Point topological defects in ordered media in dimension two

Transcription

1 Point topological defects in ordered media in dimension two David CHIRON Laboratoire J.A. DIEUDONNE, Université de Nice - Sophia Antipolis, Parc Valrose, 68 Nice Cedex, France chiron@math.unice.fr Abstract In this paper, we study topological defects in ordered media, such as liquid crystals. These defects are due to the non-trivial topology (homotopy groups of the order parameter space. We give here a general framework for the study of these singularities and then focus on point topological defects in dimension two, classified through the first fundamental group. Keywords: Topological defects; ordered media; liquid crystals; Ginzburg-Landau energy; generalized vortex; Hölder regularity; minimizing map. Introduction. Ginzburg-Landau theory and cones In this first section, we introduce a general framework for the study of topological defects in ordered media. We will focus first on the model of liquid crystals. In the theory of (nematic liquid crystals, the order parameter is the (local orientation of the molecules and is represented by a unit vector n in R 3. A typical energy which was introduced in order to describe stable configurations is the standard Dirichlet energy E(u = u, ( so that the natural space of configurations in this case is the Sobolev space H (, S (the domain R 3 represents the container of the liquid crystal. It has been established that an H (, S minimizer of E has point singularities, essentially of the form ± x x, up to a rotation (see [36], [7], [6]. This theory succesfully explains the presence of point defects in nematic liquid crystal in dimension three. However, from the physical point of view, the configurations n and n have the same optical properties, and hence represent the same physical state, so that the actual space for the order parameter is not S but instead the projective space RP S /{±}. On the other hand, one can also observe in nematic liquid crystals line defects (see e.g., [], [], [3]. However, it appears that these lines the so-called Oseen-Frank energy involving three parameters κ, κ, κ 3 (see [], [3]. The Dirichlet energy corresponds to the one constant approximation κ = κ = κ 3 =. see Appendix A for a representation of the hedgehog

2 have an infinite Dirichlet energy, hence ( has to be modified. It has been proposed (see [], [3] to replace the Dirichlet energy ( by a Ginzburg-Landau type energy E ε (u = u + V ( u ε, ( where V : R + R + is a function such that V ( =, V (t > if t and V (t + if t + (so that is the minimum. The parameter ε is the characteristic length of the defect (expected to be small from the observations. The configuration space for the order parameter is no longer RP = S /{±} but instead Σ R 3 /{±}. It possesses in particular the positive cone property and is singular at the origin 3. The metric is flat except at zero. The quantity u plays the role of a density: u far away from the defects and u u RP represents the (local direction of the molecules, and u near the defects, where u u is meaningless. The purpose of this paper is to justify the existence of these lines defects with this model, which, to our knowledge, has not been done yet. Other models in condensed matter physics are reviewed in Appendix A, and justify the introduction of a general framework we describe in the next section.. General framework We consider a compact and connected riemannian manifold N (N = RP for nematic liquid crystals, which is called the manifold of internal states, that is any state of the ordered media is uniquely described by an element of N. Next, we construct the cone over N defined as X (R + N {}, where {} is the singular point. For α X, we set α if α = and if α = (t, a R + N, we denote α t and a α α. We then define on X a natural non-smooth riemannian metric. Indeed, X is a smooth manifold except at, and we define a metric g on X \ {} = R + N in the following way. First, there is a natural identification T x X ξ (τ, ζ R T x N, and we set x g x (ξ τ + x h x x (ζ, where h is the metric on N. This metric g is well-defined in X \{}, and for this metric, the variations of the order parameter space and the variations of the modulus are orthogonal (as for a sphere. Hence, if u is smooth with values in X \ {}, we have u = u + u ( u u. (3 If N = RP, then X = ΣR 3 /{±} (see Appendix A for other models. We expect the density u to be in the absence of defects and u u describes the local state of the medium, and u = on the defect, where u u is meaningless. The distance in X. Since X \ {} is a smooth riemannian manifold, we define the length of a lipschitz curve γ : [, ] X \ {} by L(γ ( / g γ(t (γ (t dt. 3 A related model was proposed by J. Ericksen (see [], where Σ is the cone {(s, u R R 3, s = u }. Several important mathematical issues were treated by F.H. Lin and R. Hardt ([6], [8].

3 Next, for α, β X \ {}, we define d X (α, β inf L(γ, (4 where the infimum is computed over all lipschitz curves from α to β (which clearly exist in X \ {}, and if α X, β =, then d X (α, = α. Lemma The function d X is a distance on X. Moreover, if N is compact, (X, d X is a complete metric space, separable and locally compact, and for every α, β X, there exists a d X (α, β-lipschitzian map γ : [, ] X such that γ( = α and γ( = β, that is γ( = α, γ( = β and d X (α, β = L(γ. From Lemma, (X, d X is then a length space, that is a complete metric space such that for any points α, β X, there exists a lipschitz curve from α to β in X of length equal to d X (α, β. Remark We emphasize that (X, d X is not locally compact if N is not compact (since N is the sphere of center and radius in X. Moreover, there may exist more than one geodesic in X from one point to another. For instance, if N = S k R k+ is the standard sphere, then X is just (isometric to the euclidean space R k+. We naturally identify N with {} N X. Note that this embedding is not isometric but the distance induced by d X on {} N is clearly equivalent to the riemannian distance d N in N. In this paper, we are interested in the study of energy functionals of the type E ε (u u + ( u ε (5 for u : X in the asymptotic ε. More generally, one can investigate an energy like u + V ( u ε. Here, is a smooth bounded domain in R N, containing the ordered media. Ginzburg-Landau type energy. It possesses two terms: the kinetic energy u, This energy (5 is a which is defined as a Dirichlet integral with values into a metric length space as in [3] and [] (see [9] for the equivalence of the two definitions. Indeed, we emphasize that u has values into X, which is a metric space (not euclidean, except if N is a sphere, hence this term is understood through the theory of Sobolev spaces into metric space target, as developed by N. Korevaar and R. Schoen [3], after the pioneering work of M. Gromov and R. Schoen [4]. The second term is the ( u potential 4ε is well-defined since u is well-defined for u X. We are interested in the asymptotic ε, where we expect topological defects. The small parameter ε is homogeneous to a small length, corresponding to the typical size of the defect. Note that as ε, the potential term forces u to be close to, that is forces u to lie in N. In other words, the zeros of the potential term will play a fundamental role. The manifold V (u ( u {V = } N is called vacuum manifold in the physics literature. 3

