HARMONIC MAPS OF C TO GL n C IN LOW DIMENSIONS

Transcription

1 HARMONIC MAPS OF C TO GL n C IN LOW DIMENSIONS Raúl Felipe Comunicación Técnica No I-2-28/8--22 (MB/CIMAT)

2 Harmonic maps of C togl n C in low dimensions Raúl Felipe CIMAT Jalisco S/N, Valenciana, 3624 Guanajuato, Gto. APDO. POSTAL 42, 36 Guanajuato, Gto. ICIMAF Calle E No 39 esquina a 5, Vedado. La Habana, Cuba. November 2, 22 Abstract In this paper an ample class of non-trivial harmonic maps of D ½ C to GL 2C are presented. Several of them are related to some quasi-conformal mappings. These can be applied, in turn, to construct other harmonic maps fromd½ C intogl nc forn>2. Introduction Recently the harmonic maps of a compact domain D in the complex plane into a Lie group have been extensively studied [2], [3] [6] their references, in particular in [6] is given a comprehensive treatment of harmonic maps into U (n). On the other h, harmonic maps arise in physics in a natural manner as solutionsof the Chiralmodel (see [7], where only thecase n 2 wasconsidered). One of the techniques used in this area for almost 3 years has consisted of reformulating the harmonicmap equation asan integrablesystem (for instance as a Zakharov-Shabat type equation) lately, this development has been particularly strong in Great Britain. In the case, of a matrix Lie group, as is well-known, the two-dimensional Toda lattices have a relevant role. In this paper, non-trivial new harmonic maps into U (2) C GL 2 C are presented. But, our approach instead of starting with some integrable system consists of studying theharmonic map equation directly. Theidea behind our method is very simple: the solutions are expressed in the form of full Kostant-Toda matrices, which linearize the expression of the harmonic map equation by components. Hence, we make the results more explicit their proofs very clear. On the other

3 h, from our point of view harmonic maps Á : C! GL 2 C turn out to be related to quasi-conformal mappings. Let G be a matrix Lie group; the equation for a harmonic map Á : C! G is the following partial di erential equation Á Á z z + Á Á z z, () where z x + iy the overbar meanscomplex conjugation. It iswell known that the equation () is the Euler-Lagrange equation for the fuctional Á! R D kdák2 where D is a compact domain of C, i.e., Á jd satis es the harmonic map equation if only if it is a critical point of this functional. On the other h, the harmonic map equation is equivalent to the system U z V z [U;V ] (2) U z + V z where U;V : C! g. Here g is the Lie algebra of G, see [2] [3] for more details. It is well-known that the rst equation of (2) belongs to a distinguished class of PDEs known as integrable systems (it is the Zakharov-Shabat type equation). One comment is in order. The equations in (2) can be written as only one equation using a complex parameter establishing relations between harmonic maps loop groups, Riemann-Hilbert problems, etc. 2 Full Kostant-Toda harmonic maps The aim of this section is to present several non-trivial families of harmonic maps of D into GL 2 C starting from there obtain a large set of solutions of the matrix Zakharov-Shabat type equation. The results of this section can be considered as analogous to the Weierstrass formula in immersion theory, by means of which harmonic maps D! GL 2 C are described explicitly in terms of two functions (in some cases of two quasi-conformal mappings) on D. Let D be a compact domain of C ¹(z) dp dz de ned in D. We will assume that, dq dz Lemma The Beltrami type equation (z) a smooth function (z) 6 for all z 2 D, then we have (z) @z has a solution of the form z 2 D (3) u(z;z) q(z) + p(z) + u where u 2 C. 2

4 Proof. The Lemma can be proved directly. Note that when j¹j < k <, the solution u(z; z) of (3) will be a quasiconformal mapping (see []). Thanks to this simple Lemma we can nd some non-trivial solutions of (), that throughout thispaper we will call full Kostant-Toda solutions for the harmonic map equation. In what follows Á is an invertible matrix of the form a Á b c with b ac : (4) It turns out that matrices of the form (4) are generally not invertible (for instance, when c a b a 2 ), hence the condition b ac ; in this case, we have Á c b a A matrix of the form (4) with b ac will be called a full Kostant-Toda matrix a full Kostant harmonic map if this is a solution of the harmonic map equation. Let us suppose that a, b c depend on variables z z, then Á Á z therefore, we have : bz ca z c z, Á bz ca Á ba z ab z ac z z c z z ba z ab z ac z Á Á z z Á Á z z b zz c z a z ca zz c zz b z a z a z b z + ba zz ab zz a z c z ac zz b zz c z a z ca zz c zz : b z a z a z b z + ba zz ab zz a z c z ac zz Now, in order for the full Kostant-Toda matrix Á to be a solution of () we must do 2bzz c z a z c z a z 2ca zz 2c zz, (5) 2ba zz 2ab zz a z c z a z c z 2ac zz thus, we have 2c zz. Below, we choose c(z;z) g (z) + f (z) + c (6) 3

