corresponding vectors by $g_\textbf{p} \in \mathbb{R}^{2|U|}$, $h_\textbf{p} \in \mathbb{R}^{3|U|}$, $ w_\textbf{p} \in \mathbb{R}^{5|U^c|}$, and letting $ z_\textbf{p} = [r_\textbf{p},q_\textbf{p}] \in I_\delta^{3|U|} \times {I_\delta}^{5|U^c|}$, we also define

(3.41)\begin{equation} F_{{\tilde \psi}_\textbf{p }}({\tilde \psi}_\textbf{p},z_\textbf{p},L) = g_\textbf{p}({\tilde \psi}_\textbf{p},z_\textbf{p},L), \quad F_{z_\textbf{p}}({\tilde \psi}_\textbf{p},z_\textbf{p},L) = [h_\textbf{p}({\tilde \psi}_\textbf{p},z_\textbf{p},L), w_\textbf{p}({\tilde \psi}_\textbf{p},z_\textbf{p},L)]. \end{equation}

Then critical points of $\overline{\mathcal{F}}$ in the vicinity of $\mathcal{C}_0(\overline{\mathcal{F}}_4,U_+,U_-)\cap \tilde{\mathcal{U}}_\textbf{p}$ are solutions of

(3.42)\begin{equation} F_{\psi_\textbf{p}}({\tilde \psi}_\textbf{p},z_\textbf{p},L) = 0, \quad F_{z_\textbf{p}}({\tilde \psi}_\textbf{p},z_\textbf{p},L) = 0. \end{equation}

This follows from the fact that the polar-cartesian coordinate system is nonsingular in the neighbourhood of the sets $\mathcal{C}_0(\overline{\mathcal{F}}_4,U_+,U_-) \cap \tilde{\mathcal{U}}_\textbf{p}$.

The notation above also applies to the case where one of U ₁ or U ₂ is empty. In the case where U^c is empty, the $q_\textbf{p}$, and $w_\textbf{p}$ are not defined, and (3.40) is deleted. Also we use (3.41), (3.42) with $ z_\textbf{p} = r_\textbf{p } \in I_\delta^{3|U|}$.

Lemma 3.11. Let $\overline{\mathcal{F}} = L \overline{\mathcal{F}}_2 + \overline{\mathcal{F}}_4$ be defined in a tubular neighbuorhood $\cal T$ of $\mathcal{C}_0(\overline{\mathcal{F}}_4,U_+,U_-)$ as above. (i) There exists $L_0 \gt 0$ such that for every $\textbf{p} \in \mathcal{K}^{|U|}$, ${\tilde \psi}_\textbf{p } \in \tilde{\mathcal{ U}}_\textbf{p}$, and $L \in (-L_0,L_0)$ there exists a unique $\zeta_\textbf{p}({\tilde \psi}_\textbf{p}, L) \in I_\delta^{3 |U|} \times I_\delta^{5|U^c|}$ satisfying

(3.43)\begin{equation} F_{z_\textbf{p}}({\tilde \psi}_\textbf{p},\zeta_\textbf{p}({\tilde \psi}_\textbf{p},L),L) = 0, \end{equation}

and $\zeta_\textbf{p}({\tilde \psi}_\textbf{p},0) = z_{\textbf{p},0} = [0,0]$. The solution $\zeta_\textbf{p}({\tilde \psi}_\textbf{p},L)$ is real analytic in ${\tilde \psi}_\textbf{p}$, $\forall {\tilde \psi}_\textbf{p} \in \tilde{\mathcal{U}}_\textbf{p}$, and in L, $\forall L \in (-L_0,L_0)$. (ii) For $L \in (-L_0,L_0)$, solutions of the first equation of (3.42) are of the form ${\tilde \psi}_\textbf{p} = \chi$, $ z_\textbf{p} = \zeta_\textbf{p}(\chi)$, with $\zeta_\textbf{p}(\cdot)$ as in (i), and χ solutions of the equation

(3.44)\begin{equation} \mathbf{g}_{\mathbf{p}}(\chi,L) = 0, \quad \chi \in \tilde{\mathcal{U}}_\textbf{p}, \end{equation}

where

(3.45)\begin{equation} \mathbf{g}_{\mathbf{p}}({\tilde \psi}_\textbf{p},L) = D_{{\tilde \psi}_\textbf{p}} {\mathcal{G} }({\tilde \psi}_\textbf{p},L), \quad {\mathcal{G}}( {\tilde \psi}_\textbf{p},L ) := \overline{\mathcal{F}}({\tilde \psi}_\textbf{p},\zeta_\textbf{p}({\tilde \psi}_\textbf{p},L),L), \end{equation}

$\forall {\tilde \psi}_\textbf{p} \in \tilde{\mathcal{U}}_\textbf{p}$, i.e. ${\mathcal{G}}$ is the restriction of $\overline{\mathcal{F}}$ to the graph of $\zeta_\textbf{p}$, $ \forall {\tilde \psi}_\textbf{p} \in \tilde{\mathcal{U}}_\textbf{p} $, $\textbf{p} \in \mathcal{K}^{|U|}$. (iii) Let $L \in (-L_0,L_0)$, then (i), (ii) define ζ, $\bf g$, ${\mathcal{G}}$ globally on $\mathcal{M} = (\mathbb{R}\mathrm{P}^2)^{|U|}$ by $ \zeta|_{\mathcal{U}_\textbf{p}} = \zeta_\textbf{p}$, $\textbf{g}|_{\mathcal{U}_\textbf{p}} = \textbf{g}_\textbf{p}$, ${\mathcal{G}}|_{\mathcal{U}_\textbf{p}} = {\mathcal{G}}_\textbf{p}$, for all $\textbf{p} \in \mathcal{K}^{|U|} $.

