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Static Cosmic Strings on S2 and CriticalityAuthor(s): Yisong YangSource: Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 453, No. 1958 (Mar.8, 1997), pp. 581-591Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/53096 .

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Static cosmic strings on S2 and criticality

BY YISONG YANG

Department of Applied Mathematics and Physics, Polytechnic University Brooklyn, New York 11201, USA

In this paper we construct static cosmic string solutions in the Bogomol'nyi phase of the Abelian Higgs model in an S2 setting. The strings are located at the north and south poles and have identical winding numbers. This construction, in particular, solves an open problem regarding the existence of a 2-string solution on S2. Furthermore, we prove that there are no symmetric solutions realizing N strings superimposed at a single point. Such a result implies, in particular, that there is no symmetric 1-string solution on S2 and is the only known non-existence theorem for cosmic strings confined on a compact surface. The symmetry of the underlying system enables us to work on a nonlinear ordinary differential equation subject to some prescribed boundary data at t = ?oo. A shooting method is employed as an analytic tool in the study of the existence problem.

1. Introduction and main results

Developments in the areas of particle physics and cosmology in the last 15 years or so have led to the general scenario that phase transitions in the early universe can give rise to an interesting class of topological defects called cosmic strings which are believed to be responsible for accretion of matter to form galaxies (Kibble 1980, 1990; Vilenkin 1985; Brandenberger 1991; Vilenkin & Shellard 1994). Phase transitions follow a sequence of symmetry-breakings as the temperature and energy level are lowered so that in relatively later stages physical interactions are governed by superconducting cosmic strings known to arise as vortex-like solutions of the coupled Einsten and Abelian Higgs equations for which the spacetime metric assumes the following cylindrically symmetric form:

ds2 = -(dx?)2 + (dx3)2 + gjk dxdxk, j, k = 1,2, (1.1)

where g = (gjk) is the Riemannian metric of an unknown 2-surface M. The metric (1.1) leads to a remarkable reduction of the Einstein tensor G,,. In

fact, in isothermal coordinates of M, there holds

- Goo = G33 = Kg, GI, = 0 for other values of ,, v = 0,1,2,3, (1.2)

where Kg is the Gauss curvature of the surface (M, g). Thus, as a consequence of (1.2) and the Einstein equations

Gt = -47rGT/, (1.3) where G is the Newton constant of gravitation and T,, is the energy-momentum tensor of the gauge-matter sector, there is a stringent restriction to the form of T,:

? = Too = -T33, TV = 0 for other values of /, v = 0,1,2,3. (1.4)

Proc. R. Soc. Lond. A (1997) 453, 581-591 ? 1997 The Royal Society Printed in Great Britain 581 TEX Paper



In the case that the gauge-matter sector is given by the Abelian Higgs model, the condition (1.4) singles out the so-called Bogomol'nyi phase in which the masses of the Higgs and gauge bosons are identical and the coupled Einstein and Abelian Higgs equations are equivalent to the following self-dual system when M is a compact Riemann surface (Linet 1988; Comtet & Gibbons 1988; Yang 1995)

Kg = 4rGT, AA = 0, > E M, (1.5)

*gFA (82 - 112),

where the gauge field A is a real-valued connection 1-form of a unitary line bundle L -+ M, DA is the (0, 1)-component of the induced connection DA from A, (q is a cross-section of L -- M which is identified as the Higgs field, FA = dA is the curvature of L -* M, e > 0 denotes the energy scale of symmetry-breaking in the model, and g* is the Hodge dual induced from the unknown metric g on M. When M is uncompact, equations (1.5) simplify the original Einstein and Abelian Higgs equations. In fact equations (1.5) (and their conjugate, the anti-self-dual equations) are the only known structure that makes the consistency condition (1.4) stated above valid.

The second equation in (1.5) implies that q is a holomorphic section, and thus has a finite set of zeros, say Pl,... , PN where the integer N is also the first Chern

class, N = cl(L), of L -* M. Such a solution is called an N-string solution. The zeros pi,... PN are the centres of energy concentrations for the gauge-matter sector. Namely the energy density function ? assumes local peaks at these points. Using this information in the first equation in (1.5) we see that the points Pi,... ,PN also give us N curvature concentration centres. In cosmology, these points are the seeds for matter accretion in the early universe.