5 where c 2 C g (z), f (z) are smooth functions of each argument in a domain D of the complex plane. Thus, the function a(z;z) is seen to be a solution of the following dz dg dz z 2 D (7) (z) 6 in D; in the rest of this section we assume that such a condition is satis ed. On the other h, if dg dz 2b zz 2ca zz ; 2ba zz 2ab zz (8) multiplying the rst equation of (8) by a we have ab zz aca zz, now having in mind that b ac the last relation, from the second equation of (8) we have that, a zz ; also it is easy to see that b zz. Then, a b can be functions of the form a(z;z) m (z) + n(z) + a, b(z;z) h(z) + i(z) + b where a, b 2 C. By Lemma, one can readily see that the following functions are solutions of (7) a) a(z; z) g(z) f(z) + a (9) b) a(z; z) g(z) + f(z) + a Recall that, we have established a relationship between a, b c: b ac. In the case a) the function b(z;z) is given explicitly by in fact, a) b(z;z) g 2 (z) + (a + c )g (z) f 2 (z) + (a c )f (z) () + + a c (h(z) + i(z) + b ) (g(z) f(z) + a )(g (z) + f (z) + c ) (h(z) + i(z) + b ) g 2 (z) g(z)f (z) c g(z) + g(z)f (z) + f 2 (z) +c f(z) a g(z) a f(z) a c : become (). With similar arguments we can calculate b(z;z) in the case b), b) b(z; z) g 2 (z) + (a c )g(z) + f 2 (z) + (a + c )f(z) () + + a c In summary, we have the following result:. 4

6 Theorem 2 Let g f be two functions de ned on D (in particular they can be rational functions). Then, the full Kostant-Toda matrices g(z) f (z) + a g 2 (z) f 2 (z) + (a + c )g(z)+ g(z) + f (z) (a c )f (z) + + a c +c A (2) Á g(z) + f(z) + a g 2 (z) + (a c )g(z) + f 2 (z)+ g(z) + f (z) (a + c )f(z) + + a c +c A (3) are full Kostant-Toda harmonic maps of D to GL 2 C. The following Theorem establishes a very interesting property of the full Kostant-Toda harmonic maps. a Theorem 3 If Á is a full Kostant-Toda harmonic map, then b c Á a c b a is also a full Kostant-Toda harmonic harmonic map of D into GL 2 C. Proof. The role of the functions a c in (5) as also in the relation b ac, can be interchanged. In fact, both cases imply the following equations c zz ; a zz ; b zz ; a z c z + a z c z ; b ac. Corollary 4 Associated to Á Á 2 of the Theorem 2 are the following full Kostant-Toda harmonic maps Á g(z) + f (z) + c g 2 (z) f 2 (z) + (a + c )g(z)+ g(z) f (z) (a c )f (z) + + a c +a A Á a g(z) + f(z) + c g 2 (z) + (a c )g(z) + f 2 (z)+ g(z) + f (z) (a + c )f(z) + + a c +a A 5

7 Asa consequence of Theorem 2 we have the following solutions ofthematrix Zakharov-Shabat type equation. Example 5 Let H Á for g(z) z f (z) z; then H z 2z + (a + c ) H z 2z + (a c ) H So, de ning U H H z z z c z 2 z 2. + (a + c )z + (a c )z + + a c z + z a z z + a z 2 z 2 + 2zz + 2a (z z) + a 2 ; z + z a V H H z z z + a z 2 z 2 + 2zz + 2a (z z) a 2 z + z a we compute now U z V z, U z +V z, [U; V ] with a. A direct calculation gives U z + V z, on the other h we have U z V z 2 4(z z) 2 (4) nally UV 2(z z), V U 2(z z) then [U;V ] 2 4(z z) 2 (5) clearly (4) (5) imply that U z V z [U; V ]. Example 6 Beginning from the family Á 2 for g(z) z f (z) z we also have that the matrices 6