Lemma 3.11 reduces the problem of finding equilibria of the Landau–de Gennes system for L sufficiently near the origin to that of finding critical points of $\cal G$ on $(\mathbb{R}\mathrm{P}^2)^{|U|}$. The proof is based on the implicit function theorem [Reference Buffoni and Tolland5] and is omitted as it is exactly as in the case with boundary, see Lemma 4.2, [Reference Panayotaros26]. The system of coordinates defined above using the symmetry of f_B makes the computation of the derivative straightforward.

A consequence of Lemma 3.11 is that given a solution $\chi \in \tilde{\mathcal{U}}_\textbf{p}$ of the reduced equation, the components $r_\textbf{p}$, $q_\textbf{p}$ of the equilibrium are given by $\zeta_\textbf{p}(\chi,L)$. Necessary continuation conditions are obtained by the reduced equation at L = 0.

Proof of Proposition 2.3

By Lemma 3.11, for L sufficiently near 0 it is sufficient to study solutions of the reduced equation (3.45)

\begin{equation*} \textbf{g}_\textbf{p}({\tilde \psi}_\textbf{p},L) = D_{{\tilde \psi}_\textbf{p}} \overline{\mathcal{F}} ({\tilde \psi}_\textbf{p},\zeta_\textbf{p}({\tilde \psi}_\textbf{p},L),L) = D_1 \overline{\mathcal{F}}({\tilde \psi}_\textbf{p},\zeta_\textbf{p}({\tilde \psi}_\textbf{p},L),L) = 0, \end{equation*}

for some $\textbf{p} \in \mathcal{K}^{|U|}$. The functions $\zeta_\textbf{p}$ and $\overline{\mathcal{F}}$ are real analytic and we can expand $\textbf{g}_\textbf{p} $ in powers of L as

(3.46)\begin{equation} \textbf{g}_\textbf{p}(\psi,L) = \sum_{k=1}^{\infty} L^k \textbf{g}_{\textbf{p}, k}(\psi), \quad |L| \lt L_0. \end{equation}

Moreover $D_1 \overline{\mathcal{F}}_4$ vanishes identically by Lemma 3.9 (ii). The lowest order term in (3.46) is then

(3.47)\begin{equation} \textbf{g}_{\textbf{p}, 1} ({\tilde \psi}_\textbf{p}) = D_{{\tilde \psi}_\textbf{p}} \overline{\mathcal{F}}_2 ({\tilde \psi}_\textbf{p},\zeta_\textbf{p}({\tilde \psi}_\textbf{p},0)) = D_{\tilde \psi_\textbf{p}} \overline{\mathcal{F}}_2 ({\tilde \psi}_\textbf{p},0). \end{equation}

By the hypothesis of Proposition 2.3, and Lemma 3.11, we have a branch ${\tilde \Theta}(L)$, $L \in[0,\epsilon_0)$, $\epsilon_0 \leq L_0$, of solutions of $ \textbf{g}_\textbf{p}({\tilde \Theta}(L) ,L) = 0 $, and therefore

(3.48)\begin{equation} L^{-1} \textbf{g}_\textbf{p}({\tilde \Theta}(L) ,L) = 0, \quad \forall L \in (0, \epsilon_0). \end{equation}

Taking the limit $L \rightarrow 0^+$, and using (3.46), (3.47), we have that $\Theta = {\tilde \Theta}(0)$ satisfies

(3.49)\begin{equation} \textbf{g}_{\textbf{p}, 1} (\Theta) = D_{{\tilde \psi}_\textbf{p}} \overline{\mathcal{F}}_2 (\Theta, 0) = 0, \end{equation}

as stated.

The main feature of the continuation problem for graphs without boundary is that the reduced equations for $\overline{\mathcal{F}}$ have a global SO(3) symmetry. This implies that solutions of the reduced equations are in general not isolated points.

To see how the continuation argument is modified, fix $A \in SO(3)$ and define $\Psi_A^d: (\hbox{Sym}_0^3)^{|\overline G|} \rightarrow (\hbox{Sym}_0^3)^{|\overline G | } $ by

(3.50)\begin{equation} \Psi_A^d ({\overline Q})(m) = \Psi_A ({\overline Q}(m)), \quad m \in {\overline G}. \end{equation}

By (2.6) and Lemmas 3.2, 3.4 $\overline{\mathcal{F}}$ is invariant under $\Psi_A^d$, for all $ A \in SO(3)$ and L real.

Also define $T_A^d: \mathbb{R}^{5|\overline G|} \rightarrow \mathbb{R}^{5|\overline G|}$ by

(3.51)\begin{equation} (T_A^d y)(m) = T_A y (m), \quad m \in {\overline G}. \end{equation}

The manifold $\cal M$ is invariant under $T_A^d$, $\forall A \in SO(3)$. Also define $T^d:SO(3)\times \mathbb{R}^{5|\overline G|} \rightarrow \mathbb{R}^{5|\overline G|}$ by $T^d(A,y) = T^dy$. We use the polar-cartesian coordinates above in the neighbuorhood of $\mathcal{M} \subset \mathbb{R}^{5|\overline G|} $, and denote the angular, and polar components of $ T_A^d $ in the neighbuorhood of some $\tilde{\mathcal{U}}_\textbf{p}$ by $\mathcal{ R}^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p}, z_\textbf{p})$, $Z^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p}, z_\textbf{p})$ respectively. By Remark 3.8, $\mathcal{R}^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p}, z_\textbf{p})$ coincides with the restriction of $T^d_A$ to $\cal M$, and depends only on the angular part. We thus write $\mathcal{R}^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p}, z_\textbf{p}) = \mathcal{ R}^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p})$. Also let $\mathcal{R}_A^d$ denote the restriction of $T^d_A$ to $\cal M$, and denote the corresponding action of SO(3) on $\cal M$ by $\mathcal{R}^d$.