4 .3 Monotonicity formula and Hölder regularity for minimizers In this subsection, we establish the monotonicity formula and the Hölder regularity for minimizers of the functional E ε (u u + ( u e dx, for R N and u H (, X, where X is the metric cone over a smooth compact riemannian manifold without boundary. The proof of these results will be given in Appendix B. The monotonicity formula can be established for more general functionals of the type Dirichlet energy plus potential term (even if X is not the metric cone over a smooth riemannian manifold, and not only for minimizers but also for maps critical for for variations of the domain. We present also in Appendix B a regularity result for slightly more general functionals. This is detailled in Appendix B, and we have chosen in part of the paper to present only the results we need for our purpose. Observe that the notion of weakly harmonic map into a metric space does not make sense, since there does not exist in general a smooth (at least C to compute the derivative local retraction (there does not exist in general even an embedding into a finite dimensional euclidean space. We give now the definition of the normal derivative u n. Definition (Normal derivative Assume X is a metric cone over a smooth riemannian compact and connected manifold. Then, given u H (, X and x, we define The normal derivative u n u n = lim t [ t d X ( u(x + t x x x x, u(x ]. has a meaning in L loc (B d(x, where d dist(x,, and thus in particular in L ( B r (x, dh N for almost every < r < dist(x,. In order to prove the monotonicity formula, the first step is to establish the following standard Pohozaev identity. Lemma (Pohozaev identity Let < ε, u H (, X be minimizing for E ε and x. Then, for almost every < r < dist(x,, we have (N B r(x = r u dx + N B r(x ( u ε ( u ε dx dh N (x + r B r(x B r(x [ u u ] dh N (x. n Lemma is a consequence of Lemma B.3, which is settled and proved in Appendix B, section B.3. Given u H (, X, x and r >, we define the rescaled energy Ẽ ε (u, x, r r N u + ( u B r(x ε The monotonicity of the rescaled energy Ẽε(u, x,. plays a central role in the analysis of regularity in elliptic problems (see for instance [3] and [36]. This monotonicity follows easily from the Pohozaev identity. In the case X = R or a smooth compact manifold, the monotonicity formula has been established for stationary maps in [3], and for embedded metric spaces targets, one can refer to [4]. dx. 4

5 Proposition (Monotonicity formula Let < ε and u H (, X be minimizing for E ε. Then, for any x and for almost every < r < dist(x,, we have d dr Ẽε(u, x, r = r N B r(x u + n r N In particular, r Ẽε(u, x, r is non-decreasing in (, d(x,. We can now state our Hölder regularity result. B r(x ( u Theorem Let R N be bounded and smooth, and let u H (, X be minimizing for E ε. Then, u W, (, R satisfies (weakly u + ε u ( u = u. (6 Moreover, for ω, there exists K = K(, ω, N, N > and α = α(n, N (, such that and for x, y ω, u L (ω,x K ( x y α. d X (u(x, u(y K ε Finally, either u, either u ( has Hausdorff dimension N inside. Remark The last bound is well-known for the standard Ginzburg-Landau model (see [3] for instance, for α = (lipschitz bound. It does not seem easy to prove this for α =. The Hölder regularity of minimizing maps into a smooth compact riemannian manifold was first studied by R. Schoen and K. Uhlenbeck in [36], [37]. Under the hypothesis that the manifold has non-positive curvature, the lipschitz regularity (hence smoothness of minimizers was established by J. Eells and J.H. Sampson ([], using a Böchner type inequality. In the context of non-positively curved metric spaces, we refer to the works of [4], [3] and [9], which prove lipschitz regularity for minimizers. However, if the metric g on X is flat outside, the curvature of our metric cones is + at the vertex (except for the trivial case where N is a sphere and hence X is euclidean. Therefore, the regularity results in non-positively curved metric spaces do not apply to our case. We mention also the work of C. Wang ([4], who establishes the Hölder regularity of minimizers into piecewise smooth uniformly regular manifolds, which are roughly lipschitz manifolds embedded into R n having a local lipschitz retraction. Here, we will see that it is not clear whether X can be isometrically embedded into a euclidean space, but we will be able to find an embedding into a non-euclidean geometric cone (see Appendix B. The proof in Appendix B have been inspired by [4]. We conclude this section with a result on the extension problem. The natural space for a boundary datum is H (, X. It is natural to wonder whether any g H (, X has an extension inside the domain u H (, X. In other words, is the space H g (, X = {u H (, X, u = g on } ε dx. always non-empty for any g H (, X? proposition. The answer is positive, as shown by the following 5

6 Proposition Assume (X, d is the metric cone over a smooth riemannian compact manifold N. Let R N be bounded and smooth, and p < s, p be given, or s = p =. Then, for every g W s p,p (, X, there exists u W s,p (, X such that u = g on in the sense of traces and u W s,p C g s. In particular, the (non-linear mapping W p,p is surjective. tr : W s,p (, X W s p,p (, X.4 The role of the fundamental group of N In the case where N = S, and thus X C, the study for the functional (5, from the mathematical point of view, have been first done in the two dimensional case by F. Bethuel, H. Brezis and F. Helein in [3] and [4]. They exhibit, as ε, a vortex structure, that is u inside the domain, except on vortices, which are point topological defects (in dimension two: u vanishes at a point a, is of modulus / on a disk centered at a of radius of order of ε and u u : D ε (a S S has a nonzero winding number. These defects arise from topological obstructions, since the limit energy space, in the case of a Dirichlet datum g, say, H g (, N = {u H (, N, u = g on } may be empty if π (N is non trivial (see subsection.4.. In higher dimension, several authors have studied this concentration phenomenon, among which [7], [35], [5], [], [], [6]... The topological properties of N will play a fundamental role, and more specifically the homotopy groups π k (N. We will see the emergence of generalized vortices in the case where N is not simply connected. Let E ε inf{e ε (u, u H g (, X}..4. The simply connected case We assume in this subsection only that the order parameter space N is simply connected. Then by the result of [7], Theorem 6., any g H / (, N admits an extension ḡ H (, N. Let then u H g (, N be a minimizing extension. Using u as a comparison map, one infers E ε E ε (u = u, thus E ε is bounded. Let u ε be any minimizer for E ε. Then, it follows from u ε + ( u ε ε = E ε u (7 that u ε is bounded in H, thus, passing possibly to a sequence ε n, one may assume u εn weakly in H and u εn u pointwise and in L. From (7, one first infer ( u εn ε n u, u so u Hg (, N, and then, by lower semicontinuity, u lim inf u εn u u, n + 6

