EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS

Transcription

1 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS FENGBO HANG, FANGHUA LIN, AND YISONG YANG Abstract. In this paper, we present an existence theory for absolute minimizers of the Faddeev knot energies in the general Hopf dimensions. These minimizers are topologically classi ed by the Hopf-Whitehead invariant, Q, represented as an integral of the Chern-Simons type. Our method involves an energy decomposition relation and a fractionally powered universal topological growth law. We prove that there is an in nite subset S of the set of all integers such that for each N S there exists an energy minimizer in the topological sector Q = N. In the compact setting, we show that there exists an absolute energy minimizer in the topological sector Q = N for any given integer N that may be realized as a Hopf-Whitehead number. We also obtain a precise energy-splitting relation and an existence result for the Skyrme model.. Introduction In knot theory, an interesting problem concerns the existence of ideal knots, which promises to provide a natural link between the geometric and topological contents of knotted structures. This problem has its origin in theoretical physics in which one wants to ask the existence and predict the properties of knots based on a rst principle approach [N]. In other words, one is interested in determining the detailed physical characteristics of a knot such as its energy (mass), geometric conformation, and topological identi cation, via conditions expressed in terms of temperature, viscosity, electromagnetic, nuclear, and possibly gravitational, interactions, which is also known as an Hamiltonian approach to realizing knots as eld-theoretical stable solitons. Based on high-power computer simulations, Faddeev and Niemi [FN] carried out such a study on the existence of knots in the Faddeev quantum eld theory model [F]. Later, Faddeev addressed the existence problem and noted the mathematical challenges it gives rise to [F]. The purpose of the present work is to develop a systematic existence theory of these Faddeev knots in their most general settings. Recall that for the classical Faddeev model [BS, BS, F, F, FN, FN, Su] formulated over the standard (3 + )-dimensional Minkowski space of signature (+ ), the Lagrangian action density in normalized form reads L u F (u)f (u); (.) where the eld u = (u ; u ; u 3 ) assumes its values in the unit -sphere and F (u) = u (@ u u) (.) is the induced electromagnetic eld. Since u is parallel u u, it is seen that F (u)f (u) = (@ u u) (@ u u); which may be identi ed with the well-known Skyrme term [E, E, MRS, S, S, S3, S4, B] when one embeds S into S 3 SU(). Hence, the Faddeev model may be viewed as a re ned Skyrme model governing the

2 FENGBO HANG, FANGHUA LIN, AND YISONG YANG interaction of baryons and mesons and the solution con gurations of the former are the solution con gurations of the latter with a restrained range [C]. We will be interested in the static eld limit of the Faddeev model for which the total energy is given by 3X E(u) = j@ j uj + 3X jf jk (u)j dx: (.3) R 3 j= Finite-energy condition implies that u approaches a constant vector u at spatial in nity (of R 3 ). Hence we may compactify R 3 into S 3 and view the elds as maps from S 3 to S. As a consequence, we see that each nite-energy eld con guration u is associated with an integer, Q(u), in 3 (S ) = (the set of all integers). In fact, such an integer Q(u) is known as the Hopf invariant which has the following integral characterization: The di erential form F = F jk (u)dx j ^ dx k (j; k = ; ; 3) is closed in R 3. Thus, there is a one form, A = A j dx j so that F = da. Then the Hopf charge Q(u) of the map u may be evaluated by the integral Q(u) = A ^ F; (.4) 6 R 3 due to J. H. C. Whitehead [Wh]. The integral (.4) is in fact a special form of the Chern Simons invariant [CS, CS] whose extended form in (4n ) dimensions (cf. (.) below) is also referred to as the Hopf Whitehead invariant. The Faddeev knots, or rather, knotted soliton con gurations representing concentrated energy along knotted or linked curves, are realized as the solutions to the minimization problem [F], also known as the Faddeev knot problem, given as j;k= E N inffe(u) j E(u) < ; Q(u) = Ng; N : (.5) In [LY, LY4], it is shown that E N is attainable at N = and that there is an in nite subset of, say S, such that E N is attainable for any N S. The purpose of the present work is to extend this existence theory for the Faddeev knot problem to arbitrary settings beyond 3 dimensions. Our motivation of engaging in a study of the Faddeev knot problem beyond 3 dimensions comes from several considerations: (i) Theoretical physics, especially quantum eld theory, not only thrives in higher dimensions but although requires higher dimensions [GSW, P, ]. (ii) The 3-dimensional Faddeev model may be viewed naturally as a special case of an elegant class of knot energies strati ed by the Hopf invariant in general dimensions (see our formulation below). (iii) Progress in general dimensions helps us achieve an elevated level of understanding [LY3, LY5] of the intriguing relations between knot energy and knot topology and the mathematical mechanism for the formation of knotted structures. (iv) Knot theory in higher dimensions [H, K, R] is an actively pursued subject, and hence, it will be important to carry out a study of ideal knots for the Faddeev model in higher dimensions. Note that minimization of knot energies subject to knot invariants based on diagrammatic considerations has been studied considerably in literature. For example, knot energies designed for measuring knotted/tangled space curves include the Gromov distortion energy [G, G], the Möbius energy [BFHW, FHW, O, O], and the ropelength energy [B, CKS, CKS, GM, Na]. See [JvR] for a rather comprehensive survey of these and other knot energies and related interesting works. See also [KBMSDS, Kf, M, S, SKK].