There are two situations of interest. The first is when M is non-compact and conformally R2 In such a situation we have obtained a somewhat complete under- standing of the solutions of (1.5) (Chen et al. 1994; Yang 1994, 1995). In this case there are two types of obstructions. The first type comes from the finite-energy re- quirement and the second type comes from geodesic completeness of the gravitational metrics. The second situation is when M is compact. In this case we have shown in Yang (1995) that topologically M must be the sphere S2 and the string number N and the symmetry-breaking parameter e > 0 must satisfy the constraint

1 N= G (1.6) rS2G'

Furthermore, under (1.6) and the condition N > 3, we proved in Yang (1995) that equations (1.5) have an N-string solution. The main goal of the present paper is to settle the case N - 2 and give a non-existence result for the case N = 1. More precisely, we establish

Theorem 1.1. Consider the cosmic string equations (1.5) with the prescribed string number N over the standard 2-sphere S2 and assume that the symmetry- breaking scale E > 0 already satisfies the neccessary condition (1.6).

(i) When N = 2No (No > 1) is an even integer, the system (1.5) has an N-string solution so that the centres of the strings are at the north and south poles and there are exactly No strings at each of these two poles. In particular, there exists a 2-string solution with strings located at the oppossite poles.

Proc. R. Soc. Lond. A (1997)

582 Y. Yang




(ii) The system (1.5) does not have any N-string solution so that the strings are all superimposed at one point on S2 and the field configuration is symmetric about this point. In particular, there does not exist any symmetric 1-string solution at all.

These S2 solutions are in one-to-one correspondence with radially symmetric solutions on conformally Euclidean surfaces. The solutions stated in part (i) above give us a family of planar solutions which eliminate a peculiar obstruction obtained in Chen et al. (1994) when a background freedom is fixed (see ? 6). Such solutions are also presented in Linet (1990, 1994) and Kim & Kim (1994) through heuristic arguments. Here we prove their existence rigorously by a shooting method. The proof is rather different from that used in Chen et al. (1994). The non-existence result stated in part (ii), however, is a direct application of the general obstruction theorem (theorem 3.1) in Chen et al. (1994). In this sense the present paper can be viewed as an important supplement to our aforementioned paper published previously in this journal.

The rest of the paper is organized as follows. In ?2, we set up the differential equation to be solved which is defined in the complement of one of the poles on S2 and may conveniently be regarded as an equation over R2. In ?3, we construct a special family of solutions of the resulting equation to be used to produce a solution in ? 4 on the full S2 which furnishes a proof for part (i) in the main theorem. In ? 5, we prove part (ii). In ? 6, we observe some connections of our S2 solutions and the solutions in R2. In particular we show that a critical constraint originally read as a gap in the parameter range for the existence of finite-energy radially symmetric solutions must be eliminated when one allows freedom in the choice of background metrics. We then conclude the paper with some remarks.

2. Differential equation setting

Consider (1.5) defined over the base manifold (S2,g) where g is an unknown gravitational metric. Suppose that the zeros of 0 are Pi, ...,PN C S2. Then the substi- tution u = ln 112 transforms (1.5) into the equivalent form (Yang 1995):

N

Kg = -7G (E2 [eu - E2] - Age) , Agu = (eu - e2) + 47rZSe, (2.1) -=1

where Ag is the Laplace-Beltrami operator induced from g with the sign convention that all the eigenvalues of A9 are non-positive and 6p is the Dirac distribution on (S2, g) concentrated at p E S2.