Proof. To see (i) fix $L \in (-\epsilon_0,\epsilon_0)$ and let $ \zeta_\textbf{p}({\tilde \psi}_\textbf{p}) = \zeta_\textbf{p}({\tilde \psi}_\textbf{p},L) $, $\textbf{p} \in \mathcal{K}^{|U|}$. Assume first that A is such that $T_A^d$ maps a critical point $m \in \tilde{\mathcal{U}}_\textbf{p}$ to a point in the same domain. It suffices to show the local version of equivariance, namely

(3.52)\begin{equation} Z^d_{A,\textbf{p}}( {\tilde \psi}_\textbf{p}, \zeta_\textbf{p}( {\tilde \psi}_\textbf{p})) = \zeta_\textbf{p}( \mathcal{R}^d_{A,\textbf{p}}( {\tilde \psi}_\textbf{p}) ). \end{equation}

Note that $z = \zeta( \mathcal{R}^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p} ) ) $ is a solution of (3.43). Invariance of $ \overline{\mathcal{F}}$ under $\mathcal{R}_A^d$ implies that the image of a critical point of $ \overline{\mathcal{F}}$ under $\mathcal{R}_A^d$ is also a critical point of $\overline{\mathcal{F}} $, therefore $z = Z^d_{A,\textbf{p}}({\tilde \psi}_\textbf{p})$ also satisfies (3.43). By Lemma 3.11 the solution of (3.43) is unique and the two sides of (3.52) must coincide. In the case of arbitrary $A \in SO(3)$ we write A as a finite product of elements of SO(3) satisfying the above assumption.

To show (ii) we similarly work with compositions of $T_A^d$, restricted to $\cal M$, that map a subset of $\tilde{\mathcal{U}}_\textbf{p}$ to a subset of $\tilde{\mathcal{U}}_\textbf{p}$. We have

\begin{equation*}\tilde{\mathcal{G} } ( \mathcal{R}^d_A ( {\tilde \psi}_\textbf{p} ) )= \overline{\mathcal{F} } (\mathcal{R}^d_A({\tilde \psi}_\textbf{p} ), \zeta_\textbf{p} ( \mathcal{ R}_A^d ({\tilde \psi}_\textbf{p} ) ) = \overline{\mathcal{F}} (\mathcal{R}^d_A({\tilde \psi}_\textbf{p} ), Z^d_\textbf{p} (\psi_\textbf{p}, \zeta_\textbf{p} ({\tilde \psi}_\textbf{p})) = \overline{\mathcal{F} } ( {\tilde \psi}_\textbf{p}, \zeta_\textbf{p} ({\tilde \psi}_\textbf{p} ) ), \end{equation*}

using (i) and the invariance of $ \overline{\mathcal{F}}$ under $T_A^d$. Thus ${\mathcal{G} } ( \mathcal{R}^d_A ({\tilde \psi}_\textbf{p}) )= {\mathcal{G}} ( {\tilde \psi}_\textbf{p}) $.

To see (iii), we use a similar strategy using local coordinates in $ (\mathbb{R}\mathrm{P}^2)^{|U|}$ and a standard argument to see that the critical points $\tilde{\mathcal{G}}$ are mapped to critical points under $\mathcal{R}_A^d$ Furthermore $\mathcal{R}_A^d$ is a smooth $SO(3)-$action on $\mathcal{M} \approx (\mathbb{R}\mathrm{P}^2)^{|U|}$. Since SO(3) is compact, the orbit of every point $x \in \mathcal{M}$ under this action is a compact embedded submanifold $\mathcal{O}_{SO(3)} (x)$ of $\cal M$, see [Reference Lee18].

The manifolds of critical points of the reduced energy $ \mathcal{G}$ on $\mathcal{M} \approx (\mathbb{R}\mathrm{P}^2)^{d}$ can be classified by classifying the topology of orbits of the diagonal action $\mathcal{R}^d$ of SO(3) on $\mathcal{M} $, i.e. T^d of (3.51), restricted to $\cal M$. The basic fact from the theory of Lie group actions is that the orbit $\mathcal{O}(x)$ of $x \in \mathcal{M}$ under the action of SO(3) is the manifold $SO(3)/SO(3)_x$, where $SO(3)_x$ is the isotropy subgroup of x under the action, see the theorem of section 5.15 of [Reference Kolář, Michor and Slovák16]. In what follows we classify the isotropy subgroups.

By the definition of the action, the isotropy subgroup $SO(3)_x$ under $\mathcal{R}^d$ is the set of all $R \in SO(3)$ satisfying

(3.53)\begin{equation} R \; {\hat n} = \pm {\hat n}, \quad \forall {\hat n} \in \boldsymbol{\mathcal{V}}(x). \end{equation}

The set of such matrices is a subgroup of SO(3).