3 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 3 Although there are various available formulations when one tries to generalize the Faddeev energy (.3), the core consideration is still to maintain an appropriate conformal structure for the energy functional which works to prevent the energy to collapse to zero. The simplest energy is the conformally invariant n-harmonic map energy, where n is the dimension of the domain space, which is also known as the Nicole model [Ni] when specialized to govern maps from R 3 into S. Another type of energy functionals is of the Skyrme type [MRS, S, S, S3, S4, B] whose energy densities contain terms with opposite scaling properties and jointly prevent energy collapse. In fact, these terms interact to reach a suitable balance to ensure solitons of minimum energy to exist. The Faddeev model (.3) belongs to this latter category for which the solitons of minimum energy are realized as knotted energy concentration con gurations [BS, BS, FN, FN, Su]. In this paper, our main interest is to develop an existence theory for the energy minimizers of these two types of knotted soliton energies. Speci cally, we will study both the Nicole Faddeev Skyrme (NFS) type and Faddeev type knot energy (see (.4), (.5) and (.6) for de nitions). The two energy functionals have very di erent analytical properties. In particular, the conformally invariant term jruj 4n dx (.6) R 4n in the NFS model enables us to carry out a straightforward argument which shows that the Hopf Whitehead invariant Q (u) (see (.3)) must be an integer for any map u with nite NFS energy. More importantly, it allows us to get an annulus lemma (Lemma 3.) which permits us to freely cut and paste maps under appropriate energy control. In this way, as in [LY], the minimization problem ts well in the classical framework of the concentration-compactness principle [E, E, L, L]. Along this line, we shall arrive at the main result, Theorem 7., which guarantees the existence of extremal maps for an in nite set of integer values of the Hopf Whitehead invariant. The situation is di erent for the Faddeev energy (see (.6)). In this case, it seems di cult to know whether a map with nite energy can be approximated by smooth maps with similar energy control. In particular, it is not clear anymore why the Hopf Whitehead invariant (see (.3)), which is given by an integral expression, should always be an integer. Based on some recent observations of Hardt Riviere [HR] in the study of the behavior of weak limits of smooth maps between manifolds in the Sobolev spaces, and some earlier approach of Esteban Muller Sverak [Sv, EM], we are able to show that the Hopf Whitehead invariant of a map with nite Faddeev energy must be an integer (see Theorem 0.). Such a statement is not only useful for a reasonable formulation of the Faddeev model but also plays a crucial role in understanding the behavior of minimizing sequence and the existence of extremal maps. One of the main di culties in understanding the Faddeev model is that it is still not known whether an annulus lemma similar to Lemma 3. exists or not. In particular, we are not able to freely cut and paste maps with nite energy and it is not clear whether the minimizing problem would break into a nite region one and another at the in nity. That is, in this situation, the minimizing problem does not t in the framework of the classical concentration-compactness principle anymore. This di culty will be bypassed by a decomposition lemma (Lemma.) for an arbitrary map with nite Faddeev energy (in the same spirit as in [LY] for maps from R 3 to S ). Roughly speaking, the lemma says we may break the domain spaces into in nitely many blocks, each of which can be designated with some degree. By collecting those nonzero degree blocks suitably

4 4 FENGBO HANG, FANGHUA LIN, AND YISONG YANG we may have a reasonable understanding of the minimizing sequence of maps for the Faddeev energy (Theorem 3.). Based on this and the sublinear growth law for the Faddeev energy, we will obtain several existence results of extremal maps for the Faddeev energy (see Section 3.). We point out that the method to bypass the breakdown of the concentration-compactness principle is along the same line as [LY]. However, due to the fact that we do not have the tool of lifting through the classical Hopf map S 3! S in higher dimensions, we have to resort to di erent approaches to deal with the nonlocally de ned Hopf Whitehead invariant. When reduced to the Faddeev model from R 3 to S, this method gives a di erent route towards the main results in [LY]. Moreover, by establishing the subaddivity of the Faddeev energy spectrum (see Corollary 3.3), we are able to strengthen the Substantial Inequality in [LY] to an equality. That is, we are actually able to establish an additivity property for the Faddeev knot energy spectrum. We will also use the same approach to improve the Substantial Inequality for the Skyrme model to an equality (see Theorem 4.3). Here is a sketch of the plan for the rest of the paper. The rst part, consisting of Sections 7, is about the NFS model. In Section, we introduce the generalized knot energies of the Nicole type [AS, ASVW, Ni, We], the NFS type extending the two-dimensional Skyrme model [Co, dw, GP, KP, LY, PMT, PS, PS, P, SB, Wei], and the Faddeev type [F, F], all in light of the integral representation of the Hopf invariant in the general (4n ) dimensions (referred to as the Hopf dimensions). We will also obtain some growth estimates of the knot energies with respect to the Hopf number in view of the earlier work [LY3, LY5]. In Section 3, we establish a technical (annulus) lemma for the NFS model which allows truncation of a nite-energy map and plays a crucial role in proving the integer-valuedness of the Hopf Whitehead integral and the validity of an energy-splitting relation called the Substantial Inequality [LY4]. We shall see that the conformal structure of the leading term in the energy density is essential. In Section 4, we show that the Hopf Whitehead integral takes integer value for a nite-energy map in the NFS model. In Section 5, we consider the minimization process in view of the concentration-compactness principle of Lions [L, L] and we rule out the vanishing alternative for the nontrivial situation. We also show that the compactness alternative is needed for the solvability of the Faddeev knot problem stated in Section for the NFS energy. In Section 6, we show that the dichotomy alternative implies the energy splitting relation or the Substantial Inequality. These results, combined with the energy growth law stated in Section, lead to the existence of the NFS energy minimizers strati ed by in nitely many Hopf charges, as recognized in [LY]. We state these results as the rst existence theorem in Section 7. We then establish a simple but general existence theorem for both the generalized NFS model and the generalized Faddeev model in the compact case. For the Nicole model over R 3 or S 3, we prove the existence of a nite-energy critical point among the topological class whose Hopf number is arbitrarily given. The second part, consisting of Sections 8 3, is about the Faddeev model. In Section 8, we brie y describe the formulation of Faddeev model. In Section 9, various basic tools necessary for the study of Faddeev model are discussed. Section 0 is devoted to showing that for a map with nite Faddeev energy, the Hopf Whitehead invariant is well de ned and takes only integer values. We also derive a similar result for maps with mixed di erentiability (see Section 0.). Such kind of results are needed in proving the crucial decomposition lemma (Lemma.). In Section, we describe some basic rules

5 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 5 concerning the Hopf-Whitehead invariant for maps with nite Faddeev energy and the sublinear energy growth rate. Note that such kind of sublinear growth is a special case of results derived in [LY5]. The arguments are presented here to facilitate the discussions in Section, Section and Section 3. In Section, we prove a crucial technical fact: the validity of a certain decomposition lemma for a map with nite Faddeev energy. The proof of this lemma shares the same spirit as that in [LY] but is technically di erent due to the lack of lifting arguments. In Section 3, we prove the main result of the second part, namely, Theorem 3., which describes the behavior of a minimizing sequence of maps. Based on this description and the sublinear growth law, we discuss some facts about the existence of minimizers in Section 3.. In Section 4, we apply our approach in the second part to the standard Skyrme model to derive the subadditivity of the Skyrme energy spectrum and strengthen the substantial inequality to an equality. Finally, we conclude with Section 5.. Knot Energies in General Hopf Dimensions Recall that the integral representation of the Hopf invariant by Whitehead [Wh] of the classical bration S 3! S can be extended to the general case of the bration S 4n! S n. More precisely, let u : S 4n! S n (n ) be a di erentiable map. Then there is an integer representation of u in the homotopy group 4n (S n ), say Q(u), called the generalized Hopf index of u, which has a similar integral representation as (.4) as follows. Let! S n be a volume element of S n so that js n j! S n S n (.) is the total volume of S n and u the pullback map (S n )! (S 4n ) (a homomorphism between the rings of di erential forms). Since u commutes with d, we see that du (! S n) = 0; since the de-rham cohomology H n (S 4n ; R) is trivial, there is a (n )-form v on S 4n so that dv = u (! S n) (sometimes we also write u (! S n) simply as u! S n when there is no risk of confusion). Of course, the normalized volume form ~! S n = js n j! S n gives the unit volume and ~v = js n j v satis es d~v = u (~! S n). Since ~! S n can be viewed also as an orientation class, Q(u) may be represented as [GHV, Hu] Q(u) = ~v ^ u (~! S n) = v ^ u (! S js 4n n j S n): (.) S 4n The conformal invariance of (.) enables us to come up with the Hopf invariant, or the Hopf Whitehead invariant, Q(u), for a map u from R 4n to S n which approaches a xed direction at in nity, as Q(u) = js n j R 4n v ^ u (! S n); dv = u (! S n): (.3)