Case (i): N = 2No, No > 1. Suppose that pi = .= pNo = n = the north pole of S2 and Po+l =p * = PN = s = the south pole of S2. We use P = (R2, (x)) = S2-{s} and P' = (R2, (x')) S2 - {n} to cover S2 through stereographical projections from the south and north poles, respectively. For convenience, assume that, in P, the unknown metric g is conformal to the Euclidean metric. Namely gjk = e'Sjk. Then, on S2 - {s} = R2 under P, the system (2.1) becomes

A(,r + 27rGeU) = 27r2Ge7(eu-2) R (2.2 Au = e'(eu - o2) + 4rNo(x), J

where 6(x) is the Dirac distribution on R2 concentrated at the origin. For simplicity,


583



we introduce a rescaling u -u + l n 2, (2.3)

and set

a = 27re2G. (2.4) Thus, in view of (2.3), (2.4), the system (2.2) gives us

(r7/a) = u - e - 2No In Ixl + h, (2.5) where h is a harmonic function over R2 which may be assumed to be an undetermined constant. Using (2.5), we obtain from (2.2) a single scalar equation

Au = Ax -2aNoe-(u-) (eu - 1) + 47rNo5(x), (2.6)

where A > 0 is a parameter which comes from h = const. in (2.5) and may be adjusted according to our need.

3. Solution on P

In this section, we establish the existence of a solution of (2.6). Such a solution is only local for the S2 problem (2.1) because it is only defined on the coordinate patch P around the north pole. In the next section, we will show that our solutions obtained here can be used to produce solutions with strings sitting at both poles.

According to Spruck & Yang (1992), when we look for radially symmetric solutions of (2.6), the problem is equivalent to solving the following ordinary differential equation subject to a singular boundary condition at r = 0:

Urr + Ur = Ar-2aNoea(u-)(eu - 1), r > 0, r

U(r) r r > (3.1)

lim = lim rur(r) = 2N0. r-0o Inr r--40

It will be convenient to use the new variable t defined by

t=lnr, r=et. (3.2)

Thus, by virtue of (1.6) or aNo = 1, (3.1) becomes

Utt Aea(u-e)(eu - 1), -o0 < t < oo,

u(t) \(3.3) lim ) = lim ut(t) = 2No. t---oo t t--oo

We will look for a special subclass of solutions of (3.3) so that the asymptotic property

lim (t) = lim ut(t) =-2No (3.4) t-?oo t t-oo

holds true. Using the maximum principle in the differential equation in (3.3), we see that a solution of such type must be globally concave down and has a unique negative maximum. Since the equation is autonomous, we may assume that the maximum is attained at t = 0. This observation motivates our study of the following initial value problem

tt = Aea(u-eu)(eu - 1), -00 < < 00, (3.5)

u(0) = -a, a > 0, ut(o) =. J Proc. R. Soc. Lond. A (1997)

584 Y. Yang




Simple arguments show that for any a > 0, the problem (3.5) has a unique global solution. For each given a, we will find a suitable A so that the solution of (3.5) fulfils the boundary condition stated in (3.3), (3.4). Therefore we are to carry out a shooting procedure to match the 'two-point' boundary value condition stated in (3.3) and (3.4) at t = ?oo. Furthermore, we will show as a remark in the last section that a, A may be determined explicitly. This latter property makes (3.5) practical in providing numerical solutions of (3.3), (3.4).

Let u be the solution of (3.5). Of course utt < 0 and ut < 0 (t > 0). Integrating the equation in (3.5), we have

ut(t)= I Aea(u(T)-eu('r)(e(r- 1) dr.

We set formally

f(A, a) = j ea(u(T)-eu(r))(eu() - 1) dr. (3.6)

Lemma 3.1. The function f(A, a) is continuous in A > 0, a > 0.

Proof. Step 1. f(A, a) is finite. In fact, since utt < 0, we see that f(A, a) = ut(oo) for each pair A > 0,a > 0 is

either a negative number or -oo. However, the latter does not happen because

ut(t) < ut(l) = -lut(1)1, t > l (3.7)

implies that u(t) < - ut(1) I(t-1) +u(l) (t > 1) which gives the convergence of (3.6). Step 2. f(A, a) is continuous. To see this property, we first notice that the continuous dependence of u on A and

a implies that the quantity

ut(1) = Aea(u(T)-e(T)) (eU(T) - 1) d

satisfies the bound

ut(1) < -Co for A E [A, A2], a E [a, 02], (3.8)

where Aj,aj > 0 and Co > 0 only depends on Aj, aj, j = 1,2. Inserting (3.8) into (3.7), we have

u(t) < -Co(t - 1) + u(l) < -Co(t - 1), t > 1. (3.9)

Using (3.9) in (3.6), we see that (3.6) is uniformly convergent in [A1, A2] x [a1, a2]. Consequently (3.6) is continuous in A > 0, a > 0 as expected. This proves the lemma.