By (3.53) The action only affects the directions of the Q-tensors on $\Sigma_\pm$, thus a configuration on $\boldsymbol{\cal{M}}$ is determined by the directions $\{{\hat n}_j, - {\hat n}_j\}$, where j is either in $U_-$ or $U_+$. Denote by $\mathcal{V}(x)$, $x = \in (x_1, \ldots, x_{d}) \in (\mathbb{R}\mathrm{P}^2)^d$, the set distinct points of $\mathbb{R}\mathrm{P}^2$ in $\{x_1, \ldots, x_{d}\}$. Let ${\tilde d}(x)$ denote the number of values of $x \in (\mathbb{R}\mathrm{P}^2)^d$. Clearly $ 1 \leq {\tilde d}(x) \leq d$, $\forall x \in (\mathbb{R}\mathrm{P}^2)^d$. By (3.53) the isotropy subgroup of x depends on $\mathcal{V}(x)$. For l, $l'\in \mathcal{V}(x)$ we use expressions $l \perp l'$, $l^\perp$, etc., to denote orthogonality, orthogonal complement, etc., of directions in $\mathbb{R}^3$ with respect to the Euclidean inner product.

In the case ${\tilde d}(x) = 1$ we have $\mathcal{O}_{SO(3)} (x) \approx \mathbb{R}\mathrm{P}^2$ by Lemma 3.3. The isotropy subgroup below for that case describes the projective plane as a quotient of SO(3) by the isotropy subgroup of points in the diagonal of $\cal M$. The case where $U_\pm = {\overline G}$, $U_\mp = \emptyset$ and ${\tilde d} = 1$ corresponds to the constant critical points of $\overline{\mathcal{F}}$. For abelian group actions on products the orbits of points in the diagonal are diffeomorphic to all orbits. This is not necessarily the case for nonabelian group actions.

Let

(3.54)\begin{equation} R_\phi = \left[ \begin{array}{ccc} \cos \phi & - \sin \phi & 0 \\ \sin \phi & \cos \phi & 0 \\ 0 & 0 & 1 \end{array} \right], \quad A_- = \left[ \begin{array}{ccc} -1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]. \end{equation}

Note that $(A_-)^{-1} = A_-$, $R_\phi^{-1} = R_{- \phi}$, $(A_- R_\phi)^{-1} = R_{-\phi} A_- $, for all $\phi \in [0,2 \pi)$ also the matrices $A_-$, R_π, $A_- R_\pi$ describe rotations by π around the vectors ${\hat e_1}$, ${\hat e}_3$, ${\hat e}_2$ respectively, and $\{I, A_-, R_\pi, A_- R_\pi \}$, $I = R_0$, form an abelian group that contains the subgroups $\{I, A_-\}$, $\{I, R_\pi\}$, $\{I, A_- R_\pi \}$, and the trivial group $\{I \}$.

Let $\mathcal{S}_1$, $\mathcal{S}_2$ be subgroups of SO(3). We say that $\mathcal{S}_1$, $\mathcal{S}_2$ are isomorphic (under conjugacy) and we write $\mathcal{S}_1 \sim \mathcal{S}_2$ if there exists an $ {A} \in SO(3)$ such that the map $ad_A: \mathcal{S}_1 \rightarrow \mathcal{S}_2$, defined by $ad_A(g_1) = A^{-1}g_1 {A}$, $g_1 \in \mathcal{S}_1$, is a group isomorphism. Note that the isotropy subgroups of x, $y \in (\mathbb{R}\mathrm{P}^2)^d$ with $ x_j = {A} y_j $, $A \in SO(3)$, $j = 1, \ldots, d$, are isomorphic since by (3.53), $R \in SO(3)_x$ if and only if ${A}^{-1} R {A} \in SO(3)_y$.

Since the vectors ${\hat e}_j$ can be transformed to each other by a rotation the groups $\{I, A_-\}$, $\{I, R_\pi\}$, $\{I, A_- R_\pi \}$ are isomorphic. Thus the isotropy subgroups found are, up to isomorphism, the set $\mathcal{S}_r$ of (i), the abelian group of two elements $\{I, R_\pi\}$, the abelian group $\{I, A_-, R_\pi, A_-\}$, and the trivial subgroup $\{I\}$.

Proposition 2.7 on the connection between the reduced functionals appearing in the bifurcation equation and discretized Oseen–Frank functionals for the case without boundary follows from the computation of [Reference Panayotaros26] for the case with boundary.

The main observation is that equation (3.49) for critical points of $\overline{\mathcal{F}}_2$, see (2.5), restricted to $C^0 = \mathcal{C}^0(\overline{\mathcal{F}_4}, U_1,U_2)$ with $U_1 = {\overline G}$, $U_2 = \emptyset$ are critical points over configurations where the Q(m) have $i, j$th entry

(3.55)\begin{equation} {\overline Q}_{i,j}(m) = - 3 x_- \big( {\hat n}_i(m) {\hat n}_j(m) - \frac{1}{3} \delta_{i,j} \big), \quad {\hat n}(m) \in S^2, \end{equation}

for all $m \in {\overline G}$. Denoting $\overline{\mathcal{F}}_2$ restricted to C ⁰ by $\overline{\mathcal{F}}_{2,r}$ and using (2.5) we have

(3.56)\begin{equation} \overline{\mathcal{F}}_{2,r}({\hat n}) = - \frac{3 x_- }{2} \sum_{k,m \in {\overline G}} c(k,m) \big( 2 - 2 ( {\hat n}(k) \cdot {\hat n}(m) )^2 \big), \quad {\hat n}:{\overline G} \rightarrow S^2, \end{equation}

with · the standard inner product in $\mathbb{R}^3$. Thus $\overline{\mathcal{F}}_{2,r}$ does not depend on the signs of the ${\hat n}(m)$, $m \in {\overline G}$, and is well-defined as a functional on maps from $\overline G$ to $\mathbb{R}\mathrm{P}^2$.