6 6 FENGBO HANG, FANGHUA LIN, AND YISONG YANG With the above preparation, we introduce the generalized Faddeev knot energies, subclassi ed as the Nicole, NFS, Faddeev energies over R 4n, respectively, as E Nicole (u) = jruj 4n ; (.4) R 4n E NFS (u) = fjruj 4n + ju (! S n)j + jn uj g; (.5) R 4n E Faddeev (u) = jruj 4n + ju (! Sn)j ; (.6) R 4n where and in the sequel, we omit the Lebesgue volume element dx in various integrals whenever there is no risk of confusion, we use the notation jruj, jduj, and jduj interchangeably wherever appropriate, and we use n to denote a xed unit vector in R n+ or a point on S n. Besides, we use c 0 to denote the best constant in the Sobolev inequality c 0 kfk q krfk (.7) over R 4n with q satisfying =q = = =(4n ) = (4n 3)=(4n ); given by the expression (n c 0 = ([4n ][4n 3])! ) (n + ) (4n ) 4n ; (.8) (4n ) with! m being the volume of the unit ball in R m. Theorem... Let E be the energy functional de ned by one of the energy functionals given by the expressions (.4), (.5), and (.6). Then there is a universal constant C = C(n) > 0 such that E(u) CjQ(u)j 4n 4n : (.9) In the case when E is given by (.6), the constant C has the explicit form Proof. Recall the Sobolev inequality over R 4n From the pointwise bound C(n) = n (c 0 js n j ) 4n 4n n n : (.0) of the form C(n; p)kfk q krfk p ; < p < 4n ; q = (4n )p 4n p : (.) ju (! S n)j C jruj n ; (.) and assuming dv = u (! S n) and v = 0, where is the codi erential of d which is often denoted by d as well, we have C jrvj 4n n R 4n R 4n ju (! S n)j 4n n C 3 R 4n jruj 4n ; (.3) where we have used an L p -version of the Ga ney type inequality [ISS, Sc] for di erential forms (we thank Tom Otway for pointing out these references). Choose p = (4n )=n so that q = (4n )=(n ) in (.). The conjugate exponent q 0 with respect to q is q 0 = q=(q ) = (4n )=n. Thus the Hölder inequality and

7 (.3) lead us to EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 7 js n j jq(u)j kvk q ku (! S n)k q 0 4n Ckrvk (4n )=n ku (! S n)k (4n )=n C jruj 4n R 4n 4n ; (.4) which establishes (.9) for the energy functional given by (.4) or (.5). Consider now the energy functional E p (u) = jruj p + ju (! Sn)j : (.5) R 4n In [LY5], we have shown that, when the exponent p in (.5) lies in the interval 4n(4n ) < p < ; (.6) 4n + there holds the universal fractionally-powered topological lower bound E p (u) C(n; p)jq(u)j 4n 4n ; (.7) where the positive constant C(n; p) may be explicitly expressed as C(n; p) = (c 0 js n j ) 4n 4n (n) p (4n p) (4n p) 4n (4n )(8n p) p(4n + ) (4n )(8n p) p(4n+) 8n(4n p) : (.8) It is seen that our stated lower bound for the energy de ned in (.6) corresponds to p = 4n so that C(n; 4n ) is given by (.0) as claimed. For the earlier work in the classical situation, n =, see [KR, Sh, VK]. Note that the energy E AF (u) = ju (! S n)j 4n n (.9) R 4n is also of interest and referred to as the AF model [AF] when n =. Combining (.3) and (.4), we have CjQ(u)j ku (! S n)k (4n )=n; (.0) which implies that the energy E AF de ned in (.9) satis es the general fractionallypowered topological lower bound (.9) as well. We next show that the lower bound (.9) is sharp. Theorem... Let E be de ned by one of the expressions stated in (.4), (.5), (.6), and (.9). Then for any given integer N which may be realized as the value of the Hopf Whitehead invariant, i.e., Q(u) = N for some di erentiable map u : R 4n! S n, and for the positive number E N de ned as E N = inffe(u)je(u) < ; Q(u) = Ng; (.)

8 8 FENGBO HANG, FANGHUA LIN, AND YISONG YANG we have the universal topological upper bound where C > 0 is a constant independent of N. E N CjNj 4n 4n ; (.) Proof. In [LY5], we have proved the theorem for the general energy functional E(u) = H(ru) dx; R 4n where the energy density function H is assumed to be continuous with respect to its arguments and satis es the natural condition H(0) = 0. Hence the theorem is valid for the energy functionals (.4) and (.6). For the energy functional (.5), there is an extra potential term ju nj. However, this term does not cause problem in our proof because the crucial step is to work on a ball in R 4n of radius jnj 4n and u = n outside the ball. Therefore, the potential term upon integration contributes a quantity proportional to the volume of the ball, which is of the form CjNj 4n 4n. In the following rst few sections, we will concentrate on the energy functional (.5). 3. Technical Lemma Let B be a subdomain in R 4n and consider the knot energy (.5) restricted to B, E(u; B) = fjruj 4n + ju (! S n)j + ju nj g: (3.) B We use B R to denote the ball in R 4n centered at the origin and of radius R > 0. The following technical lemma plays an important part in our investigation of the rst part of this paper. Lemma 3... For any small " > 0 and R, let u : B R n B R! S n satisfy E(u; B R n B R ) < ". Then there is a map ~u : B R n B R! S n such that (i) ~u = u R, (ii) ~u = n R, (iii) E(~u; B R n B R ) < C", where C > 0 is an absolute constant independent of R; ", and u. The same statement is also valid when ~u is modi ed to satisfy ~u = n R and ~u = u R. To obtain a proof, it will be convenient to work on a standard small domain. First, for the map stated in the lemma, de ne Hence y B n B and u R (y) = u(ry) for x = Ry B R n B R : (3.) " > E(u; B R n B R ) = fjr y u R (y)j 4n + j(u R ) (! S n)(y)j R + R 4n ju R (y) nj g dy: (3.3) B nb Consequently, we have 3= " > dr ds r fjru R j r + j(u R ) (! S n)j R + R 4n ju R nj g: (3.4)