7 the last inequality being a consequence of the minimizing property of u. Therefore, u εn u in H, where u is a minimizing extension of g in, u εn u ( u εn and. In this case, we have no singular limit, since the energy remains bounded: E ε E R +. For minimizers in the simply connected case, the potential term does not play an important role, and we may replace the problem E ε by the limiting problem { } E inf u, u Hg (, N. This is what is done in the S model for liquid crystals, since the sphere S is simply connected. However, this will no longer be the case for non-simply connected manifolds, like RP, as shown in subsection.4. below..4. The non simply connected case In the case where N is non-simply connected, the infimum E ε may diverge when ε. For instance, let be the unit disk in R and g : = S N a smooth map non-homotopic to a constant in N, that is g is a non-trivial element of π (N. Assume then that lim inf ε E ε <. Then, there exists a sequence ε n and u n Hg (, X such that u n + ( u n ε n independent of n. One may then repeat the argument used in subsection.4. to prove the existence of u Hg (, N. Therefore, one may consider a minimizing v Hg (, N. By the regularity results in [3] (or [36] and [37] this minimizer v is smooth, which contradicts the fact that g = v S is a non-trivial element of π (N. In this case, we have then a singular limit Energetic costs of a defect E ε + as ε. First, let us recall the way to classify topological defects. We expect an order parameter for instance continuous in the plane except at a point, say in D ( \ {}, or on a curve, which reduces to the -dimensional case. Therefore, we may consider u restricted to a small circle around this point as a map ũ from S to N. Topological defects are singularities that we can not remove from local surgery. In particular, the homotopy class of ũ is the good characterization of the topological defect. It is well known (see [38] that, given a base point b N, the homotopy classes from S to N, that is the (path-connected components of C(S, N, are in one-to-one correspondence with the conjugacy classes of π (N, b, that is the equivalence classes of π (N, b for the equivalence relation C ε n g g h π (N, b s.t. g = hgh. In the case where the fundamental group π (N, b is abelian, then there is a natural identification of all the first homotopy groups π (N, b for b N, then denoted by π (N, and the conjugacy class of 7

8 γ is reduced to {γ}, thus the topological defects are characterized by an element of the group π (N. We denote Υ(N { homotopy classes from S to N }, which can be identified with the conjugacy classes of π (N, b for any b N. In general, even though π (N, b has a group structure, and may be non-abelian, Υ(N has no natural group structure. The first aim of this subsection is to associate a length to any element of Υ(N. Let γ Υ(N. To any c γ H (S, N, seen as a closed curve in N, we may define its length ( π c (t dt, (8 S where denotes tangential differentiation. By Cauchy-Schwarz, this length is greater than or equal to the usual length S c (t dt. We define then the length λ(γ as the infimum of all these lengths for c γ H (S, N, that is { ( λ(γ inf π c (t dt S }, c γ H (S, N. (9 Note that by standard regularization, γ H (S, N, and even γ C (S, N, is dense in γ for the C norm. Furthermore, the existence of minimizers for (9 follows directly from the compact embedding H (S, N C, and it is easily seen that the minimizers c are geodesics, and then π c = λ(γ is constant, and then (8 is the usual length of the (lipschitz closed curve c. Product of homotopy classes. We denote P(Υ(N the set of all subsets of Υ(N. First, we define the product g f of two maps f, g C(S, N such that f( = g( as the map f(e iθ if θ π, g f(e iθ = g(e iθ if π < θ < π, which clearly defines a continuous map. If c C([, ], N is a continuous path in N, then c denotes the path with reverse time, that is c (t = c( t for t [, ]. The definition of the product easily extends to c f (if c( = f( and g c (if c( = g( for any continuous path c, giving a path from f( to c( (and from c( to g(. We then define the following map Υ(N Υ(N P(Υ(N (α, β α β by the formula: { } α β = homotopy class of ((c g c f, f α, g β, c path from f( to g(. The set α β is then a union of homotopy classes, which can consist in strictly more than one homotopy class if π (N is non-abelian. Given a large open disk containing two small disjoint closed disks, and given any f α and g β, then consider all maps v with values in N continuous on the closed large disk minus the two small disks such that v = f and v = g on the two small disks: then α β consists in all the homotopy classes of v restricted to the boundary of the large disk. In other words, with the terminology of topological defects, α β consists in all the possible defects you may obtain by the combination of the defects α and β. In particular, even if π (N is non-abelian, we have the commutativity relation α β = β α, 8

9 but this set does not, in general, consists in a single homotopy class. In terms of conjugacy classes, for g, h π (N, b, this comes from the simple relation g h = g (h g g, that is g h and h g are in the same conjugacy class, and this overcomes the fact that π (N may be non-abelian. For Γ, Γ two subsets of Υ(N, that is two unions of homotopy classes, we define Γ Γ as the union of all homotopy classes of γ γ for γ Γ and γ Γ. Note that the operation is not in general a group operation (except when π (N is abelian, but is associative: if α, β and γ Υ(N, then (α β γ = α (β γ, and therefore we may write it α β γ. To compute α α... α m, we proceed as in the figure below, where we fix f i α i for i m and consider all the maps v C( D \ m i= D i, N such that v = f i on D i. Here, D is the large disk, and the D i s are the small pairwise disjoint disks. The product α α... α m is then the union of all possible homotopy classes of v restricted to D. We emphasize that the order of the disks D,...,D m is not important, even if π (N is non-abelian. In the case where π (N is abelian, then Υ(N π (N and α β is just the group operation in π (N. f f f m D D D m D Figure : Computation of α α... α m In the case of N = S, the combination of two vortices of degrees d and d gives a vortex of degree d + d. When π (N is non-abelian, the combination of two defects of homotopy classes γ and γ may give any element of the product γ γ, and in particular, we emphasize that this result is not unique (see the end of the Appendix. Energetic costs of a defect. It can be energetically favourable, as it is the case for minimizing maps, to decompose a nontrivial defect in less energetic defects. Indecomposable defects will be called simple defects. Therefore, we define also for any γ Υ(N, λ (γ { n 4π inf λ(γ i, γ i= n γ i, γ i Υ(N, γ i i= } non trivial, ( where, we recall, the order in the product is not relevant. In the case N = S, we easily obtain, for d Υ(S π (S Z, λ(d = π d and λ (d = π d, and the infimum in ( is achieved only for n = d and γ i = sign(d = ± Z. The quantities λ(γ and λ (γ are energetic costs. Indeed, as a single vortex of degree d has a cost πd log ε (up to a term of order one as ε, a topological defect γ Υ(N has a cost λ(γ 4π log ε. Moreover, in the standard Ginzburg-Landau model, minimizers for a Dirichlet datum of degree d are known to have an energy π d log ε (up to a term of order one as ε, corresponding to a splitting of the degree 9