We are now ready to invoke a shooting argument to prove the following lemma.

Lemma 3.2. For any a > O, there is a A = A(a) > 0 so that the solution u(t) of (3.5) satisfies

lim ut(t) = -2No. (3.10) t- oo

Proof. By the property -oo < u < -a < 0, we have e-a < e-aeu < 1. Hence the equation in (3.5) gives us the inequality

Ae-aeau(eu - 1) > Utt > Aeau(eu - 1), t > 0. (3.11)


585



The right-hand side of (3.11) leads us to

utt > -eae, t > 0. (3.12)

Since ut < 0 for t > 0, we obtain by multiplying (3.12) by ut and integrating in t > 0 the upper bound

A t

(Ut (t))2 < _ au(t) 2 aC a

(e-a( _ eau(t)) - (e^ -e a

Therefore

0 > ut(t) > -(2A/a)(e-aa - ea(t)). (3.13)

On the other hand, the left-hand side of (3.11) implies

utt < Ae-a(e- - l)eau, t > 0. (3.14)

Multiplying (3.14) by ut < 0 and integrating in t > 0, we obtain the lower bound

(t(t))2 > Ae-a - )(eau - e-a) a

= e-a -(e-)(e-a - eaU), t > 0. a

Thus

ut(t) < -/(2A/a)e-a(1

- e-)(e-ac

- eau(t)). (3.15)

From (3.13) and (3.15), we find after setting t = oo the inequalities

f f(A,.a) >~_ C2 _, a } (3.16)

f(A, a) - e--e-a(l -e-l )e-a".

By (3.16) we can find A1 > 0 and A2 > 0 so that

f(Ai, a) <-2No < f(A2, a).

Since f(., a) is continuous (see lemma 3.1), we conclude from the above that there exists some A = A(a) between A1 and A2 so that f(A(a), a) = -2No. The proof of the lemma is complete. U

Lemma 3.3. Any solution of (3.5) must be an even function.

Proof. Given a > 0, let u(t) be the unique solution of (3.5). Define a new function u by setting

u(t), t- ' t) (-t), t < 0.

It is easily checked that u is an even function which also solves (3.5). By uniqueness, u = u and the lemma follows. 1

In view of lemmas 3.2, 3.3, we see that for any a > 0 there is a A = A(a) > 0 so


586 Y. Yang




that the problem (3.5) has a unique solution which also satisfies (3.3), (3.4). Using the transformation (3.2) with r = Ix , we obtain a radially symmetric solution of (2.6) which gives rise to a solution of (2.1) in the coordinate patch P so that the first No strings are all located at the north pole. The rest of the problem is to show that this solution also gives rise to No strings at the south pole.

4. Solutions on the full S2

Let u(x) be a radially symmetric solution of (2.6) found in the last section as a solution of (3.3), (3.4). Then, as a function of r = Ixl, u satisfies both the boundary condition in (3.1) and the additional property

lim rur(r)= -2No. (4.1) r--oo

We will show in this section that the property (4.1) enables us to extend u originally constructed over P = S2 - {s} to the entire S2 to give rise to a full S2-solution with No strings sitting at both north and south poles. To this end, we recall that equation (2.1) ~under the coordinate patch P' = (R2, (x')) around the south pole may be reduced into an equation similar to (2.6):