9 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 9 Hence, there is an r (; 3=) such r fjru R j 4n + j(u R ) (! S n)j R + R 4n ju R nj g ds r < ": (3.5) In what follows, we x such an r determined by (3.5). Consider a map v R : R 4n! R n de ned by v R = 0 in B n B r ; (3.6) v R = u R r ; v R = n : (3.7) Then, for p = (4n ) =(4n ), we have, in view of (3.6) and (3.7), the bound which in terms of (3.5) leads to krv R k L p (B nb r) Ckru R k L 4n (@B r); (3.8) B nb r jrv R j (4n ) (4n ) C " 4n 4n : (3.9) Since (4n ) > 4n(4n ), we have p > 4n. So the Hölder inequality with conjugate exponents s and t gives us jrv R j 4n jb n B r j t jrv R j p s ; (3.0) B nb r B nb r where 4ns = p = (4n ) =(4n ) and t = s=( s). Therefore, we have, in view of (3.9) and (3.0), jrv R j 4n C " 4n 4n : B nb r (3.) Recall that, since R, we also have r ju R nj ds r < ". Hence, for any q >, we have r ju R nj q ds r C r ju R nj ds r C ". Since the ball is in R 4n, we see that for q = 4n(4n )=(4n ) (of course, q > ), we have kv R nk L 4n (B nb r) Cku R nk L q (@B r) C " q : (3.) Therefore, we have seen that (v R n) has small W ;4n (B n B r )-norm. Using the embedding W ;4n (B n B r )! C(B n B r ) (noting that dim(b n B r ) = 4n < 4n), we see that (v R n) has small C(B n B r )-norm. As a consequence, we may assume n v R > on B n B r : (3.3) Since v R is harmonic, jv R nj is subharmonic, jv R nj 0, on B n B r. Hence jv R nj C jv R nj ds r "C : (3.4) B nb r R4n To get a map from B n B r, we need to normalize v R, which is ensured by (3.3). Thus, we set w R = vr jv R j on B n B r : (3.5)

10 0 FENGBO HANG, FANGHUA LIN, AND YISONG YANG Then w R S n. We can check that jw R nj < 4jv R nj and j@ j w R j < 4j@ j v R j in view of (3.3). Therefore we have R 4n B nb r jw R nj 8C"; (3.6) R j(w R ) (! S n)j C B nb r B nb r jrw R j 4n C B nb r jrv R j 4n C " 4n 4n ; (3.7) B nb r jrv R j 4n C jb n B r j t B nb r jrv R j 4n s ; (3.8) where t = s=(s ) and s = 4n=(4n ). The bounds (3.) and (3.8) may be combined to yield jrw R j 4n B nb r C 3 ": (3.9) Thus, we can summarize (3.6), (3.7), and (3.9) and write down the estimate fjrw R j 4n B nb r + R j(w R ) (! S n)j + R 4n jw R nj g < C": (3.0) w R = n; r, w R = u R =ju R j = u R. De ne ~u(x) = w R R x for x B R n B rr ; ~u(x) = u(x) for x B rr : (3.) We see that the statements of the lemma in the rst case are all established. The proof can be adapted to the case of the interchanged boundary conditions ~u = u on B R and ~u = n on B R. Hence, all the statements of the lemma in the second case are also established. 4. Integer-Valuedness of the Hopf Whitehead Integral As the rst application of the technical lemma established in the previous section, we prove Theorem 4... If u : R 4n! S n is of nite energy, E(u) <, where the energy E is as given in (.5), then the Hopf Whitehead integral (.3) with v = 0 is an integer. Let the pair u; v be given as in the theorem and f" j g be a sequence of positive numbers so that " j! 0 as j! and fr j g be a corresponding sequence so that R j! as j! and E(u; R 4n n B Rj ) < " j, j = ; ;. Let fu j g be a sequence of modi ed maps from R 4n to S n produced by the technical lemma so that u j = u in B Rj and u j = n on R 4n n B Rj. Then Q(u j ) = v js n j j ^ u j(! S n) (4.) R 4n is a sequence of integers. We prove that Q(u j )! Q(u) as j!. We know that fju j(! S n)jg is bounded in L (R 4n ) and L 4n n (R 4n ) due to the structure of the knot energy (.5), the de nition of u j, and the relation (.). By

11 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS interpolation, we see that the sequence is bounded in L p (R 4n ) for all p [ 4n ; ]. n From the relations dv j = u j(! S n) and v j = 0, we see that fjrv j jg is bounded in L p (R 4n ) for all p [ 4n ; ] as well. Using the Sobolev inequality n C(m; p)kfk q krfk p (4.) in R m with q = mp=(m p) and < p < m, we get the boundedness of fv j g in L q (R 4n ) for q = (4n )p=(4n p) with 4n p, which gives the range for q, n q(n) 4n n To proceed, we consider the estimate q (4n ) 4n 3 : (4.3) js n j jq(u) Q(u j )j = (v ^ u (! S n) v j ^ u j(! S n)) R 4n (v ^ u (! S n) v ^ u j(! S n)) + (v ^ u R R j(! S n) v j ^ u j(! S n)) 4n 4n I () j + I () j : To show that I () j which (4.4)! 0 as j!, we look at the bottom numbers (for example) for u j(! S n)! u (! S n) weakly in L p (R 4n ) (4.5) for p = 4n so that the conjugate of p is p 0 = p = 4n = q(n), as de ned in (4.3). n p n Hence the claim I () j! 0 immediately follows from (4.5). On the other hand, since q(n) >, we see that fv j g is bounded in W ; (B) for any bounded domain B in R 4n. Using the compact embedding W ; (B)! L (B) and a subsequence argument, we may assume that fv j g is strongly convergent in L (B) for any bounded domain B. Thus, we have I () j kv v j k L (B)E(u j ) + (kvk 4n n + kv j k 4n ) n R 4n nb n ju j(! S n)j 4n 4n n C kv v j k L (B) + C E(u j ; R 4n n B) n 4n : (4.6) It is not hard to see that the quantity E(u j ; R 4n n B) may be made uniformly small. Indeed, for any " > 0, we can choose B su ciently large so that E(u; R 4n n B) < ". Let j be large enough so that B Rj B. Then E(u j ; R 4n n B) E(u; R 4n n B) + E(u j ; B Rj n B Rj ) " + C" j ; (4.7) in view of Lemma 3.. Using (4.7) in (4.6), we see that I () j! 0 as j!. Consequently, we have established Q(u j )! Q(u) as j!. In particular, Q(u) must be an integer because Q(u j ) s are all integers.