10 d, that we may assume positive, into d vortices of degree. The simple defects are the defects of degree ±, which are those that appear for minimizers in the Ginzburg-Landau model (see [4]. Here, we will show that the energy of minimizers for a Dirichlet datum of homotopy class γ Υ(N is (up to a term of order one as ε, λ (γ log ε, corresponding to a splitting of the defect γ into a configuration achieving (. 3 Concentration in dimension two 3. Statement of the results In the sequel, we will restrict ourselves to the dimension N =, and we provide a study for the minimizers for the energy (5. The aim is to generalize the study for the case where N = S, for which u is complex-valued, to the case where N is any manifold of internal state, as in the examples reviewed in section A.3 in the Appendix. Our main interest concerns the behavior of minimizers for such models, in dimension, and we generalize some results of E. Sandier (see [34] and R.L. Jerrard (see [9]. We will follow closely the lines of [34], which has the advantage of not requiring a lower bound at small scales. We recall the notion of radius used there to measure the size of a set in R. Let K R be compact. We define the radius of K, denoted K, by { n } K inf r i, K n i=d ri (a i. i= In this subsection, is a bounded smooth and simply connected domain in R. As a consequence, is homeomorphic to the disk D. We will consider boundary maps g in the space H (, N C (, N, so that its homotopy class γ is clearly defined. If π (N is abelian, then one may define the homotopy class of g as the sum of the homotopy classes of g restricted to all the connected components of. This allows to include the case where is non-simply connected if π (N is abelian. The first result is a lower bound for maps from an open set R with values in a compact riemannian manifold N. Proposition 3 Let R be a bounded simply connected domain and let ω be compact at distance greater than ρ > from the boundary. Then, given any v H ( \ ω, N with trace on of homotopy class γ, we have ( ρ v λ (γ log. \ω ω Moreover, there exist K >, depending only on N, such that, for any ω η ρ, there exist k disks D i ( i k of radii at most η such that k Kλ (γ and ( η v λ (γ log. ω ( ( k i= D i\ω The second statement in this proposition is a concentration property of the energy. We can now state our lower bound result. Our first theorem is about a lower bound for this energy in terms of the homotopy class of the Dirichlet datum g : N. This lower bound is the analogue of the well-known bound E ε (u π d log ε C for a Dirichlet datum of degree d and in H (, S, which has been proved in [34] and [9], and also in [4] as a consequence of the study of minimizers for a fixed boundary datum.

11 Theorem Let g : N be a given H boundary map of homotopy class γ Υ(N. Then for any u H g (, X, we have E ε (u, λ (γ log ε C, where C is a constant depending only on and g. This lower bound, as for the usual Ginzburg-Landau energy is optimal for minimizers, up to a term of order one, as shown in the following lemma. Lemma 3 Assume g : N is a given H boundary map of homotopy class γ Υ(N. Then, there exists C such that for < ε /, where C depends only on and g. min{e ε (u, u H g (, X} λ (γ log ε + C, The next Proposition states that maps u ε H g (, X satisfying the upper bound E ε (u ε λ (γ log ε + C, which is, as we have just seen in Lemma 3, valid for minimizers, and optimal up to order one by Theorem, enjoy some compactness and concentration properties, namely the energy of the maps u ε concentrates on a finite number of disks of radius of order ε. Proposition 4 Let g : N be a H boundary map of homotopy class γ Υ(N. u ε Hg (, X satisfies E ε (u ε, λ (γ log ε + C. Then, there exists C, depending on C, and g such that uε + ( u ε ε C. Assume Moreover, for any C η Cε, there exists an integer k Kλ (γ and k disjoint disks (D i i k of radius η such that E ε (u ε, \ ( k i=d i λ (γ log η + C. The last statement implies that the energy of u ε concentrates on a bounded number of disks of radius of order ε. The main result is then the study of the asymptotics as ε for maps satisfying the bound E ε (u ε λ (γ log ε + C, for instance, the minimizers. Theorem 3 Let g : N be a H boundary map of homotopy class γ Υ(N. Let be given a sequence ε n as n + and assume that for a sequence u n H g (, X, there exists C > such that for all n E εn (u n, λ (γ log ε n + C. ( Then, up to a subsequence, there exist an integer k depending on γ, points a i, i k and u Hg,loc ( \ {a,..., a k }, N Wg,p (, N ( p < such that u n u weakly in H g,loc ( \ {a,..., a k }, X

12 and u n u weakly in W,p g (, X for any p <. Moreover, if γ i Υ(N denotes the homotopy class of u around a i, then γ i is non-trivial, γ γ... γ k and 4πλ (γ = k λ(γ i, ( i= that is the configuration (γ i achieves the infimum in (. Finally, if u n is a minimizer for all n, then u n u strongly in H g,loc ( \ {a,..., a k }, X and u C g ( \ {a,..., a k }, N W,p (, N ( p <. Remark 3 The bound ( in itself does not prevent u n from having trivial defects, that is u n vanishes inside a disk D, does not vanish on the boundary D, and its homotopy class γ there is trivial. However, our theorem states that u n weakly converges in H around such a point. If u n is a minimizer, then we show that the convergence is actually strong in H (around this point. Remark 4 Since H (, X is not a linear space, we have to clarify the notion of weak convergence employed in this theorem. Given p <, a sequence (v k W,p (, X and v W,p (, X, we will say that v k v in W,p (, X as k + if and only if and sup v k W,p < k N v k v in L p (, X as k +. This last statement has a clear meaning since X is a metric space. For a compactness result, see Theorem.3 in [3]. Similarly, we will say that v k v as k + in H if v k v in H and v k v as k +. Remark 5 In the case N = S, for which we know that λ = π and λ = π in π (S = Z, ( is just the fact that if g has degree d >, then the vortices of a minimizer are of degree. Remark 6 Concerning the precise definition of the homotopy class of u around each a i, this relies on the work of R. Schoen and K. Uhlenbeck [36] (see also [8] for a work in the VMO space. In the case where N = S, the configuration (a i, γ i has to minimize the so-called renormalized energy introduced in [4]. For general manifolds N, we presume that no expression depending directly of the singular points a i s and the topological defect γ i s may be found. Remark 7 Finally, we would like to mention that these results can be extended to more general potentials than ( u. The proof of the results in this section only requires a potential of the form f( u ε ε, with f : R + R + smooth, non-negative, satisfying f( =, f(t > for all t and log f L (,.