8 U z + z + a z 2 z 2 + 2zz + 2a (z z) a 2 ; z z a V are solutions of (2). z + z + a z 2 z 2 + 2zz + 2a (z z) + a 2 z z a We consider now full Kostant-Toda harmonic maps induced by the shift matrix its transpose T. Theorem 7 There exist full Kostant-Toda harmonic maps Á Á T, such that Á (Á ) z Á (Á ) z (6) Á T (Á T ) z T Á T (Á T ) z (7) Proof. The proof of the Theorem reduces to determine Á Á T such a that (6) (7) are satis ed. Let Á be a Kostant matrix, such b c that, (Á ) z Á. In this case we have az a b z c z b (8) from (8) it follows that a, b z c z b, we now take b ; it implies that c(z;z) z + h(z) where h(z) is a smooth function. On the other h, the condition (Á ) z Á asserts that c z 7

9 therefore h(z) z + h where h 2 C, thus Á z + z + h : Now, in the same way it is easy to see that Á T z + z + h z + z + h + Notice that the full Kostant-Toda harmonic maps Á Á T are not included in thefamily of solutions of thetheorem 2 the Corollary 4. Its easy to show that the other element of the base of sl2c, H does not generate a full Kostant-Toda harmonic map, since jhj 6. Corollary 8 The following full Kostant-Toda matrices Á a z + z + h, Á a T are full Kostant-Toda harmonic maps of D to GL 2 C. Proof. It follows from the Theorem 3. Remark 9 Consider the matrix S s z + z + h + z + z + h s. Le us denote L S S, from (6) follows that, then is obvious that S Á S Á S z L S zs (9) Á S Á S z L S zs (2) thus, (9) (2) imply that if L z + L z S z S z + S z S z (2) then Á S will be a harmonic map. After a straightforward calculation we can see that the equation (2) is equivalent to the following equation 8

10 2s zz (sz + s z ) 2s(s z + s z ) s z + s z hence, s z z + s or s z z + s. In this case L L 2 z z + s (z z + s ) 2 (z z + s ) z z + s (z z + s ) 2 (z z + s ) The corresponding full Kostant-Toda harmonic maps are Á L Á L2 : z z + s z 2 z 2 ; + (s + h )z + (s h )z + + s h z + z + h z + z + s z 2 + z 2 : + (s h )z + (s + h ) z + + s h z + z + h The solutions Á L Á L2 belong to the family Á Á 2 of the Theorem 2, respectively. Therefore, dressing we do not obtain new solutions of the hamonic map equation. Remark Choosing two 2 2 constant matrices A B such that [A; B], we can construct a harmonic map that satis es Á (Á) z A, Á (Á) z B; (22) clearly if Á satis es (22) then Á is a harmonic map. Notice that, before we considered the cases A B A B T, full Kostant-Toda harmonic maps were constructed. But, ingeneral fora B arbitrary constant matrices then the corresponding Á are not full Kostant-Toda harmonic maps. For instance, let us consider the Pauli matrix The matrix ¾ : 9

11 a e Á ¾ z+z b e (z+z) c e z+z d e (z+z) is a harmonic map if a b d 6, because c Á ¾ (Á ¾ ) z ¾ Á ¾ (Á) z ; but Á ¾ is not a full Kostant-Toda harmonic map. On the other h we want to remark that the full Kostant-Toda harmonic maps determined by the family Á Á 2 of Theorem 2 Corollary 4 do not satisfy (2) unless g f are both constant functions. 3 More full Kostant-Toda harmonic maps In the previous section, we have seen that using harmonic functions of the form : u(x; y) c f (x + iy) + c g (x iy) + c 2 (23) where c ; c c 2 are complex numbers, it is possible to construct harmonic maps of D into GL 2 C. But we do not have to restrict ourselves to this case, in fact, as we shall now show the functions a, b c can also have a very di erent appearance to (23), i.e, we can also consider functions a, b c of a more general nature. These harmonic maps are connected with more analytic methods. We recall that the equation (5) shows that if c zz ; a zz ; b zz ; a z c z + a z c z ; b ac. (24) then Á given by (4) will be a harmonic map of D into GL 2 C. Let D be a domain in C, with its compact in C. The domain D is said to be regular if the Dirichlet problem has a solution for every k 2 C ( ), i.e, if given k 2 C ( ) it is possible to nd a function F that is harmonic in D, continuous in the closure D satis es F (z) k (z), z 2 D (see [5]) Theorem Let D be a regular domain c 2 C 2 (D) a harmonic function in D, i.e, c zz ; then for every z 2 D there is a ball B (z ) jz z j < ² two functions a b not unique, such that in B (z ). a zz ; b zz ; a z c z + a z c z ; b ac (25)