12 FENGBO HANG, FANGHUA LIN, AND YISONG YANG 5. Minimization for the Nicole Faddeev Skyrme Model Consider the minimization problem (.) where the energy functional E is de ned by (.5). Let fu j g be a minimizing sequence of (.) and set Then we have f j (x) = (jru j j 4n + ju j(! S n)j + jn u j j )(x): (5.) f j L(R 4n ); kf j k CjNj 4n 4n ; (5.) and kf j k E N + (say) for all j. Use B(y; R) to denote the ball in R 4n centered at y and of radius R > 0. According to the concentration-compactness principle of P. L. Lions [L, L], one of the following three alternatives holds for the sequence ff j g: (a) Compactness: There is a sequence fy j g in R 4n such that for any " > 0, there is an R > 0 such that sup f j (x) dx < ": (5.3) j R 4n nb(y j ;R) (b) Vanishing: For any R > 0, lim j! sup yr 4n B(y;R) f j (x) dx = 0: (5.4) (c) Dichotomy: There is a sequence fy j g R 4n and a positive number t (0; ) such that for any " > 0 there is an R > 0 and a sequence of positive numbers fr j g satisfying lim j! R j = so that We have the following. B(y j ;R) R 4n nb(y j ;R j ) f j (x) dx tkf j k < "; (5.5) f j (x) dx ( t)kf j k < ": (5.6) Lemma 5... The alternative (b) (or vanishing) stated in (5.4) does not happen for the minimization problem when N 6= 0. Proof. Let B be a bounded domain in R m and recall the continuous embedding W ;p (B)! L mp m p (B) for p < m. We need a special case of this at p = : W ; (B)! L m m (B) (m > ): (5.7)

13 B EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 3 Hence, for any function w, we have m jwj k m m m CB (jwj k + jwj k B C B jwj k + jwj (k B B C jwj k + jwj (k B B jrwj) ) m m ) m m + m m jrwj m m B jrwj m B ) m m (if jwj is bounded, k, (k, then) C jwj + jrwj : m (5.8) B B m Now taking m = 4n so that = 4n >, k = 4, w = u m 4n j n, and B = B(y j ; R), we have from (5.8) the inequality B(y j ;R) (4n ) ju j nj n C We now decompose R 4n ju j nj + B(y j ;R) B(y j ;R) into the union of a countable family of balls, + jru j j 4n 4n : (5.9) R 4n = [ i=b(y i ; R); (5.0) so that each point in R 4n lies in at most m such balls. Then de ne the quantity a j = sup ju j nj + jru j j : 4n (5.) i B(y i ;R) B(y i ;R) Thus the alternative (b) (vanishing) implies a j! 0 as j!. Therefore ju j nj (4n ) X (4n ) n ju j nj n R 4n i= B(y i ;R) X a 4n j C ju j nj + jru j j 4n i= B(y i ;R) B(y i ;R) ma 4n j C (ju j nj + jru j j 4n ) R 4n ma 4n j CE(u j )! 0 as j! : (5.) De ne the set A j = fx R 4n j ju j (x) nj g (say). Then (5.) implies lim ja jj = 0; (5.3) j! where ja j j denotes the Lebesgue measure of A j. Since Q(u j ) = N 6= 0, we see that u j (R 4n ) covers S n (except possibly skipping n). The de nition of A j says u j (A j ) contains the half-sphere below the equator of S n. Consequently, A j ju j(! S n)j dx ju j (A j )j jsn j; (5.4)

14 4 FENGBO HANG, FANGHUA LIN, AND YISONG YANG where js n j is the total volume of S n. However, the Schwartz inequality and (5.3) give us ju j(! S n)j dx ja j j ju j(! S n)j A j R 4n as j!, which is a contradiction to (5.4). ja j j (EN + )! 0; (5.5) Suppose that (a) holds. Using the notation of (a), we can translate the minimizing sequence fu j g to fu j ( y j )g = f~u j ()g (5.6) so that f~u j g is also a minimizing sequence of the same Hopf charge. Passing to a subsequence if necessary, we may assume without loss of generality that f~u j g weakly converges in a well-understood sense over R 4n to its weak limit, say u. Of course, E(u) lim inf j! fe(u j)g = E N : (5.7) Lemma 5... The alternative (a) (or compactness) stated in (5.3) implies the preservation of the Hopf charge in the limit described in (5.7). In other words, Q(u) = N so that u gives rise to a solution of the direct minimization problem (.). Proof. Let " and R be the pair stated in the alternative (a). Then sup fjr~u j j 4n + j~u j(! S n)j + j~u j nj g < ": (5.8) j R 4n nb R Besides, for the weak limit u of the sequence f~u j g, we have fjruj 4n + ju (! S n)j + ju nj g " (5.9) R 4n nb R and js n j jq(u) Q(~u j )j I j + J + K j ; (5.0) where I j = v ^ u (! S n) ~v j ^ ~u j(! S n) ; B R B R J = v ^ u (! S n) ; R 4n nb R K j = ~v j ^ ~u j(! S n) : (5.) R 4n nb R It is not hard to see that the quantities J and K j are small with a magnitude of some power of ". In fact, (.5) and (.) indicate that j~u j(! S n)j is uniformly bounded in L p (R 4n ) for p [ 4n ; ]. Then the relation d~v n j = ~u j(! S n), ~v j = 0, and the Sobolev inequality (4.) imply that ~v j is uniformly bounded in L q (R 4n ) for q [ 4n (4n ) ; ] n 4n 3 (see (4.3)). Using (.) again, we have K j k~v j k L 4n n (R 4n nb R ) k~u j(! S n)k L 4n n (R 4n nb R ) CE(~u j ; R 4n n B R ) n 4n C" n 4n : (5.)

15 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 5 By the same method, we can show that the quantity J obeys a similar bound as well. For I j, we observe that since ~u j(! S n) converges to u (! S n) weakly in L (B R ) and ~v j converges to v strongly in L (B R ), we have I j! 0 as j!. Summarizing the above results, we conclude that Q(~u j )! Q(u) as j!. In the next section, we will characterize the alternative (c) (dichotomy). 6. Dichotomy and Energy Splitting in Minimization Use the notation of the previous section and suppose that (c) (or dichotomy) happens. Then, after possible translations, we may assume that there is a number t (0; ) such that for any " > 0 there is an R > 0 and a sequence of positive numbers fr j g satisfying lim j! R j = so that f j (x) dx te(u j ) < "; (6.) B R f j (x) dx ( t)e(u j ) < ": (6.) R 4n nb Rj For convenience, we assume R j > R for all j. Therefore, from the decomposition E(u j ) = f j (x) dx + f j (x) dx + E(u j ; B Rj n B R ); (6.3) B R R 4n nb Rj and (6.), (6.), we have E(u j ; B R n B R ) E(u j ; B Rj n B R ) < "; E(u j ; B Rj n B Rj =) E(u j ; B Rj n B R ) < ": (6.4) Using Lemma 3., we can nd maps u () j and u () j from R 4n to S n such that u () j = u j in B R, u () j = n in R 4n n B R, and E(u () j ; B R n B R ) < C"; u () j = u j in R 4n n B Rj, u () j = n in B Rj =, and E(u () j ; B Rj n B Rj =) < C". Here C > 0 is an irrelevant constant. Use the notation F (u) = v ^ u (! S n). Since F (u) depends on u nonlocally, we need to exert some care when we make argument involving truncation. In view of the fact that u j and u () coincide on R 4n n B Rj, we have j coincide on B R and u j and u () j ju j(! S n) (u () j ) (! S n) (u () j ) (! S n)j 4n n R 4n C(E(u j ; B Rj n B R ) + E(u () j ; B R n B R ) + E(u () j ; B Rj n B Rj =)) C": Consequently, using the relations dv j = u j(! S n), v j = 0, dv (i) j (6.5) = (u (i) j ) (! S n); v (i) j = 0, i = ;, we have in view of (6.5) and (4.) with p = (4n )=n and q = (4n )=(n ) that kv j v () j v () j k 4n n Cku j(! S n) (u () j ) (! S n) (u () j ) (! S n)k 4n n C " n 4n : (6.6)