13 Finally, let us mention that it would be interesting to consider the full Oseen-Frank energy for liquid crystals (without external fields κ (div n + κ ( n curl n + κ 3 n curl n (3 where the κ i s, i 4 and τ are material (positive constants. Generally, the constants give rise to an elliptic integrand. The investigation for this functional does not really make sense in two dimensions. Indeed, as shown by E. Sandier (see [33], in general, minimizers for an energy like (3 in a cylinder do depend on the component along the axis of the cylinder. Therefore, we hope that the understanding in two dimensions of the Dirichlet integrand will enable us to understand this type of energy density in dimension three. 3. Energetic costs λ and λ for some models in condensed matter physics We now give the values of λ and λ for the various models we have reviewed in the Appendix, section A.3. Planar Spins. We recall that in this case, We also compute for d Z = π (S N = S, π (S = Z Υ(S and π (S = {}. λ(d = π d and λ (d = π d. The simple defects are those of degree ±, and the infimum in ( is achieved only for γ i = sign(d. This is the well-known case of the standard Ginzburg-Landau model, for which the minimizers are known to have vortices of degree ± (see [4], [9], [34]. Ordinary Spins. For Ordinary Spins, we recall that N = S = SU(/U(, π (S = {} Υ(S and π (S = Z. Therefore, we are in the simply connected case arising in the S model for liquid crystals. Nematics. In the example of interest for us in liquid crystals, the parameter space is the projective space RP = S /{±}, for which, denoting H = U( ( U( = I, U(, N = RP SU(/H, π (RP = Z/(Z Υ(RP and π (RP = Z. We have also, for γ = d {, } = Z/(Z Υ(N, where d = or, λ(γ = λ (γ = πd. Biaxial Nematics. The order parameter space for biaxial nematics is identified with SU(/H, where H {±I, ±σ x, ±σ y, ±σ z } is the non-abelian group of quaternions. We recall that N = SU(/H, π (SU(/H = H, π (SU(/H = {} and { Υ(N {I }, { I }, {±σ x }, {±σ y }, {±σ z } }. 3

14 We easily compute if γ = {I }, λ(γ = π if γ = { I }, π otherwise and λ (γ = if γ = {I }, π 8 if γ = { I }, π 6 otherwise. Moreover, all the (non-trivial defects except { I } are simple defects, and the infimum in ( for γ = { I } is achieved only for the decompositions { I } γ, where γ {{±σ x }, {±σ y }, {±σ z }} is any simple defect. Superfluid Helium-3. For Superfluid Helium-3 in the dipole-locked phase, we have N = SO(3 SU(/{±I } = S 3 /{±} = RP 3, π (N = Z/(Z and π (N = {}. We easily deduce that for γ = d {, } = Z/(Z Υ(N, where d = or, λ(γ = λ (γ = πd. In the case of Superfluid Helium-3 in the dipole-free phase, we had N = (SO(3 SO(3/Σ (SU( SU(/H, π (N = H/H Z/(4Z Υ(N and π (N = π (H Z. We finally easily compute λ(γ = if γ =, π if γ = or 3, π if γ = and λ (γ = if γ =, π 6 if γ = or 3, π 8 if γ =. Moreover, the simple defects are and 3 =, and the infimum in ( for γ = is achieved only for the decompositions = + and = =. 3.3 Small energy minimizers Let us now state some remarks concerning the main difference between these models and the usual Ginzburg-Landau model. First, in the case N = S, the topological defects can be detected through the jacobian of u (see [] and [], which is no longer the case for general manifold N : for instance π (RP = Z/(Z is not seen by real cohomology. Another difference comes from the fact that the fundamental property of Clearing-Out (or η-ellipticity is no longer true. In the standard Ginzburg-Landau model, the typical energetic cost of a vortex is log ε. The Clearing-Out Theorem (see [7] or [5] for instance states that there exists η > (small such that if u is a solution of the Ginzburg-Landau equation u + ε u( u = in the ball B R N (N verifying E ε (u, B η log ε, then u (. This property ensures us that the modulus of u is far from provided the energy is small enough compared to the typical energy log ε. This allows to write down elliptic equations for the phase and 4

15 the modulus of u (we can then write u = ρe iϕ and then use elliptic theory. In our case, for instance when π (N {}, the typical energetic cost is still, as we have seen, log ε, but the Clearing-Out Theorem is false. We refer to Remark III.3 in [5] (they consider g : B = S S as the identity (hence of degree, and using Theorem 7. in [7], justify that the minimizers of E ε in Hg (B, R 3 converge in H to the hedgehog u (x = x x, which prevents the Clearing-out Theorem to be true in this case. In this example, this is the fact that π (S = Z is non-trivial that makes the Clearing-Out Theorem false. 3.4 The distance d X (Proof of Lemma The triangle inequality for d X and the symmetry of d X are straightforward. Let us prove that (4 is achieved. If α = (or β = by symmetry, this is clear since for γ a lipschitz curve from to β, we have ( / L(γ = g γ(t (γ (t dt γ(t dt β, {γ } {γ } and for the particular path γ(t tβ, there is equality, thus d X (, β = β is achieved. We now assume α and β non zero. Let (γ n be a minimizing sequence for (4. Assume that for a subsequence, γ n passes through for all n, and denote t n (, any time for which γ n (t n =. Then, we have L(γ n = {γ n } ( / tn g γn(t(γ n(t dt γ n (t dt + γ n (t dt α + β. t n Passing to the limit yields d X (α, β α + β. Since the reverse inequality is always true (take the path γ (t = ( tx for t γ (t = (t y for t, or use the triangle inequality, we have and d X (α, β = α + β, and it is achieved by the particular path γ. Now, assume that, for a subsequence if necessary, γ n never passes through for all n. If lim inf n + (min γ n >, then there exists δ > such that for n large enough, γ n δ, that is the paths γ n lie far away from the singularity of X, and the existence of geodesics follow from classical arguments. We are then left with the case where γ n approaches. In this case, we may slightly modify γ n in such a way that γ n passes through and (γ n is still a minimizing sequence, which is the first case we have treated. It follows then that (4 is always achieved, and that there always exists a geodesic from α to β either consisting in two straight lines from α to and then from to β, either a geodesic lying in the smooth part of X. The property d X (α, β = if and only if α = β then follows. Let us now prove the completeness of (X, d X. Consider x n = ( x n, xn x a Cauchy sequence in X. Then, ( x n n is a Cauchy sequence in R, hence converges. If lim n + x n =, then x n in (X, d X (since d X (x n, = x n. If lim n + x n = t >, then ( xn x n is a bounded sequence in N, thus up to a subsequence, since N is compact, we may assume xn x n y N, and we easily see that x n = x n xn x ty in (X, d n X. Therefore, (x n is a Cauchy sequence having a convergent subsequence (in X, hence converges. 5