Au = Alx'l- aNoeau-e) (e - 1) + 47rNo6(x), x' E R2, (4.2)

so that, with r' = 1x'l, its radially symmetric version is of the form

ur/ + urT = Arf-2aNoea(ueu)(eu - 1), ' > 0

ur r'- (4.3) lim =im ml r'uTr,(r') = 2No. r-,o Inr' r,--Io

On the other hand, since P = (R2, (x)) and P' = (R2, (x')) are represented by the stereographical projections with respect to the north and south poles, respectively, we have the relation

rr'= 1, (4.4) where the fact that S2 has unit radius has been used. With (4.4) in mind, we deduce the simple identities

r'ur, = -rur, }

^ "-T-'I inPflP'. (4.5)

T2 Ur'r' + rir' = r21rr + r in '

Ur ()

Using (1.6) or aNo = 1, (4.1), and (4.5) in (4.3), we see immediately that u can indeed be extended to an N = 2N0-string solution of (2.1) so that there are precisely No strings superimposed at each of the two poles. Therefore part (i) of theorem 1.1 is proven.

5. Non-existence of symmetric solutions

In this section, we show that (2.1) does not have any symmetric solution when the strings are all superimposed at a single point. Namely, pi = PN = P n = the north pole (say). To this end, we first state a simple criterion.

Lemma 5.1. Let (u,g) be a solution of (2.1). Then u < ln 2.


587



Proof. Let Sj be the complement of the 6-neighbourhoods of the points pi,... , PN in S2. Thus, for 6 > 0 sufficiently small, we have u(x) < lnc2 when x c dOS. Using this property and the maximum principle to the equation Agu = (eu - e2), x E Sj, we have maxXEs2 u(x) < Ine2 as expected. U

Lemma 5.2. In (2.1), let Pi = = PN = n = the north pole. Under this circumstance the system (2.1) has no solution on the full S2 which is symmetric with respect to the string point n.

Proof. Consider (2.1) in the open set S2- {s} = U. The metric g on U = R2 is globally conformal to the standard Euclidean metric of R2. Hence gjk = e7bjk and (2.2) holds with No = N. We use the translation (2.3) and (2.4) to simplify the notation. Since we are only interested in symmetric solutions, the unknown pair (u, r7) may be assumed to be radially symmetric with respect to the origin of R2 which corresponds to the north pole n of S2. Therefore the harmonic function h in (2.5) (with No = N) must be a constant by virtue of the radial symmetry. Hence we arrive consecutively at (2.6) and then (3.1) with No = N. The change of variable (3.2) may again be used to obtain the ordinary differential equation