16 6 FENGBO HANG, FANGHUA LIN, AND YISONG YANG Since the numbers p; q above are also conjugate exponents, we obtain from (6.6) the bound jf (u j ) F (u () j ) F (u () j )j B R [fr 4n nb Rj g = Applying (6.7), we have B R [fr 4n kv j v () j v () j(v j v () nb Rj g j k 4n n j v () j ku j(! S n)k 4n n ) ^ u j(! S n)j C" n 4n : (6.7) js n j jq(u j ) (Q(u () j ) + Q(u () j ))j jf (u j ) F (u () j ) F (u () j )j B R [fr 4n nb Rj g + jf (u j )j + jf (u () j )j + B Rj nb R B R nb R C " n 4n B Rj nb Rj = jf (u () j )j + C (E(u j ; B Rj n B R ) n () 4n + E(u ; B R n B R ) n () 4n + E(u ; B Rj n B Rj =) n C" n 4n : j j 4n ) (6.8) Since " > 0 can be arbitrarily small and Q(u j ); Q(u () j ); Q(u () j ) are integers, the uniform bound (6.8) enables us to assume that On the other hand, since (.9) implies that N Q(u j ) = Q(u () j ) + Q(u () j ); 8j: (6.9) jq(u () j )j 4n 4n CE(u () j ) = C(E(u j ; B R ) + E(u () j ; B R n B R )) CE(u j ) + C "; (6.0) we see that fq(u () j )g is bounded. We claim that Q(u () j ) 6= 0 for j su ciently large. Indeed, if Q(u () j ) = 0 for in nitely many j s, then, by going to a subsequence when necessary, we may assume that Q(u () j ) = 0 for all j. Thus we see that Q(u () j ) = N in (6.9) for all j and E(u () j ) E(u j ; R 4n n B Rj ) + C" = f j (x) dx + C": (6.) R 4n nb Rj As a consequence, we have in view of (6.) and (6.) that E N lim sup E(u () j ) ( t) lim E(u j ) + " + C" j! j! ( t)e N + C ": (6.) Since 0 < t < and " is arbitrarily small, we obtain E N = 0, which contradicts the topological lower bound E N CjNj 4n 4n (N 6= 0) stated in (.9).

17 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 7 Similarly, we may assume that Q(u () j ) 6= 0 for j su ciently large. Of course, fq(u () j )g is bounded as well. Hence, extracting a subsequence if necessary, we may assume that there are integers N 6= 0 and N 6= 0 such that Q(u () j ) = N ; Q(u () j ) = N ; 8j: (6.3) Furthermore, for the respective energy in ma at the Hopf charges N ; N ; N, we have E N + E N E(u () j ) + E(u () j ) = E(u j ; B R ) + E(u j ; R 4n n B Rj ) + E(u () j ; B R n B R ) + E(u () j ; B Rj n B Rj =) E(u j ) + C": (6.4) Since " > 0 may be arbitrarily small, we can take the limit j! in (6.4) to arrive at E N + E N E N ; N = N + N : (6.5) We can now establish the following energy-splitting lemma. Lemma 6... If the alternative (c) (or dichotomy) stated in (5.5) and (5.6) happens at the Hopf charge N 6= 0, then there are nonzero integers N ; N ; ; N k such that E N E N + E N + + N k ; N = N + N + + N k ; (6.6) and that the alternative (a) (or compactness) stated in (5.3) takes place at each of these integers N ; N ; ; N k. Proof. If the alternative (c) happens at N 6= 0, we have the splitting (6.5). We may repeat this procedure at all the sublevels wherever the alternative (c) happen. Since (.9) and (.0) imply that there is a universal constant C > 0 such that E` C for any ` 6= 0. Hence the above splitting procedure ends after a nitely many steps at (6.6) for which the alternative (c) cannot happen anymore at N ; N ; ; N k. Since the alternative (b) never happens because N s 6= 0 (s = ; ; ; k) in view of Lemma 5., we see that (a) takes place at each of these integer levels. The energy splitting inequality, (6.6), is referred to as the Substantial Inequality in [LY4] which is crucial for obtaining existence theorems in a noncompact situation. 7. Existence Theorems We say that an integer N 6= 0 satis es the condition (S) if the nontrivial splitting as described in Lemma 6. cannot happen at N. De ne S = fn j N satis es condition (S)g: (7.) It is clear that, for any N S, the minimization problem (.) has a solution. As a consequence of our study in the previous sections, we arrive at Theorem 7... Consider the minimization problem (.) in which the energy functional is of the NFS type given in (.5). Then there is an in nite subset of, say S, such that, for any N S, the problem (.) has a solution. In particular, the minimum-mass or minimum-energy Hopf charge N 0 de ned by N 0 is such that E N0 = minfe N j N 6= 0g (7.)