12 Proof. The purpose isto build solutions of (25) through a known harmonic function c in D. The system of partial di erential equations in (25) has an abundance of solutions having the desired property b ac. In fact, let c be a given harmonic function in D (we recall that D is regular). The solution of the equation a z c z + a z c z : (26) will be required to have one of the following two properties: ) a z c z, a z c z, (27) 2) a z c z, a z c z, in some closed ball of D with center z. It is clear that those a for which (27 ) or (27 2) is true are solutions of (26). Let us write (27 ) as a systemoffour equations for bu bv (a bu + ibv) : bu x v y, bu y v x, bv x u y, bv y u x, (28) where c u + iv. Let z 2 D, let B (z ) be a circular neighborhood of z such that B (z ) ½ D. Hence, dbu v y dx + v x dy dbv u y dx u x dy are exact di erentials in B (z ), bu bv can be found by evaluating the following linear integrals Z (x;y) (x ;y ) [ v t (s;t)ds + v s (s; t)dt] bu(x;y) bu(x ; y ); (29) Z (x;y) (x ;y ) [u t (s;t)ds u s (s;t) dt] bv (x; y) bv (x ;y ); (3) along a line segment in B (z ), where bu(x ;y ) bv (x ;y ) can be chosen arbitrarily. But then bu(x; y) bv (x; y) are uniquely determined in B (z ). Now, since c is a harmonic function also in B (z ), c @v @u, is an analytic function in B (z ), then the Cauchy-Riemann equations for the function c (32)

13 Hence, from (28) (32) we @x (33) (34) In view of (33) (34) the function a z will be an analytic function in B (z ). Thus, a zz in B (z ). Let us de ne b + ac, then b zz a z c z + a z c z + a zz + c zz : It remains to take ² such that B (z ;²) ½ B (z ). The construction of a b in the case (27 2) is reduced to the case (27 ), replacing a by a. Theorem 2 Let D be a regular domain in C, with its compact in C. Then for every z 2 D there is " > for which one can nd at least two families of full Kostant-Toda harmonic maps of B (z ;²) into GL 2 C. Proof. This is a consequence of the Theorem 8. 4 Full Kostant-Toda harmonic maps intogl 3 C For n > 2 one cannot hopeto construct full Kostant-Toda harmonicmapseven in the case n 3 by meansof some simple algorithm, however theresultsof the sections 2 3 allow us to also construct full Kostant-Toda harmonic maps of D into GL 3 C starting with a full Kostant-Toda harmonic map for n 2. We have a Proposition 3 Let Á be a full Kostant-Toda harmonic map, b c where b ac, then 2

14 a b c is a full Kostant-Toda harmonic map of D into GL 3 C. Proof. It is easy to show that, A ³ c b a a A, then, with a straightforward calculation we obtain ³ bá báz ³ bá b z ca z c z ba z ab z ac b z ca z c z ba z ab z ac z A, : (35) A (36) Since Á is a harmonic map, by (35) (36) Á b will be a harmonic map. Now, a similar method enables one to obtain the following less evident result a Proposition 4 Let Á be a full Kostant-Toda harmonic map, b c where as always let us assume that b ac. Then, the matrix a b c itself is a full Kostant-Toda harmonic map of C into GL 3 C. Proof. Recall rst that Á is a harmonic map, then A c zz ; a zz ; b zz ; a z c z + a z c z ; b ac. On the other h, we have 3