In contrast, the Oseen–Frank functional $\mathcal{F}_{OF} $ of (2.3) can be discretized by

(3.57)\begin{align} \overline{\mathcal{F}}_{OF} ({\overline{\textbf{n}}}) & := \frac{1}{2} \sum_{k,m \in {\overline G}} c(k,m) |{\overline{\textbf{n}}}(k) - {\overline{\textbf{n}}}(m)|^2 \nonumber\\& = \frac{1}{2} \sum_{k,m \in {\overline G}} c(n,m) \big(2 - 2 ({\overline{\textbf{n}}}(k) \cdot {\overline{\textbf{n}}}(m) ) \big), \quad {\overline{\textbf{n}}}:{\overline G}\rightarrow {S^2}. \end{align}

This functional is not invariant under arbitrary changes of sign of $\overline {\bf{n}}$ at the different sites, see discussion in [Reference Panayotaros26] comparing (3.56), (3.57).

The proof of Proposition 2.8 for the case without boundary is as in the case without boundary in [Reference Panayotaros26] and is omitted.

4.1. Discrete geometries: boundary and initial conditions

For a set G of N sites, the Q-tensor Q(k) at site $k \in \{1, \ldots, N\}$ is described by its components $y_j(k)$, $j \in \{1, \ldots, 5\}$, defined in section 3. The gradient equations

\begin{equation*} \dot Q = - \nabla \mathcal{G} (Q), \end{equation*}

$\mathcal{G}(Q) = \mathcal{F}_{LG}(Q)$ (case without boundary), $ \mathcal{G}(Q) = \overline{\mathcal{F}}_{LG}(Q;Q_b)$ (case with boundary), $\nabla = \nabla_Q$, have the general form

(4.1)\begin{equation} {\dot y}_j(k) = - L (\Delta y_j)(k) - \frac{\partial f_B(Q(k))}{\partial y_j(k) }, \quad j \in \{1, \ldots, 5\}, \quad k \in \{1, \ldots, N\}. \end{equation}

The second term is the gradient of the onsite energy

(4.2)\begin{equation} f_B(Q(k)) = - \frac{a^2}{2} \hbox{tr}(Q(k)^2) - \frac{b^2}{3} \hbox{tr}(Q(k)^3) + \frac{c^2}{4} (\hbox{tr}(Q(k)^2))^2, \end{equation}

(4.3)\begin{equation} \hbox{tr}(Q(k)^2) = y_1(k)^2 + y_2(k)^2 + y_3(k)^2 + y_5(k)^2 + y_6(k)^2, \end{equation}

\begin{align*} \hbox{tr}(Q^3(k)) & = \frac{3}{\sqrt{6}} y_2(k)^2 y_5(k) + \frac{3}{\sqrt{2}} y_2(k) y_3(k) y_6(k) + \frac{3}{2\sqrt{2}} (y_3(k)^2 y_1(k) - y_6(k)^2 y_1(k) )\\ & \quad - \frac{3}{2\sqrt{6}} (y_3(k)^2 y_5(k) + y_6(k)^2 y_5(k)) - \frac{1}{\sqrt{6}} y_5(k)^3 + \frac{3}{\sqrt{6}} y_1(k)^2 y(k)_5, \end{align*}

for all $k \in \{1, \ldots, N \}$, and has the same form at all interior sites.

The first term $ (\Delta y_j)(k)$ at site k is the linear coupling to other sites and contains the information about the geometry of the set G.

Geometry A. Chain with two boundary points. In the case of a chain with nearest-neighbour interactions and boundary values y_L, $y_R \in \mathbb{R}^5$, e.g. setting $y(0) = y_L$, $y(N+1) = y_R$, we have

(4.4)\begin{equation} (\Delta y_j)(k) = y_j(k-1) + y_j(k+1) - 2 y_j(k), \quad k \in \{2, \ldots, N-1\}, \end{equation}

(4.5)\begin{equation} (\Delta y_j)(1) = y_{L,j} + y_j(2) - 2 y_j(1), \quad (\Delta y_j)(N) = y_{j}(N-1) + y_{R,j} - 2 y_j(N), \end{equation}

with $y_L = \{y_{L,j} \}_{j=1}^5$, $y_R = \{y_{R,j} \}_{j=1}^5$, for all $ j \in \{1, \ldots, 5\}$.

Geometry B. Closed chain. In the case of a chain with nearest-neighbour interactions and periodic boundary conditions

(4.6)\begin{equation} (\Delta y_j)(k) = y_j(k-1) + y_j(k+1) - 2 y_j(k), \quad k \in \{2, \ldots, N-1\}, \end{equation}

(4.7)\begin{equation} (\Delta y_j)(1) = y_j(N) + y_j(2) - 2 y_j(1), \quad (\Delta y_j)(N) = y_j (N-1) + y_j(1) - 2 y_j(N), \end{equation}

for all $ j \in \{1, \ldots, 5\}$.

We integrate numerically the equations for the two systems with initial conditions that are not uniaxial.

Geometry A. In the case of the open chain, we choose uniaxial boundary conditions Q_L, Q_R

(4.8)\begin{equation} Q_j = - 3 x_- \left ({\hat n}(j) \otimes {\hat n}(j) - \frac{1}{3}I \right), \quad j = L, R, \end{equation}

$x_-$ the negative solution of $-a^2 + 6c^2 x + b^2x^2 = 0$.