18 8 FENGBO HANG, FANGHUA LIN, AND YISONG YANG is an element in S. Furthermore, for any nonzero N, we can nd N ; ; N k S such that the substantial inequality (6.6) is strengthened to the equalities E N = E N + E N + + N k ; N = N + N + + N k ; (7.3) which simply express energy and charge conservation laws of the model in regards of energy splitting. Proof. Use the Technical Lemma (Lemma 3.) as in [LY] to get (7.3). The rest may also follow the argument given in [LY]. Next, we show that, in the compact situation, the minimization problem (.) has a solution for any integer N. For this purpose, let E(u) denote the energy functional of the NFS type or the Faddeev type given as in (.5) or (.6) evaluated over S 4n for a map u from S 4n into S n. Namely, E NFS (u) = fjduj 4n + ju (! S n)j + jn uj g ds; (7.4) S 4n E Faddeev (u) = jduj 4n + ju (! Sn)j ds: (7.5) S 4n The Hopf invariant Q(u) of u is given in (.). We have Theorem 7... For any nonzero integer N which may be realized as a Hopf number, i.e., there exists a map u : S 4n! S n such that Q(u) = N, the minimization problem E N = inffe(u) j E(u) < ; Q(u) = Ng over S 4n has a solution when E is given either by (7.4) or (7.5). Proof. Let fu j g be a minimizing sequence of the stated topologically constrained minimization problem and v j be the potential (n )-form satisfying dv j = u j(! S n); v j = 0; j = ; ; : (7.6) Passing to a subsequence if necessary, we may assume that there is a nite-energy map u (say) such that u j! u, du j! u, and u j(! S n)! u (! S n) weakly in obvious function spaces, respectively, as j!, which lead us to the correct comparison E(u) E N by the weakly lower semi-continuity of the given energy functional. To see that Q(u) = N, we recall that the sequence fv j g may be chosen [Mo] such that it is bounded in W ; (S 4n ) by the L (S 4n ) bound of fu j(! S n)g. Hence v j! some v W ; (S 4n ) weakly as j!. Therefore v j! v strongly in L (S 4n ) as j!. Of course, dv = u (! S n) and v = 0. Consequently, we immediately obtain Q(u) = js n j and the proof is complete. S 4n v ^ u (! S n) = js n j lim j! S 4n v j ^ u j(! S n) = N; (7.7) Note that the existence of global minimizers for the compact version of the Nicole energy (.4), E(u) = jduj 4n ds; (7.8) S 4n was studied by Riviere [Ri] for n =. See also [L] and [DK]. In particular, he showed that there exist in nitely many homotopy classes from S 3 into S having energy minimizers.

19 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 9 We now address the general problem of the existence of critical points of (7.8) at the bottom dimension n = whose conformal structure prompts us to simply consider it over R 3. Thus we are led to the Nicole model. Speci cally, for a map u : R 3! S, the Nicole energy [Ni] is given by E(u) = jruj 3 : (7.9) R 3 For convenience, we may use the stereographic projection of S! C from the south pole to represent u = (u ; u ; u 3 ) by a complex-valued function U = U + iu as follows, U = u + u 3 ; U = u + u 3 ; (7.0) where u 3 = p u u for u belonging to the upper or lower hemisphere, S. Following [AF] (see also [ASVW, HS]), we use the toroidal coordinates (; ; ') to represent a point x = (x ; x ; x 3 ) in R 3 by x = q sinh cos '; x = q sinh sin '; x 3 = q sin ; (7.) where q = cosh cos and 0 < < ; 0 ; '. The AF ansatz [AF, ASVW, HS] reads U(; ; ') = f() e im'+in ; m; n ; (7.) where the undetermined function f satis es the normalized boundary condition f(0) = lim!0 f() = 0; f() = lim! f() = ; (7.3) so that the Hopf map is given by the choice f() = sinh with m = n =, or U(; ; ') = sinh e i+i' : (7.4) After some calculation, it can be shown [AF, HS] that the Hopf invariant of u designated by (7.0) (7.3) is given as Q(u) = mn: (7.5) Besides, with the new variable t = sinh ; (7.6) the function f becomes a function of t, which is still denoted by f(t) for simplicity, so that the Nicole energy (7.9) takes the form [ASVW] E(f) = 3 0 t( + t ) f t ( + f ) + + t m t + n f ( + f ) 3 dt; (7.7) and the boundary condition (7.3) is reinterpreted in terms of t given in (7.6). The Euler Lagrange equation of (7.7) is [ASVW] t ( + t )( + f )(t [ + t ]f t + [m + n t ]f )f tt 4t 4 ( + t ) ff 4 t + t 3 ( + 3t )( + t )( + f )f 3 t t ( + t )(m + n t )f 3 f t + t 3 (m + n [ + t ])( + f )f f t (m + n t ) f 3 ( f ) = 0: (7.8) It is important to note that the advantage of using the AF ansatz (7.) is that it is a compatible ansatz [ASVW], meaning that (7.8) gives rise to the critical points of the original Nicole energy (7.9). More precisely, the critical points of (7.7) subject to the boundary condition f(t)! 0 as t! 0, f(t)! as t!, give rise to the critical points of the Hopf number (7.5) for the Nicole energy through (7.0) (7.)

20 0 FENGBO HANG, FANGHUA LIN, AND YISONG YANG and (7.6). Although (7.8) looks complicated, it has a nontrivial solution f(t) = t when m = n =, which implies that the Hopf map is an explicit critical point [ASVW]. Our purpose below is to show that, for any m; n, the equation (7.8) has a nite-energy solution satisfying the stated boundary condition at t = 0 and t =. In fact, such a solution also minimizes the energy (7.7). To proceed, we introduce another new variable Then the boundary condition for f becomes 0 g = arctan f: (7.9) g(0) = 0; g() = ; (7.0) and the energy (7.7) is converted into the simpli ed form given as I(g) = t( + t ) gt + m + t t + tan 3 g n dt; (7.) ( + tan g) where we have suppressed an irrelevant constant factor. It is seen that the Hopf map, de ned by g(t) = arctan t, is of nite energy for any integers m; n. We now de ne the admissible space as A = fg(t) j g(t) is absolutely continuous over the interval (0; ); satis es the boundary condition (7.0), and I(g) < g; (7.) and consider the associated minimization problem I 0 inffi(g) j g Ag: (7.3) Let fg j g be a minimizing sequence of (7.3). We may assume that I(g j ) I 0 + (say) for all j = ; ;. We will show that fg j g contains a subsequence which converges in a well-de ned way to an element in A, g 0 (say), and I(g 0 ) = I 0. In fact, collectively writing P (g) = tan g ( + tan g) ; (7.4) we see that P () is a periodic even function of period, whose singularities at oddinteger multiples of = are removed if we understand P ( ) = lim g! P (g) = 0, etc. In the sequel, we always observe such a convention for P (). Therefore, for any g A, the modi ed function jg(t)j; if jg(t)j < ~g(t) = ; ; if jg(t)j ; (7.5) lies in A and satis es 0 ~g and I(~g) I(g). Hence, with suitable modi cations if necessary, we may assume that our minimizing sequence fg j g satis es the same boundedness condition 0 g j, j = ; ;. On the other hand, near t = 0 and t =, we have, respectively, t 0 g j (t) (s 3 3 t 3 3 ) ds s dg j 3 ds ds t 3 (I(gj )) 3 ; (7.6)