16 3.5 The extension problem The proof of Proposition relies on the non-isometric embedding Φ : (X, d X (Y, d Y provided in section B.6. Let g W s p,p (, X. The map Φ g W s p,p (, Y W s p,p (, R q has, by classical results on extension of Sobolev maps, an extension U W s,p (, R q, with U W s,p (,R q C Φ g W s p,p C g W s p,p. Moreover, one may choose U with values into Y, since Y is convex (take for instance the extension of Φ g by mean value. By (B.3, we infer U W s,p (, Y, and then, since ρ : (Y, d Y (X, d X is lipschitzian (and s, u ρ U W s,p (, Y and u W s,p (,X C g W s p,p. Finally, we easily check that tr u = ρ tr U = ρ g = g, hence u has the desired property. 4 Lower bounds and concentration in dimension two Let us first state some remarks. First, note that we have { } inf λ(γ, γ Υ(N, γ non trivial K >, (4 for otherwise, there would exist a sequence c j H (S, N of closed paths in N such that S c j as j + and the homotopy class of c j is non-trivial for all j N. Since N is compact and in view of the embedding H (S C (S, c j converges as j + in C to a c H verifying S c =, thus the non-trivial paths c j converge uniformly to a constant path (thus trivial, which is absurd. Moreover, since we always have λ (γ λ(γ /(4π (take n =, γ = γ, we infer from (4 that for any n and non-trivial γ i s ( i n such that γ γ... γ n, n λ(γ i n K. i= In particular, in the infimum (, one can restrict to n Kλ(γ and λ(γ i λ(γ /(4π. Therefore, the infimum in ( is always achieved. The proofs for the two-dimensional case follow very closely the lines of the work of E. Sandier ([34]. 4. Proof of Proposition 3 The first ingredient in the proof of Proposition 3 is the following simple lemma. Lemma 4. Let < r < R and v H (D R \ D r, N. We assume that on D R, v is an H map of homotopy class γ. Then, v DR\Dr λ(γ ( R 4π log. r Proof of Lemma 4.. By the density result of [36], stating that smooth maps are dense in H (, N for open subset of R, it suffices to prove the lemma for a smooth v : D R \ D r N. The homotopy 6

17 class of its restriction to any circle D ρ for r ρ R is then γ. We denote (ρ, θ polar coordinates. We have v = v + v D R \D r D R \D r ρ ρ θ π R v dρdθ r ρ θ = R dρ π v dθ. ρ θ By definition of λ, so and the lemma is proved. π π r v dθ λ(γ, θ R v DR\Dr λ(γ dρ 4π r ρ = λ(γ ( R 4π log r Proof of Proposition 3. The proof follows literally the proof of [34] (in the case f(r, s = log( s r. The only inequality used concerning the degree is d k i= d i if d = k i= d i, which has to be replaced by λ (γ k i= λ (γ i if γ k i= γ i for γ, γ i Υ(N. This later fact comes directly from the definition of λ. 4. Proof of Theorem Let u be a minimizer for the problem inf{e ε (u, u H g (, X}. then, we know by Theorem 4 in [9] that u is (Hölder-continuous in. Moreover, u. Indeed, consider ũ : X defined by { u(x if u(x, ũ(x if not. u(x u(x Then, we have ũ = g on, ũ u and 4ε ( ũ = 4ε { u } ( u 4ε ( u with strict inequality unless u a.e. in. Since u is minimizing, u. Furthermore, using Lemma 8 in [9] (stating that u = a.e. in {u = } and the continuity of u (hence { u > t} is open for any t >, u u = { u >} u u = lim u u, t { u >t} by monotone convergence. Using the definition of the metric, we infer for any t > u u = u ( u u, { u >t} { u >t} 7

18 and it follows u dx = u dx + lim t { u >t} u ( u u (5 We will make use of the following notations. For t, set t {x, u(x > t}, ω t {x, u(x t}, Γ t t \ = ω t For t, consider the functions Θ(t ( u t u dx and ν(t u dh. Γ t Since u is in H and continuous, the coarea formula and (5 give E ε (u, = By Cauchy-Schwarz inequality, + and from the definition of the radius, [ Γt u + ( t Γ t ε u dh ] t Θ (t dt. u dh H (Γ t, ν(t H (Γ t diam(γ t 4 ω t (if u, v Γ t are such that diam(γ t = u v, then ω t D((u + v/, u v / and thus ω t u v = diam(γ t. It follows from the inequality (a + b / ab that E ε (u, + + ν(t + 8( t ω t ε dt ν(t ε t ω t dt + + t Θ (t dt. t Θ (t dt We integrate by parts the last term. Since u, Θ has a support included in (, ] and is locally W, in R + (note that Θ( = +. Since Θ, for any η >, + η t Θ (t dt = + η tθ(t dt + η Θ(η Letting η, we have by monotone convergence (Θ, Θ a.e. + From Proposition 3, we deduce t Θ (t dt + tθ(t dt + η tθ(t dt. tθ(t dt. Θ(t λ (γ log( ω t C. (6 Indeed, it suffices to extend the boundary datum g in a neighborhood δ of, which is possible with an energy controlled by the H norm of g on, and then to apply Proposition 3 to this extension, since then ω t is at distance greater than δ from the boundary. Hence, E ε (u, ε ( t ω t tλ (γ log( ω t dt C. 8