We use the values $ b^2 = \sqrt{6}$, $a^2 = \frac{1}{2}$, $c^2 = \frac{2}{3}$, then $x_- = \frac{1}{\sqrt{6}}$.

The unit vectors ${\hat n}(L)$, ${\hat n}(R)$ are at an angle. The corresponding Q-tensors are obtained from the polar representation (3.24) of the vectors ${\hat n}(L)$, ${\hat n}(R)$, via (3.18).

The corresponding vectors y_L, y_R are obtained via (3.11)-(3.14), i.e. the respective Q_L, Q_R are as in (3.10) The initial condition at the interior points is chosen to be

(4.9)\begin{equation} y(k) = \frac{k}{N+1} (y_R - y_L), \quad k = 1, \ldots, N. \end{equation}

The interior values interpolate linearly between the boundary values, and are not generally uniaxial. In cases of boundary conditions expected to lead to more than one attracting equilibria, we use small perturbations of (4.9) that may lead to different asymptotic states.

Geometry B. In the closed chain we use uniaxial initial conditions of the form

\begin{equation*} Q(k) = s(k) \left( {\hat n}(k) \otimes {\hat n}(k) - \frac{1}{3}I \right), \quad k \in \{1, \ldots, N\}, \end{equation*}

with different choices of the functions s, $\hat n$. The choice $s = -3 x_-$ at all sites, $x_-$ the negative solution of $-a^2 + 6c^2 x + b^2x^2 = 0$, corresponds to an initial condition of uniaxial tensors on the manifold of minima of f_B.

We use the values $ b^2 = \sqrt{6}$, $a^2 = \frac{1}{2}$, $c^2 = \frac{2}{3}$, then $x_- = \frac{1}{\sqrt{6}}$.

Also, we consider

\begin{equation*}s(k) = \sigma((k-1)/N), \quad k \in \{1, \ldots, N\},\end{equation*}

for σ smooth 1−periodic functions that oscillate around the value $-3 x_-$, and

\begin{equation*}{\hat n}(k) = {\hat \nu}( (k-1)/N), \quad k \in \{1, \ldots, N\},\end{equation*}

where ${\hat \nu}: [0,1] \rightarrow S^2$ are smooth parameterized curves.

We are interested in two classes of curves $\hat \nu$, corresponding to initial conditions expected to converge to states that are discrete analogues of the two homotopy classes of $\mathbb{R}\mathrm{P}^2$.

First, we consider closed curves that are continuous deformations of curves whose image is a subset of an open hemisphere, see e.g. Figure 1(a). The corresponding initial conditions are expected to converge to discrete analogues of homotopicaly trivial curves on $\mathbb{R}\mathrm{P}^2$, i.e. curves that can be continuously deformed to point.

Also, we consider curves that are continuous deformations of curves that join any two antipodal points of S ², see e.g. Figure 1(b). The corresponding initial conditions are expected to converge to discrete analogues of homotopicaly nontrivial curves on $\mathbb{R}\mathrm{P}^2$, i.e. curves that cannot be continuously deformed to a point.

Figure 1.

Examples of closed curves of the two homotopy classes of $\mathbb{R}\mathrm{P}^2$, visualization using the 2−sphere. When antipodal points are identified, (a) depicts a homotopicaly trivial curve on the projective plane, while (b) depicts a homotopicaly non-trivial curve on the projective plane.

An example of an initial condition expected to converge to a configuration that is a discrete analogue of a homotopicaly trivial curve in $\mathbb{R}\mathrm{P}^2$ is

(4.10)\begin{equation} \phi(k) = \frac{\pi}{2} + \frac{1}{5} \cos \frac{2 (k-1) \pi}{N}, \quad \theta(k) = \frac{\pi}{2} + \frac{1}{5} \sin \frac{2 (k-1) \pi}{N}, \quad s(k) = \frac{3}{\sqrt{6}}; \end{equation}

$ k =1, \ldots, N$. The corresponding Q(k), and y(k) are obtained via (3.18), (3.11)-(3.14), (3.10). Figure 1(a) depicts the vectors $\hat n$ of the curve defined by (4.10). This is an homotopicaly trivial curve on the projective plane since it is continuously contractible to a point.

Examples of initial conditions that are expected to converge to discrete analogues of homotopicaly nontrivial curves in $\mathbb{R}\mathrm{P}^2$ are

(4.11)\begin{equation} \phi(k) = \frac{(k - 1)\pi}{N}, \quad \theta(k) = \frac{\pi}{2}, \quad s(k) = \frac{3}{\sqrt{6}}; \end{equation}

(4.12)\begin{equation} \phi(k) = \frac{(k - 1) \pi}{N}, \quad \theta(k) = \frac{\pi}{2} + \frac{1}{5}\sin \frac{2 \pi(k-1) }{N}, \quad s(k) = - \frac{3}{\sqrt{6}}+ \frac{1}{5} \sin \frac{2 \pi (k-1)}{N}; \end{equation}

$ k =1, \ldots, N$. The corresponding Q(k), and y(k) are obtained via (3.18), (3.11)-(3.14), (3.10). Figure 1(b) depicts the vectors $\hat n$ of the curve defined in (4.11). The first discretizes a curve on the 2−sphere that connects antipodal points and cannot be continuously contracted to a point. In (4.12) we add a deformation, and vary the radius s.