21 and EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS g j (t) t s 3 ds 3 t s 3 dg j ds 3 3 ds 3 t 3 (I(gj )) 3 ; (7.7) which indicates in particular that fg j g satis es the boundary condition (7.0) uniformly. The structure of the energy I given in (7.) shows that for any numbers 0 < a < b <, the sequence fg j g is bounded in W ;3 (a; b). Using a diagonal subsequence argument, we may assume without loss of generality that fg j g is weakly convergent in W ;3 (a; b) for any 0 < a < b <. We use g 0 to denote the so-obtained weak limit of fg j g over the entire interval (0; ). We need to prove that g 0 A and I(g 0 ) = I 0. For convenience, we set b J(g; h; a; b) = t( + t ) gt + 3 m + t t + n P (h) dt; (7.8) a where g; h are absolultely continuous over (0; ) and P () is de ned by (7.4). We note that P 0 (h) = tan h( tan h) ( + tan h) ; h 6= odd-integer multiple of ; P 0 (h) = 0; h = odd-integer multiple of : (7.9) Hence, P 0 is bounded. Besides, we may check that J(; h; a; b) is convex for xed h; a; b. Therefore, we have lim (J(g j; g j ; a; b) J(g j ; g 0 ; a; b)) = 0; (7.30) j! and the weakly lower semicontinuity of J(; g 0 ; a; b) implies that Consequently, we get I 0 = lim j! I(g j ) J(g 0 ; g 0 ; a; b) lim inf j! J(g j; g 0 ; a; b): (7.3) lim inf j! J(g j; g j ; a; b) = lim j! (J(g j ; g j ; a; b) J(g j ; g 0 ; a; b)) + lim inf j! J(g j; g 0 ; a; b) J(g 0 ; g 0 ; a; b): (7.3) Letting a! 0 and b! in (7.3), we see that I(g 0 ) = J(g 0 ; g 0 ; 0; ) I 0 as claimed. The fact that g 0 satis es the boundary condition (7.0) follows from the uniform bounds (7.6) and (7.7). Thus, g 0 A. The Euler Lagrange equation of (7.) is t( + t ) gt + m + n P (g) gt = t g t + + t + t m t t + n t m P (g) t + n P 0 (g): (7.33)

22 FENGBO HANG, FANGHUA LIN, AND YISONG YANG With the help of this equation, we may show that g 0 satis es 0 < g 0 (t) < ; 0 < t < : (7.34) In fact, if there is a point t 0 > 0 such that g 0 (t 0 ) = 0 or g(t 0 ) = =, then the property 0 g 0 (t) = implies that g0(t 0 0 ) = 0. In view of the uniqueness theorem for the initial value problem of an ordinary di erential equation, we infer that g 0 (t) 0 or g 0 (t) = since g = 0 and g = = are two trivial solutions of (7.33). This conclusion contradicts the boundary condition (7.0) enjoyed by the function g 0 obtained earlier. The property (7.34) ensures the invertibility of the transformation (7.9) so that we obtain a critical point for the original energy (7.7). We may summarize our study above in the form of the following existence theorem. Theorem For any N, the Nicole energy (7.9) has a nite-energy critical point u in the topological class Q = N. More precisely, for any m; n, the energy functional (7.9) has a nite-energy critical point u represented in terms of the toroidal coordinates through the expressions (7.0) (7.3) so that its Hopf invariant satis es Q = mn, its associated con guration function f de ned in (7.) is positive-valued with range equal to the full interval (0; ) and minimizes the reduced one-dimensional energy (7.7) in the variable t = sinh. As mentioned already, since (7.9) is conformally invariant, it covers the spherical energy (7.8) when n =. Therefore, Theorem 7.3 establishes the existence of a critical point of the energy (7.8) at n = among the topological class Q = N for each N. 8. Generalized Faddeev Knot Energy In the subsequent sections, we shall study the topologically constrained minimization problem of the generalized Faddeev knot energy in arbitrary (4n ) dimensions. The generalization we will be focused on is de ned by the energy E(u) = fjduj 4n + ju! S nj g; (8.) R 4n where, for convenience, we have absorbed the unimportant coe cient in (.6) to unity. One may argue that a more natural generalization of the Faddeev knot energy should take the original quadratic form so that E(u) = fjduj + ju! S nj g: (8.) R 4n However, at this moment, the energy (8.) seems to be too hard to approach. Indeed for n and a map u : R 4n! S n with R fjduj + ju! R 4n S nj g <, it is not necessary that u! S n is a closed form. On the other hand, it is worth mentioning that (8.) may be viewed as a natural extension of the Faddeev energy as well because (i) both energy density terms are quadratic when n =, and (ii) with respect to the rescaling of coordinates, x 7! x ( > 0), the two energy terms respond with and, respectively, as in the classical Faddeev model. As mentioned in the introduction, one of the main di culties in understanding the Faddeev model is that it is still not known whether an annulus lemma similar to Lemma 3. exists or not. In particular we are not able to freely cut and paste maps with nite energy and it is not clear the minimizing problem would break into a nite region one

23 EXISTENCE OF FADDEEV KNOTS IN GENERAL HOPF DIMENSIONS 3 and a problem at the in nity, that is, the minimizing problem does not t in the frame of classical concentration compactness principle anymore. This di culty will be bypassed by a decomposition lemma (Lemma.) for an arbitrary map with nite Faddeev energy (in the same spirit as in [LY] for maps from R 3 to S ). 9. Some General Facts and Useful Properties and Relations In this section we collect some basic facts which will be used frequently later. We will use the algebraic notations in [F, Chapter ]. Assume n is an integer and k n. Then we denote (n; k) = f = ( ; ; k ) j i s are integers and < < k ng : (9.) If = ( ; ; k ) (n; k), k < n, then (n; n k) is obtained from the complement f; ; ng n f ; ; k g. If x ; ; x n are the coordinates on R n, then we write dx = dx ^ dx ^ ^ dx k : If, for every (n; k),! is a distribution on an open subset of R n, then we call! = X! dx (n;k) a (k-form) distribution. Occasionally, we will need to verify some weak di erential identities. It is convenient to have the following basic rule. Lemma 9... Assume that p ; p ; p 3, is an open subset of R n, L p loc () is a k-form, L p0 loc () is another form such that d Lp loc (), Lp0 loc (), Lp 3 loc () and d L p0 3 loc (). Then, in sense of distribution, we have d ( ^ ) = d ^ + ( ) k ^ d: Here p 0 = p p is the conjugate power of p. Similar for p 0 and p 0 3. Proof. First assume p ; p ; p 3 <. By mollifying arguments we may nd a sequence of smooth k-forms i C () such that i! in L p loc () and Lp 3 loc (), d i! d in (). Taking a limit in the equation L p loc d ( i ^ ) = d i ^ + ( ) k i ^ d; the lemma follows. For the remaining cases, without loss of generality, we assume p =, p ; p 3 <. Then we may nd a sequence of smooth k-forms i C () such that i * in L loc (), i! in L p 3 loc () and d i! d in L p loc (). Then i ^! ^ in sense of distribution and d i ^! d ^, i ^ d! ^ d in (). The same limit process as above implies the lemma. L loc For a smooth map u, the exterior di erential d commutes with the pullback operator u. It remains true under suitable integrability condition on the derivatives of u when it is only weakly di erentiable. Lemma 9... Assume that R n is open, is a smooth k-form on R l with compact support, u W ;k+ loc ; R l. Then in sense of distribution du = u d: