Bubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n

Transcription

1 Acta Mathematica Sinica, English Series Jul., 2010, Vol. 26, No. 7, pp Published online: June 15, 2010 DOI: /s Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2010 Bubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n Xiao Huan MO Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing , P.R.China moxh@pku.edu.cn Fang SUN Key Laboratory of Pure and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing , P.R.China nusgnaf@gmail.com Abstract We show that any harmonic sequence determined by a harmonic map from a compact Riemannian surface M to CP n has a terminating holomorphic (or anti-holomorphic) map from M to CP n, or a bubble tree limit consisting of a harmonic map ˆf : M CP n and a tree of bubbles h μ λ : S2 CP n. Keywords bubble tree, harmonic sequence, non-isotropic, energy identity MR(2000) Subject Classification 58E20 1 Introduction Consider a sequence of harmonic maps f k from a compact Riemannian surface M toacompact Riemannian manifold N. It is important to give a nice description of the behavior of the sequence {f k } as k. In [1], the authors told us that given a sequence of harmonic maps f k with bounded energy from a fixed Riemann surface M to a compact Riemannian manifold N, there is a subsequence of these maps which converges uniformly in C 1 away from a finite number of points {p 1,...,p l } M to a limiting harmonic map ˆf : M N. The points correspond to harmonic maps from the standard two-sphere S 2 into N whose presence is due to energy concentrating near each p i as k tends to infinity. These harmonic maps from S 2 into N are the bubbles in the theory. Furthermore, for sequence of harmonic maps from a fixed Riemann surface to a compact Riemannian manifold there is no energy loss [2] and there are no necks [3], i.e. the energy of the bubble tree map is the limit of the energies of the sequence and the images of the bubbles are at zero distance from the bubble preceding it in the tree. Received November 26, 2008, Accepted April 9, 2009 Supported by National Natural Science Foundation of China (Grant No )

2 1278 Mo X. H. and Sun F. In this paper, we shall discuss this problem when N is a complex projective space with the Fubini Study metric of constant holomorphic sectional curvature 4, denoted by CP n. Let M be a compact Riemann surface. An important tool in the study of a harmonic map f from M to CP n is its harmonic sequence ( complex Frenet frame) [4 7]. If this is finite, the map is called (complex) isotropic and is constructible from holomorphic maps; otherwise, the harmonic sequence is infinite. The map is called non (complex) isotropic. There are lots of non-isotropic harmonic maps from compact Riemann surfaces to CP n [6 10]. The map is said to be full if f(m) lies in no proper projective subspace. For more details, see Section 2. A natural question is to consider the compactness of the harmonic sequences determined by a non (complex) isotropic harmonic map f : M CP n. For an example, consider harmonic map f 0 : T 2 = C/10πZ 2 CP 5 defined by f 0 (x, y) = [ f 0 (x, y)] where ( f 0 (x, y) = cos x, 5 5 ) sin x, cos s, sin s, cos t, sin t, s = 3 5 x y, t = 3 5 x 4 5 y. Then the harmonic sequence f 0 f1 f2 determined by f 0 is infinite. Furthermore, there is an α [0, 2π) such that after selection of a subsequence, f k converges pointwise to [ ] ˆf(x, y) = cos x, sin x, 1α e cos s, e 1α sin s, e 1α cos t, e 1α sin t 5 5 on T 2. For more details, see Section 3. The purpose of this paper is to prove the following bubble tree convergence result: Theorem 1.1 Let f : M CP n be a full non-isotropic harmonic map from a compact Riemannian surface M and with harmonic sequence {f k }. Then there are a finite set of bubble points {p 1,p 2,..., p l } M, a harmonic map ˆf : M CP n and a finite set of nontrivial harmonic maps h μ λ : S2 CP n, λ = 1,..., l; μ = 1,..., L λ such that after selection of a subsequence, f k converges uniformly in C to ˆf on M\{p 1,p 2,..., p l } and the following holds : lim E(f k)=e( ˆf)+ l L λ E(h μ λ ). (1.1) k λ=1 μ=1 Moreover all necks converge to points in CP n. Consequently, the images of f k : M CP n converge pointwise to this image { ˆf, h μ λ }. As usual, we call (1.1) the energy identity. See [11 14] for some recent developments. Note that the non-isotropic harmonic surfaces in complex projective spaces have been discussed in [15 17] recently. See Section 5 for the proof of Theorem 1.1.