19 Now, for fixed ε > and t (,, the function r ε ( t r tλ (γ log(r has a minimum for r = / ελ (γt( t, thus E ε (u, ( ελ (γt tλ (γ tλ (γ log dt C = λ ( t (γ log ε C (7 since t t log(t( t L (,. This concludes the proof. Before going further, let us state a corollary of Theorem, which allows the boundary map to have a modulus different form but greater than or equal to /. It is the analogue of the Corollary (p. 395 in [34]. Corollary 4. Let be a smooth bounded and simply connected domain in R and v : X be such that v + ( v ε = e ε (v < + and v / on. Let γ be the homotopy class of v v. Then, for any u H (, X such that u = v on and < ε < ε (, v, E ε (u λ (γ log ε C, where C depends only on the smoothness of and e ε(v. Proof of Corollary 4.. It suffices to extend v in a map ṽ in the same way as in the lemma (p. 395 in [34] to a δ-neighborhood δ of in such a way that ṽ = on δ, the energetic cost of the extension and δ ṽ is controlled by e ε(v and the smoothness of. Then, we apply Theorem to the extended map ṽ in δ. 4.3 Proof of Proposition 4 We follow closely the lines of the proof of Theorem 3 in [34]. First, we can assume u since the projection v of u on the unit ball B ( X (defined by v = u if u and v = u u if u satisfies v = g on, e ε (v e ε (u, thus by Theorem, by hypothesis. Consequently, λ (γ log ε C E ε (v E ε (u λ (γ log ε + C, E ε (v E ε (u C and any localization property of the energy at order for v is inherited by u (at order. We denote T T tθ(t dt, + I t Θ (t dt, N From the proof of Theorem, we know that and also that ε ω t ( t dt, J u ( u + ε dx, ( tλ (γ log dt C. ω t J T T, I N (8 E ε (u T + N I + J λ (γ log ε C. (9 9

20 Moreover, we have by hypothesis In particular, one infers, E ε (u = T + N λ (γ log ε + C. ( T J C and N I C ( for a constant C depending on, g and C. Writing = t dt, we deduce from (8 T λ (γ log ε J λ (γ log ε and since t log( t L (,, T λ (γ log ε λ (γ t dt = λ (γ ( ε t log dt C, ω t ( t log ε ω t ( t dt C. By Jensen inequality applied with the concave function log and the interval [, ] with measure t dt (hence the total mass of [, ] is, We therefore deduce ( t log ε ω t ( t dt log ( T λ (γ log ε λ (γ log ( ( log t ε ω t ( t dt ε ω t ( t dt + log. ε ω t ( t dt C = λ (γ log I C, where C depends only on, g and C. Since N I and using (, we infer and thus λ (γ log ε + C T + N λ (γ log ε + N λ (γ log N C, ( We easily deduce from this relation that N λ (γ log N C. N C (3 for a constant C depending only on, g and C. This is the first assertion of the Proposition. Concerning the second one, we first infer from (3 that ε ω t ( t dt = I N C. (4 Moreover, we have by ( (note that the integrand is uniformly bounded from below from (6 in the proof of Theorem tθ(t + tλ (γ log( ω t dt C. (5 Hence, by (4, (5 and the mean-value formula, there exists / t such that ω t Cε and Θ(t + λ (γ log( ω t C. (6

21 Applying Proposition 3, taking C larger if necessary, for any C η Cε, there exist k disjoint disks D i, i k, with radii η and k Kλ (γ such that Comparing with (6, we obtain k i= D i \( k i= D i ( u u λ (γ log η ω t. ( u u λ (γ log η + C. Therefore, since u and using the bound (3 for the modulus, we have E ε (u, \ k i=d i λ (γ log η + C. This concludes the proof of Proposition Proof of Lemma 3 The purpose of this subsection is to construct a comparison map close to the minimizer. We choose n and γ k ( k n achieving the minimum in (, that is γ γ... γ n and 4πλ (γ = n k= λ(γ k. For each k n, we fix a minimizer c k γ k for (9, that is such that λ(γ k = π S c k (t dt. Fix k distinct points x k in and let r > be such that the disks D r (x k are mutually disjoint and do not intersect. We consider the map v : ( n k= D r(x k N defined by v (x g(x on and v (x k + rω c k (ω for ω S. By definition, since γ γ... γ n, we may extend v to a map v H ( \ ( n D k= r (x k, N. We therefore define u : X by u v on \ ( n D k= r (x k and, denoting x x k = ρω, ρ r and ω S, we set u(x ρ ε c k(ω if ρ ε, c k (ω otherwise. We then compute, for k n, E ε (u, \ ( n D k= r (x k = \( n k= D(x k v = C, E ε (u, D ε (x k C and E ε (u, D r (x k \ D ε (x k = ( r dρ ( c k ε ρ λ(γ k log ε + C, S 4π where C depends only on r. Combining these three inequalities concludes the proof. 4.5 Proof of Theorem 3 The proof of the weak Hloc convergence is similar to the one of Theorem 4 in [34] (using the compactness result given in [3], Theorem.3, with X locally compact since N is compact. We just have to replace ( there by E εn (u n, \ D Dn(ηD λ (γ log + C. (7 η

22 This step already exhibits the points (a n i for i k (since k Kλ (γ, we may assume, up to a subsequence, that k is independent of n. By construction, the a n i s are the centers of the disks in u (7, assuming in addition that the homotopy class of n u n on the boundary of the disk is non-trivial (this additional requirement can be imposed without changing the proof of [34]. The points (a i are limits (up to another subsequence of (a n i. We then have a i for i k. Proof of the W,p bound. Concerning the bounds in W,p, we argue as in [4]. Fix p <. We consider η j j and denote m the smallest integer such that η m C and N the largest integer such that η N ω t. Note that m depends only on, g and C, whereas N + as ε since ω t Cε. We first bound \ω t u n p, where t is the one given in the proof of Proposition 4, writing j k i= D i(η j and \ω t u n p N j=m+ Using Hölder inequality, we obtain \ω t u n p u n p + u n p + u n p. j \ j \ m N \ω t N ( p/ Area p/ ( j \ j u n j \ j ( p/ + Area p/ ( \ m u n \ m ( p/. + Area p/ ( N \ ω t u n N \ω t j=m+ In view of (7, the middle term is bounded by C, and for the other two terms, we use Proposition 4 with η = η j to obtain \ω t u n p C ( + N j=m+ ( p/j ( + log( j p/ + ω t p/ ( log ω t p/. Since the right-hand side is uniformly bounded in ε, that is in N m +, we obtain \ω t u n p C p, (8 where C p depends only on p <,, g and C. We are then left with the L p bound on ω t, for which it suffices to write Hölder inequality ( p/, u n p Area p/ (ω t u n ω t ω t and to conclude with inequalities ( and (6 that Combining (8 and (9 yields the W,p bound u n p C p ω t u n p Cε p log ε /p C p. (9