Utt = Ae2(1-aN)tea(u-e) (eu - 1), -oo < t < oo,

~~~~~u(t) } >~~~~(5.1) lim (t) lim ut(t) - 2N.

t-o00 t t->-oo

Recall the condition (1.6). Namely, aN = 2. Then the new variable T =-t in (5.1) gives us the system

UTT = Ae2rea(U-e)(eu - 1), -oo <T < oo

u 1 \ (5.2) lim u() lim u(T) =-2N.

T-I-00 T T--00 /

If there is a solution for the original string problem, then, by lemma 5.1, the cor-

responding solution u of (5.2) satisfies u < 0 everywhere. Consequently the integral

ro ] e2ea(-e") (eu - 1) dr

-00

is convergent. This result implies that the number

c = lim Ur(T) (5.3) T->-00

is a finite number. Using the relation

UlT-_oo = U\x=sCS2 = a finite number,

we see that the only situation we can have in (5.3) is a = 0. Therefore, using theorem 3.1 in Chen et al. (1994), we find

2N E (4/a, oo)

or 7re2GN > 1, which contradicts the condition (1.6). The lemma is proven. 1

In conclusion, the proof of part (ii) of theorem 1.1 is complete.


Y. Yang 588




6. Criticality and background choice

The purpose of this section is to clarify an obstruction theorem obtained earlier in Chen et al. (1994) for radially symmetric solutions of (1.5) when M is conformal to R2. In this situation we have the following radial version of (2.1) with strings superimposed at the origin of R2:

A(r + 27Geu) = 272Ge(eu - 2), (6.1)

Au = eV7(eu - e2) + 4rN(x),

Hence it follows from (6.1) that there holds the relation

7q = 27e2G(u - eu - 2N n r) + c (6.2)

between the metric exponent r1 and the amplitude function u, where c is an arbitrary constant which clearly defines a 'background' metric. It is reasonable to assume that c in (6.2) is independent of the gravitational constant G. Therefore, if we require that the flat Minkowskian metric

ds2 = -(dx?)2 + (dx3)2 + (dxl)2 + (dx2)2

be recovered in the event that G = 0 (vanishing gravity and decoupling of the gauge- matter fields), we are led to the assumption that c = 0 which fixes the background freedom. Such a point of view is taken in Chen et al. (1994) and it is shown that there is a finite-energy solution if, and only if, the string number N satisfies N < 1/TrGe2 but stays away from

N= 2 G (6.3) ~WE2 G' The critical condition (6.3) seems rather peculiar. It has already been shown in Yang (1995) that, when one abandons the above background-fixing assumption and allows c in (6.2) to be an undetermined parameter, finite-energy solutions exist in the sector u = 0 at infinity. The study of the present paper establishes the existence of finite-energy solutions under the condition (6.3) which satisfy

u(7) lim () = lim rur(r)=-2N. r-oo in r r-

These results indicate that (6.3) should not be regarded as an obstacle to existence whenever background metric is not assumed to be fixed by the above procedure. On the other, however, the existence of finite-energy solutions in the parameter regimes away from (6.3) is not sensible to the choice of c in (3.16) as may directly be checked following the steps given in Chen et al. (1994). Summarizing these results, we obtain:

Theorem 6.1. In the Bogomol'nyi limit, the coupled Einstein and Abelian Higgs equations have a finite-energy radially symmetric N-string solution over a conformally Euclidean 2-surface M if, and only if, the string number N, the gravitational constant G, and the symmetry-breaking scale e > 0 satisfy the condition

N < . (6.4) 7re2G

We conclude this paper with a few remarks. First, we observe that our results in Yang (1995) and in the present paper reveal


589



that the existence of a multi-string solution over a compact 2-surface depends cru- cially on the local winding numbers of the strings. In Yang (1995), it is shown that a solution exists under the necessary condition (1.6) when the local string winding numbers are all smaller than half of the total string number N. The result stated in part (i) of theorem 1.1 says that the existence is still ensured when the strings are

sitting at the two opposite poles with equal winding numbers. This is an interesting borderline situation. As we go further and assume that the strings are all superimposed at a single point, still under the condition (1.6), there will be no symmetric solutions at all and the local winding takes its largest possible value, the global winding number N. Due to such a picture, we are led to believe that local windings of the strings play an important role to global existence on a compact surface.

Next, we would like to point out that it may be a reasonable conjecture that there is no solution on S2 for N strings superimposed at a single point. In particular, we conjecture that there is no 1-string solution at all. The key to this problem is of course to show that when pi =. = PN = p, any solution of (2.1), namely,

Kg = -rG(2[eu - 2] - A ge), AgU = (eu - e2) + 4rNSp, (6.5)