3 Bubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n Harmonic Sequences In this section we give a concise account of the theory of harmonic sequences of maps into CP n. This account is specially tailored to our needs, but for a fuller description of the more general theory of harmonic sequences of maps into complex Grassmannians, the reader is referred to [7]. Let M be an (oriented) Riemannian surface and f : M CP n be a non-constant harmonic map. Denote the Riemannian metric on M by ds 2 M = ϕ ϕ, whereϕ is a complex-valued oneform; ϕ is defined up to a complex factor of absolute value 1. Choose a field of unitary frame Z A so that Z 0 spans f(x), x M. Throughout this paper, our index conventions are as follows: 0 A, B,... n, 1 i, j,... n. Then f ω 0ī = a 0ī ϕ + b 0ī ϕ, (2.1) where ω 0ī are the Maurer Cartan forms of U(n + 1) satisfying dz A =Σ B ω A BZ B. The energy of the map f is by definition E(f) = tr(f ds 2 ) 1 M where ds 2 is the Fubini Study metric on CP n with the constant holomorphic sectional curvature 4 and the trace is taken with respect to the metric on M. It is easy to see that ds 2 =Σ i ω 0ī ω 0ī. (2.2) A map which is a critical point of the energy functional is called harmonic. The metric ds 2 M has a connection form ρ satisfying dϕ = 1ρ ϕ. Define the covariant derivative of a 0ī by Da 0ī := da 0ī + a 0 jω jī 1a 0ī ρ. (2.3) We get therefore the following criterion for the harmonicity of f, which we will apply in the next section. Lemma 2.1 (See [18]) The property that f is a harmonic map is expressed by the following condition : Da 0ī 0mod ϕ. Define the mappings : V W T (1, 0), : V W T (0, 1) by (ξz 0 )=ξa 0ī Z i ϕ, (ξz0 )=ξb 0ī Z i ϕ

4 1280 Mo X. H. and Sun F. where V (resp., W ) is the vector bundle with fibers f(x) (resp.,f(x) )andt (1, 0) (resp., T (0, 1) ) is the cotangent bundle of M of type (1, 0) (resp., type (0, 1)). We shall call these the fundamental collineations. It is easy to show the following: Lemma 2.2 (See [18 19]) Let f : M CP n be a harmonic map. Then f and f are harmonic maps. Repeating the constructions of Lemma 2.2 we get two sequences of harmonic maps f 0 (= f) f 1 f2 f 0 f 1 f 2 (2.4) whose image spaces are connected by fundamental collineations. harmonic sequences determined by f. Let Such sequences are called f = f f1 2 s 1 f2 s fs be a harmonic sequence where each f k is a map M CP n or, what is the same, a rank one vector bundle (a line bundle) over M. Then the map k is a holomorphic bundle map [19]: f k k fk+1 T (1, 0). Similarly, we have anti-holomorphic bundle map k : f k f k 1 T (0, 1).If k 0 but k 1 0 (resp., k 0 but k+1 0) then the sequence terminates with f k at the right- (resp., left-) hand side. In this case, f 0 is generated by a holomorphic map, the members of the sequence are orthogonal to each other, and the sequence has length at most n + 1. Such a harmonic map is called isotropic. A non-isotropic harmonic map f is characterized by the property that k (and k ) are non-zero holomorphic (anti-holomorphic) map for every k Z. Notice that two consecutive elements in the harmonic sequence are always orthogonal. In general, we say f has isotropy order r if f 0 f k,for1 k r, but f 0 is not orthogonal to f r+1 [20]. It is convenient to say that an isotropic map has isotropy order =. Thus,ifr<, then1 r n. If f is a map from a compact oriented surface M to CP n,thedegree of f deg(f) is the image of the generator of H 2 (CP n, Z) inh 2 (M, Z) =Z by the homomorphism induced by f. We define the holomorphic or energy of f by 1 E() = Σ j a 2 0 j 2 ϕ ϕ. Similarly, the antiholomorphic or energy of f is by definition E( ) 1 = Σ j b 2 0 j 2 ϕ ϕ. Let E( k ) be the holomorphic or energy of f k and E( k ) the antiholomorphic or energy of f k. By using (2.1) and (2.2), we have E(f k )=E( k )+E( k ). (2.5) The Kähler form of ds 2 is κ = 1 2π Σ jω 0 j ω 0 j. (2.6)