23 and the weak convergence follows from u n and the compactness result in [3] (Theorem.3. Proof of a i. We now prove that the a i s are not on the boundary. Our proof adapts the proof of Lemma 3 in [4] and our Lemma 4.. Let r min{ a i a j, i j} >. We can prove as in Lemma 4. that, if and < r < R, then for v H (D R \ D r, N, v λ(γ R (D R \D r π ( + o R ( log(. r Indeed, we just note that if, for instance, the tangent line to at is {} R, the angle θ runs between π + o R ( and π + o R (. Consequently, for < r < r, we have, arguing as in the proof of Proposition 3, u \( k i= Dr(a i ( k i= λ(γ i β i ( + o r ( log r C, (3 4π where β i = if a i and β i = if a i, and C depends on g and. On the other hand, by weak Hloc convergence in \ {a,..., a k } and (7, u lim inf E ε n (u n, \ ( k n + i= D r(a n i λ (γ log r + C. (3 \( k i= Dr(a i Comparing inequalities (3 and (3 as r, we infer In particular, since λ(γ i > and k β i λ(γ i 4πλ (γ. i= k λ(γ i 4πλ (γ, i= we deduce β i = for all i k, that is a i, and k λ(γ i = 4πλ (γ, that is the configuration (γ i achieves the minimum in (. i= The minimizing case. We assume now that u n is a minimizer for all n, thus satisfying the bound ( (by Lemma 3. We then know that u n. We have to prove that u n u strongly in H loc ( \ {a,..., a k }. Let a \ {a,..., a k }. We argue as in the proof of Theorem in [3]. First, in view of the bound (7, there exists R > such that D R (a \ {a,..., a k } and E εn (u n, D R (a C(a, R. (3 Let D D R (a. Since H ( D C is compact, we may assume, up to a subsequence, that v n u n D v uniformly on D. By (3, one then infers that u n uniformly on D, thus u n / on D for n sufficiently large. In particular, since D \ {a,..., a k }, we may write, at least for n sufficiently large, v n = v n π(ζ n on D, 3

24 with / v n, ζ n H ( D, ˆN, and π : ˆN N is the universal covering of N (ζn is unique up to the action of an element of π (N. Indeed, since u n / on D for n sufficiently large, the u homotopy class of n u n is necessarily trivial for otherwise, there would exist a singular point a i inside D. We construct then a comparison map ũ n : D X by considering ρ n the solution of { ε n ρ n + ρ n = in D, ρ n = v n on D and let ξ n H (D, ˆN be a minimizing harmonic extension of ζ n from D into ˆN. By the result of Morrey (see [3], we know that this extension is smooth in D. We claim that we may assume ζ n ζ in C and weakly in H ( D, where ζ is a lift of v. Indeed, since N is compact, there exists a compact set K ˆN such that π(k = N (cover N by a finite number of balls on which we can trivialize the covering π. Then, we can choose the lifting ζ n such that ζ n ( D K. Since ζ n has its C,/ semi-norm bounded by C independently of n, then ζ n ( D is included in a fixed compact set: the CR / -(closed neighborhood of K. Therefore, we may assume that ζ n is uniformly bounded. We have ζ n ζ strongly in H / and therefore, by the well-known fact that minimizers of the Dirichlet integral are strongly precompact in H (see [7] for instance, we may assume (up to a subsequence that ξ n ξ strongly in H (D, N, where ξ is a minimizing harmonic extension of ζ. We then set ũ n ρ n π(ξ n. Similarly to [3], Theorem, one proves that D ρ n + ( ρ n ε n as n +. On the other hand, since ξ n ξ strongly in H (D, N, we infer ũ n + ( ũ n D ε n D ξ as n +. Using ũ n as a comparison map on D (ũ n = u n on D, we deduce lim sup E εn (u n, D ξ. (33 n + Moreover, u n u weakly in H (D and ξ is a minimizing extension of u D = v, thus D ξ u lim inf D E ε D n + n (u n, D. (34 The combination of inequalities (33 and (34 yields from which it comes and 4ε n lim n + E ε n (u n, D = D D ξ, ( u n as n + u n u strongly in H (D, where u is a minimizing extension of u D. By the result of Morrey (see [3], 5.4, we then know that u is smooth in D. In the case of any compact subset K of \ {a,..., a k }, we extract from the covering D R(a (a, a K a finite number of disks and the conclusion follows from the convergence on each disk. 4

25 Appendix A: Topological defects in ordered media A. Topological defects In this appendix, we focus on topological defects in some ordered media. The topological theory of these defects is explained in details by N.D. Mermin in [8]. An ordered medium is a region of the space described by a map which assigns to each point in this region an order parameter, that is an element of an order parameter space (also called manifold of internal states N. A first well-known example is the case N = S, modeling planar spins. Another basic example is the RP model of liquid crystals, in particular uniaxial nematics. A molecule of a liquid crystal may be seen as a thin rod, and the order parameter is the local orientation of the molecule (see [] and [3] for the french reader, and is called the director. In this case, N = S. However, n and n characterize the same physical state, hence, the parameter space S must be replaced by the projective space N = RP S /{±}. In condensed matter physics, we expect that the order parameter varies continuously in the region occupied by the medium, except at points or on lines. In dimension two, for instance, the order parameter may be continuous in a neighborhood of a point minus this point. Its restriction on a small circle around the point may be a map non-homotopic to a constant in N, that is a non-trivial element of π (N. Hence, the singularity at that point can not be removed by local and continuous surgery and we call this singularity a topological defect. In the case of planar spins, the order parameter space is N = S, and the fundamental group π (S is Z. The topological defects are called in this case vortices and are classified by the degree, also called the winding number, that is the number of turns of the order parameter. This type of defects appear for instance in the Ginzburg-Landau model (for superconductivity or superfluidity. This underlines the role of the topological properties of the order parameter space N, for instance the first homotopy group. For liquid crystals, we emphasize that the main difference between the two models, S and RP, is of topological type. Indeed, we have π (S = {} and π (RP = Z/(Z and the important fact is that S is simply connected, whereas RP is not. In particular, in dimension two, there can not exist topological defects in the S model (since S is simply connected, any singularity can be removed by local and continuous surgery, whereas it does occur in the RP model (RP is not simply connected. The only non-trivial defect in this model is called disclination, and looks as follows Figure : The disclinaison In dimension three, the order parameter may be continuous except on a curve giving a non-trivial element of π (N on some small circle around the curve. In liquid crystals, these line defects are 5