15 ³ c c + ac a A, thus we see that ³ eá But then it follows from (37) that ca z b z c z ca z + b z c z ba z ab z ac z A. (37) ³eÁ eáz c za z + ca zz b zz c zz c z a z ca zz + b zz c zz b z a z + ba zz a z b z ab zz a z c z ac c za z c z a z b z a z a z b z a z c z A, A(38) also, in a similar manner it follows at once that ³eÁ c za z + ca zz b zz c zz c z a z ca zz + b zz c zz z b z a z + ba zz a z b z ab zz a z c z ac c za z c z a z A b z a z a z b z a z c z A(39) now, the conclusion of the Theorem is clear from (38) (39) taking into account the equations at the begining of the proof. In this part wecompletethestudy of full Kostant-Toda harmonicmapsfrom a compact D from C into GL 3 C. The following Lemma is immediately veri ed: Lemma 5 The equation uu zz u z u z z 2 D (4) admits the following two solutions u + (z;z) ¹ (z)e kz u (z; z) ¹ 2 (z)e kz, where ¹ (z) ¹ 2 (z) are functions de ned in D di erent from zero k 2 C. We now construct more full Kostant harmonic maps 4

16 Proposition 6 Let c be a harmonic function in D. Then, the following matrices Á + c c u + (z;z) A + c c u (z; z) where u + (z;z) u (z;z) are the solutions of (4) described in the Lemma 5, are full Kostant-Toda harmonic maps from C into GL 3 C. Proof. In fact, given a matrix of the form a b c f for which b ac f 6, we have A, A, then, it should be noted that c f a b a f f A, Á Á z b z ca z c z fz f ba z ab z ac z a fz f f z f C A (4) Á Á z b z ca z c z fz f ba z ab z ac z a fz f f z f C A (42) from (4) (42) it follows that in the particular case: a, then b c + obviously b is a harmonic function in D. On the other h, in order for Á to be a full Kostant-Toda harmonic map, is necessary that ff zz f z f z z 2 D: Now the Proposition is an inmediate consequence of the Lemma 5. 5

17 5 Other harmonic maps intogl 3 C In this section, we will construct other harmonic maps by a procedure that we call dimensional extension. The central idea is already found in section 4. We have the following a Lemma 7 Let Á D! GL b c 2 C be a harmonic map, then the following eg(z;z) Á Á e F(z;z) A (43) a ¾e E(z;z) b c A (44) will be harmonic maps of D into GL 3 C, whenever ; ¾ are complex numbers di erent from zero G, F E are harmonic functions in D. Proof. Let 6, T (z;z) Á A where Á D! GL 2 C is a harmonic map, we shall choose T (z;z) such that is a harmonic map. T (z; z) Á A ( ) (T (z; z)) z Á z A; ( ) (T (z; z)) z Á z A then 6

18 similarly ( ) T (z;z)(t (z;z)) z Á Á z A (45) ( ) from (45) (46) we see that if T (z; z)(t (z; z)) z Á Á z A (46) (lnt (z; z)) zz (47) then is a harmonic map. Thus, equation (47) gives T (z;z) e G(z;z) where G (z; z) is a harmonic function in D. Similar arguments can be used to prove that the matrix ¾ are harmonic maps: The previous results of the present paper allow us to construct many more harmonic maps from D into GL 3 C. For instance, from (43) (44) it follows z + z + h e eg(z;z) z + z + h z + z + h + A, A z + z + h ¾e E(z;z) z + z + h + A are harmonic maps of D into GL 3 C. 6 Acknowledgements This research was partially supported under CONACYT Grant 37558E partially supported by CITMA-ICIMAF through project 88. 7

19 7 References [] Ahlfors, L.V. Lecture on Quasiconformal mappings D. Van Nostr Company, Inc. 966 [2] Cherednik, I. Basic methods of soliton theory. Advances Series in Mathematical Physics. vol 25. World Scienti c, 996 [3] Guest, M.A. Harmonic Maps, Loop Groups Integrable Systems. London Mathematical Society. Student Text 38, 997 [4] Hille, E. Analytic function theory. Chelsea Publishing Company. 973 [5] Nikishin, E.M. Sorokin, V.N. Rational approximations orthogonality. American Mathematical Society. Providence, V 92 [6] Uhlenbeck, K.K. Harmonic maps into Lie groups (classical solutions of the chiral model). J.Di.Geom. 3(989)-5 [7] Ward, R.S. Classical solutions of the chiral model, unitons holomorphic vector bundles. Commun. Math. Phys. 28(99)