must be symmetric about p. A possible approach to this problem is to show first that in a punctured S2, which may be identified with R2, the unknown metric g is conformally Euclidean. Then one can reduce (6.5) into a scalar equation with radially symmetric coefficients and one proceeds to show that all its solutions must be radially symmetric as in the work of Chen & Li (1993) on a perturbed Liouville-type equation.

In the proof of the part (i) of theorem 1.1, we have seen that, to ensure the solv- ability of (3.3), (3.4) from the initial value problem (3.5), the parameter A depends on the value of u at t = 0, u(0) = -a. We remark that such a dependence can explicitly be determined. In fact, multiplying the differential equation in (3.3) by Ut and integrating over the interval (-oo, t), we have

u2(t) = 4N2 - (2A/a)ea((t-eu(t)) -0 < t < oo. (6.6)

Using u(0) = -a and ut(0) = 0 in (6.6) we obtain

A = 4,e2GN ea~ . (6.7)

Thus we can state that a solution of (3.5) gives rise to a solution of (3.3), (3.4) (strings concentrated at two opposite poles) if and only if (6.7) is fulfilled. The condition (6.7) is of obvious importance when one wants to obtain numerical solutions of (3.3), (3.4).

The formula (6.6) and the concavity of u have yet another interesting mathematical implication. To see it, we rewrite (6.6) as

LN2 -- 2 a(u((t)-e)) t O,

ut(t)= a

(6.8) t _j /4N2 - 2Aea(u(t)-eu(t)) t >0,

where A satisfies (6.7) and u(0) = -a. In view of (6.7), we see that ua -a is an equilibrium solution of (6.8). On the other hand, our solution to (3.3), (3.4) also solves (6.8) subject to the same initial condition. Therefore the solutions of (6.8) under the initial condition u(0) = -a suffer non-uniqueness. For this reason the original second-order problem (3.5) is more useful than its first-order reduction (6.8).

We have seen that the cosmic strings on S2 are closely related via stereographical


590 Y. Yang




projections to strings on the full plane. When the string number N is even and N = 2No (No > 1), a solution realizing No strings located at each of the two poles corresponds to the critical condition (6.3) for N and (6.3) is no longer an obstacle to the existence of a finite-energy N-string solution on the plane. When all the strings are superimposed at one point on S2, the condition (6.4) for the existence of planar finite-energy solutions prohibits the existence of symmetric solutions on S2 because the topology of S2 requires (1.6) which makes (6.4) invalid.

It is a pleasure to thank Emeric Deutsch whose numerical simulation of the 2-string solutions convinced the author that such solutions should exist as a consequence of non-uniqueness in equation (6.8). This work was supported in part by NSF under grant DMS-9400243.

References

Brandenberger, R. H. 1991 Cosmic strings and the large-scale structure of the universe. Phys. Scripta T 36, 114-126.

Chen, W. & Li, C. 1993 Qualitative properties of solutions to some nonlinear elliptic equations in R2. Duke Math. J. 71, 427-439.

Chen, X., Hastings, S., McLeod, J. B. & Yang, Y. 1994 A nonlinear elliptic equation arising from gauge field theory and cosmology. Proc. R. Soc. Lond. A446, 453-478.

Comtet, A. & Gibbons, G. W. 1988 Bogomol'nyi bounds for cosmic strings. Nucl. Phys. B 299, 719-733.

Kibble, T. W. B. 1980 Some implications of a cosmological phase transition, Phys. Rep. 69, 183-199.

Kibble, T. W. B. 1990 Cosmic strings-an overview. In The Formation and evolution of cosmic strings (ed. G. Gibbons, S. Hawking & T. Vachaspati), pp. 3-34. Cambridge University Press.

Kim, C. & Kim, Y. 1994 Vortices in the Bogomol'nyi limit of Einstein-Maxwell-Higgs theory with or without external sources. Phys. Rev. D 50, 1040-1050.

Linet, B. 1988 A vortex-line model for a system of cosmic strings in equilibrium. Gen. Relat. Grav. 20, 451-456.

Linet, B. 1990 On the supermassive U(1) gauge cosmic strings. Class. Quantum Grav. 7, L75- L79.

Linet, B. 1994, Les equations de Bogomol'nyi pour le modele abelien de Higgs en relativite generale. Preprint, Universite Pierre et Marie Curie.

Spruck, J. & Yang, Y. 1992 The existence of non-topological solitons in the self-dual Chern- Simons theory. Commun. Math. Phys. 149, 361-376.

Vilenkin, A. 1985 Cosmic strings and domain walls. Phys. Rep. 121, 263-315.

Vilenkin, A. & Shellard, E. P. S. 1994 Cosmic strings and other topological defects. Cambridge University Press.

Yang, Y. 1994 Obstructions to the existence of static cosmic strings in an Abelian Higgs model. Phys. Rev. Lett. 73, 10-13,

Yang, Y. 1995 Prescribing topological defects for the coupled Einstein and Abelian Higgs equations. Commun. Math. Phys. 170, 541-582.

Received 20 April 1996; accepted 9 May 1996


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