5 Bubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n 1281 Thedegreeoff k can be computed from (2.1) and (2.6) as follows (see [19, 21 22]): deg(f k )= 1 fk κ = 1 [ E(k ) E( π π k ) ]. (2.7) 3 Examples M In this section we study the compactness of the harmonic sequence determined by some explicit examples. Example 3.1 (Cyclic harmonic sequence) Considering the harmonic map φ : T 2 CP 2 from the Clifford torus [7, 10, 23], Wolfson showed that φ is non-isotropic and generates a cyclic harmonic sequence of order 3 with φ 3k+t = φ t for all k, t 0 [7, 23]. Hence the harmonic sequence φ k determined by φ has its compactness obviously. Example 3.2 (Non-cyclic harmonic sequence) The map f 0 : T 2 = C/10πZ 2 CP 5 defined by f 0 (x, y) =[ f 0 (x, y)] ( ) f 0 (x, y) = cos x, sin x, cos s, sin s, cos t, sin t, 5 5 where s = 3 5 x y, t = 3 5 x 4 y. (3.1) 5 By a simple calculation, we have Da 0ī 0mod ϕ where Da 0ī is defined in (2.3). Thus by Lemma 2.1, f 0 is harmonic. We put a := 1 5 (3 + 14), J := 0 1 (3.2) 1 0 and J A := āj. (3.3) aj Let f 0 f1 f2 be a harmonic sequence determined by f 0. By straightforward computations one obtains t fk = A k t f0, k =1, 2,..., (3.4)

6 1282 Mo X. H. and Sun F. where f k is a local lift of f k, k =1, 2,... to C 6. It is easy to see from Lemma 6.3 in [6] that the harmonic sequence determined by f 0 is infinite and not cyclic. Note that a is the complex number of unit modulus. We set a =e 1θ, θ [0, 2π). (3.5) It follows that By (3.2), we see that ā =e 1θ. J 4 = 1 0 = I 2, 0 1 from which together with (3.3) and (3.5) we obtain I 2 A 4k = e 4 1kθ I 2, k N. e 4 1kθ I 2 Plugging this into (3.4) yields [ f 4k (x, y) = cos x, 5 5 e 4 1kθ cos t, e 4 1kθ sin t sin x, 1kθ e 4 cos s, e 4 1kθ sin s, ] where s and t are defined in (3.1). Except at cos x = 0, in terms of the affine coordinate on CP n we may write f 4k (x, y) = ( tan x, 5 e 4 1kθ cos s e 4 1kθ sin t cos x )., cos x, 5 e 4 1kθ sin s 14 cos x, 5 e 4 1kθ cos t 14 cos x, Note that e 4 1kθ S 1 and S 1 is compact. Then there is an α [0, 2π) such that after selection of a subsequence of k, e ±4 1kθ converges to e ± 1α.Set [ ] ˆf(x, y) = cos x, sin x, 1α e cos s, e 1α sin s, e 1α cos t, e 1α sin t. 5 5 Except at sin x =0,intermsoftheaffinecoordinateonCP n we may write ( 5 ˆf(x, y) = cot x, e 1α cos s 14 sin x, 5 e 1α sin s 14 sin x, 5 e 1α cos t 14 sin x, 5 e 1α sin t ). 14 sin x It is easy to see f 4k converges pointwise to Im ˆf when cos x 0. Similarly, we have f 4k converges pointwise to Im ˆf when sin x 0. Hence f 4k converges pointwise to Im ˆf on T 2. In fact, f 4k converges uniformly in C to ˆf on T 2.