To verify the approach to near uniaxial Q-tensors at all sites k of $\overline G$ or G, we obtain Q(k) at each time t from the numerically computed y(k) at time t using the inverse of the linear transformation defined by (3.11)-(3.14), and (3.10). We further compute the eigenvalues and eigenvectors of Q(k) used in the polar representation (3.2) of Q(k), and the quantities

(4.13)\begin{equation} s(k) = 2 \lambda_1(k) + \lambda_2(k), \quad r(k) = 2 \lambda_2(k) + \lambda_1(k), \end{equation}

where $\lambda_1(k) \geq \lambda_2(k) \geq \lambda_3(k) $ are the eigenvalues of Q(k) at each time. The normalized eigenvector that corresponds to $\lambda_1(k)$ is denoted by $\bf n$, and the notation

(4.14)\begin{equation} \textbf{n}(k) = [{n}_1(k),{n}_2(k),{n}_2(k)], \end{equation}

for its components. The dependence on time is not explicit in this notation. The eigenvalues should satisfy strict inequalities in the vicinity of $\Sigma^3_-$.

4.2. Numerical observations: summary

The main observations after several integrations of the gradient flow (4.1) are the following:

(i) All initial conditions for geometries A, and B lead to trajectories for which the Q-tensor becomes near-uniaxial after a few time units, with s, r converging to values in the vicinity of $s = -3 x_-$, r = 0. The velocity of approach to near-uniaxial states is not seen to depend significantly on the number of sites N.

(ii) All initial conditions for geometries A, and B lead to an equilibrium configuration. This occurs at a time scale that is much longer than the time scale of approach to near-uniaxial Q-tensors. The time to approach the equilibrium depends on the number of sites N and the initial condition.

(iii) In the case of the closed chain, geometry B, we see two attracting asymptotic states, the constant state, and a state that connects two antipodal points that lie on a plane. The constant state is homotopicaly trivial and is the global minimum of the energy. The second state is a discretization of a homotopicaly non-trivial curve, similar to Figure 1(b). Multiple (at least two) stable equilibria are also observed for geometry A with some special boundary conditions. These equilibria are discrete analogues of non-homotopic curves on the projective plane and may occur under more general boundary conditions.

In what follows we describe in more detail some of numerical experiments with geometries A and B. The results below are typical a larger set of experiments performed.

4.3. Geometry A: open chain

We consider the evolution of the open chain system of (4.1) with boundary conditions y_L, y_R as in (4.8), and $\hat n$ as in (3.24) with $\theta = \pi/2$, ϕ = 0, and $\theta = \pi/2 - 0.5$, ϕ = 0.5 respectively. Also $x_-= - 1/\sqrt{6}$. The initial condition is obtained from (4.9).

The rapid approach to a near uniaxial Q-tensor is seen over a range of coupling constants $0.01 \leq L \leq 0.2$. Table 1 shows results using N = 65, and L = 0.01. The value of r_max at t = 0 indicates the departure from uniaxiality at the interior sites. The time scale of approach to near-axisymmetric tensors indicated of a few units is typical of what we saw in all numerical experiments with geometries A and B and $30 \leq N \leq 85$. The parameter s similarly approaches the value s ≈ 1.22 at all sites, as expected from $s=-3 x_-=\frac{3}{\sqrt{6}} \approx 1.2247$ for the radius of the critical manifold of f_B.

Table 1.

r_max, $s_{\max}$ at time t for an open chain of N = 65 sites, L = 0.001. $r_{\max}$, $s_{\max}$ are numbers that satisfy $ |r(k)| \leq r_{max}$, $s(k)\geq s_{max}$ for all $k \in \{1, \ldots, N \}$ at time t, see (4.13). Geometry A, boundary conditions (4.8), ${\hat n}(L)$, ${\hat n}(R)$ as in (3.24) with $\theta = \pi/2$, ϕ = 0, and $\theta = \pi/2 - 0.5$, ϕ = 0.5 respectively; initial condition as in (4.9). Parameter values $ b^2 = \sqrt{6}$, $a^2 = \frac{1}{2}$, $c^2 = \frac{2}{3}$, $x_- = - \frac{1}{\sqrt{6}}$

In Figure 2, we consider the same initial and boundary conditions with L = 0.01 and three values of N = 30, 65, 80. The initial conditions are obtained from the boundary values and (4.9). The plots show that the parameter s reaches the same approximate asymptotic value s ≈ 1.22 indicating that the Q-tensor is uniaxial. The dependence of the asymptotic values of s on the site is not significant. The time taken to reach the near-uniaxial state is t ≈ 5 for the three chains considered.

Figure 2.

s v.s. t for open chains with N = 30, 65, 80, L = 0.01, see (4.13) for definition of s. Geometry A, boundary conditions (4.8), ${\hat n}(L)$, ${\hat n}(R)$ as in (3.24) with $\theta = \pi/2$, ϕ = 0, and $\theta = \pi/2 - 0.5$, ϕ = 0.5 respectively; initial condition as in (4.9). Parameter values $ b^2 = \sqrt{6}$, $a^2 = \frac{1}{2}$, $c^2 = \frac{2}{3}$, $x_- = - \frac{1}{\sqrt{6}}$. (a) Evolution of s, site k = 17, N = 30. (b) Evolution of s, site k = 28, N = 30. (c) Evolution of s, site k = 36, N = 65. (d) Evolution of s, site k = 36, N = 65 open chain,site 58, n=65. (e) Evolution of s, site k = 36, N = 80. (f) Evolution of s, site k = 68, N = 80.

The approach to equilibrium in these experiments is indicated in Figure 3 where we show the evolution of the first component $\hat n_1(k)$ of the normalized eigenvector of the largest eigenvalue of Q(k) at some representative sites $ k \in \{1, \ldots, N\}$. In all the plots we see an initial rapid increase or decrease in the value of n ₁ followed by a slow approach to the asymptotic value. For the case of N = 30, we show the behaviour at sites k = 17 and k = 28. The value of n ₁ differs slightly between the two: at site 17, the asymptotic values is $ n_1 \approx 0.9289$, while at site 28, the asymptotic $ n_1 \approx 0.810 $. The different values reflect the spatial pattern of the equilibrium state, and its dependence on the boundary condition. Similar time evolution patterns are observed for systems with N = 65 and N = 80 sites, see Figure 3. Figure 4 shows the spatial asymptotic spatial configurations for the N = 30, 65, 80 chains.

Figure 3.

n ₁ v.s. t for open chains of N = 30, 65, 80 sites, L = 0.01, see (4.14) for definition of n ₁. Geometry A, boundary and initial conditions, and parameter values as in Figure 2. (a) Evolution of n ₁, site k = 17, N = 30. (b) Evolution of n ₁, site k = 28, N = 30. (c) Evolution of n ₁, site k = 36, N = 65 site 36, N = 65. (d) Evolution of n ₁, site k = 58, N = 65. (e) Evolution of n ₁, site k = 36, N = 80. (f) Evolution of n ₁, site k = 68, N = 80.

Figure 4.

Numerical asymptotic (equilibrium) spatial configurations, open chain experiments of Figures 2, 3. Spatial configurations shown at times $t = 80,000$, (a), N = 30, and $t=200,000$ for (b), (c), N = 65, 80 sites respectively

Figure 5 shows the evolution from initial condition

(4.15)\begin{equation} y(k) =- \frac{k}{N+1} (y_R - y_L), \quad k = 1, \ldots, N. \end{equation}

Figure 5.

n ₁ v.s. t for open chains of N = 30, 65, 80 sites, L = 0.01, see definition of n ₁ in (4.13). Geometry A, boundary conditions and parameter values as in Figures 2, 3. Initial conditions are obtained from (4.15), different from those of Figures 2, 3. (a) Evolution of n ₁, site k = 17, N = 30. (b) Evolution of n ₁, site k = 28, N = 30. (c) Evolution of n ₁, site k = 36, N = 65. (d) Evolution of n ₁, site k = 58, N = 65. (e) Evolution of n ₁, site k = 36, N = 80. (f) Evolution of n ₁, site k = 68, N = 80.

This configuration is expected to be further from the equilibrium, and the time to the reach the asymptotic values of the $\hat n (k)$ is longer than the time of Figure 3. The time scale is however comparable.

Figures 3, 5 indicate that the approach to equilibrium is achieved at a much slower rate, e.g. a time scale of $ t \approx 10^5$. Moreover the time to equilibrium increases with the number of sites.

We also see examples of boundary conditions that lead to at least two stable equilibria. An example is shown in Figure 6 where we see two different equilibrium states for uniaxial boundary conditions of the form (3.19) with $\phi_L = 0$, $\theta_L = \pi/2$; $\phi_R = \pi/2$, $\theta_R = \pi/2$, i.e. the directors at the two ends are perpendicular. We may expect the existence of equilibria where the interior directions interpolate between the two perpendicular directions at the two edges by rotating in opposite directions. To see these two equilibria we use initial conditions are two different small perturbations of (4.9).

Figure 6.

Evidence for bistability for a uniaxial boundary condition where the directions at the two boundary points are perpendicular, open chain geometry A, N = 65 sites. Boundary conditions for (a), (b) given by Q-tensor of (3.19), with $\textstyle\phi_L=\;\mathrm\pi/2$, $\textstyle\theta_L=\mathrm\pi/2$; $\phi_{R\;}=0$, $\theta_R=\mathrm\pi/2$ for left and right boundaries respectively. Initial conditions for (a), (b) two are different perturbations of (4.9) at the leftmost interior site k = 1. Figures (a), (b) show the corresponding vector n at t = 100000. In both cases lies on on the same plane at all sites and we show values (in radians) of direction angles at sites k $\textstyle=\;1,\;32,\;65$. The angle difference at k = 32 is 1.47, close to $\pi/2\approx1.57$, thus the configurations of (a), (b) are different, and have opposite helicity.

In the asymptotic state shown in Figure 6(a) we used initial condition (4.9), perturbed by adding 0.1 to all $y_i(k)$, $i = 1, \ldots, 6$, at site k = 1. We show the vector $\bf n$ at the interior sites $k = 1, \ldots N$, N = 65, at time $t = 100,000$. The vector $\bf n$ at different sites is on the same plane, and we indicate the angles ϕ at sites $1,\ 32$, and 65 (0.02, 0.72 and 1.55 respectively in radians). (The angle θ is approximately $\pi/2$ in all the sites.) In the asymptotic state shown in Figure 6(b), we used initial condition (4.9), perturbed by adding −0.07 to all $y_i(k=65)$, $i = 1, \ldots, 6$ at site k = 65. We show the vector $\bf n$ at the interior sites $k = 1, \ldots N$, N = 65, at time $t = 100,000$. Again, the vector $\bf n$ at different sites is on the same plane, and we indicate the angles ϕ angles sites $1,\ 32$, and 65 (1.55, 0.86, 0.02, respectively, in radians). We see that the two configurations are different, e.g. the angle difference at k = 32 is 1.47, close to $\pi/2 \approx 1.57$. Each configuration also exhibits different helicity as the direction rotates in opposite directions as we move from the left to the right end of the chain. which are