7 Bubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n Cyclic Properties of the Sequence of Energy In [6], the author presents that for any full non-isotropic harmonic map f : M CP n of isotropy order r< the sequence of degree {degf k } of the harmonic sequence of f has mean zero, precisely, he proved the following: Lemma 4.1 (See [6]) Let f : M CP n be a full harmonic map of isotropy order r<, and with harmonic sequence {f k } k Z.Then s deg(f l+k )=0, for all l Z, (4.1) k=0 where s +1 is a common divisor of r +1 and n +1. Let f : M CP n be a full non-isotropic harmonic map and let {f k } k Z be its harmonic sequence. In the following we consider possible cyclic properties of the sequence {E(f k )} k Z required in the proof of Theorem 1.1. Lemma 4.2 Let f : M CP n be a full harmonic map of isotropy order r<, andwith harmonic sequence {f k } k Z. Assume that the fundamental collineation k 1 of f k 1 is nondegenerate. Then E(f k )=E(f k 1 )+π[deg(f k 1 )+deg(f k )], (4.2) and, in particular, if s {0, 1} E(f k )=E(f k 1 ), (4.3) where s +1 is a common divisor of r +1 and n +1. Proof Recall Theorem 2.2 (c) in [18]. The map k k 1 f k 1 is f k 1 itself. It is an immediate consequence of this result E( k )=E( k 1 ), from which together with (2.5) and (2.7) we obtain E(f k )=E( k )+E( k ) = πdeg(f k )+2E( k ) = πdeg(f k )+2E( k 1 ) = πdeg(f k )+E( k 1 )+πdeg(f k 1 )+E( k 1 ) = E(f k 1 )+π[deg(f k 1 )+deg(f k )]. If s {0, 1}, then 1 deg(f l+k )=0, for all l Z, k=0 from Lemma 4.1. Plugging this into (4.2) yields (4.3). Proposition 4.3 Let f : M CP n be a full harmonic map of isotropy order r<, and with harmonic sequence {f k } k Z. Then the sequence {E(f k )} k Z is cyclic, of period s+1. That is, E(f s+l+1 )=E(f l ), for all l Z,

8 1284 Mo X. H. and Sun F. where s +1 is a common divisor of r +1 and n +1. Proof A non-isotropic harmonic map f is characterized by the property that k (and k )are non-zero holomorphic (anti-holomorphic) map for every k Z (cf. [6]). Together with (4.1) and (4.2) we conclude that [ E(f s+l+1 )=E(f l )+π deg(f l )+2 Proposition 4.4 = E(f l )+π s k=1 s deg(f l+k )+π k=0 ] deg(f l+k )+deg(f s+l+1 ) s deg(f l+k+1 )=E(f l ). k=0 Let f : M CP n be a full harmonic map of isotropy order r<, andwith harmonic sequence {f k } k Z. Suppose that n +1 2 is a prime number or r {1, n 2, n 1}. Then the sequence {E(f k )} k Z is cyclic of period 1. That is, E(f k )=constant. Proof If n +1 2 is a prime number or r = n 1, then It follows that s =0. Ifr {1, n 2}, then (r +1,n+1)=1. (r +1,n+1) {1, 2}. It follows that s {0, 1}. Our result can be obtained from (4.3). 5 Proof of Theorem 1.1 Given a sequence of harmonic maps from surface f k : M N. SackandUhlenbeckmade a key observation that the lack of compactness is caused by the concentration of the energy at isolated point. By rescaling near these points of concentration, they obtain nonconstant harmonic maps from 2-sphere, usually referred to as bubbles [1]. Now we are going to show that harmonic sequence determined by a non-isotropic harmonic map f : M CP n has a well-known Sacks Uhlenbeck limit consisting of a harmonic map ˆf : M CP n and some bubbles -harmonic maps S 2 CP n obtained by a renormalization process. Proof of Theorem 1.1 Set Λ= max 0 k s {E(f k)}, where s + 1 is a common divisor of r +1 and n +1, andr is the isotropy order of f. Then, from Proposition 4.3, there exists k 0 {0, 1,..., s} with E(f k )=E(f k0 ) Λ < + for each k Z. It means that {f k } has uniform bounded energy. Now Theorem 1.1 can be obtained from Lemma in [2] or [3].

9 Bubble Tree Convergence for the Harmonic Sequence of Harmonic Surfaces in CP n 1285 Remark 5.1 Similarly, harmonic sequence (2.4) determined by a non-isotropic harmonic map f : M CP n has a bubble tree limit. Remark 5.2 Let M g be a compact Riemann surface of genus g. Examples of non-isotropic harmonic maps to CP n were given in [6 10]. If g 1and0 d g 1, Lemaire has constructed Riemann surfaces M g and non-isotropic harmonic maps M g CP 1 [9]. Jensen and Liao gave full non-isotropic harmonic maps M 1 CP n whose harmonic sequence is not cyclic [8]. In [10], the authors found many examples of non-isotropic harmonic map M g CP 3 for any g 1. References [1] Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of 2-spheres. Ann. of Math., 113, 1 24 (1981) [2] Jost, J.: Two-dimensional geometric variational problems. Pure and Applied Mathematics. A Wiley- Interscience Publication. John Wiley and Sons, Ltd., New York, Chichester, 1991, x+236 pp [3] Parker, T. H.: Bubble tree convergence for harmonic maps. J. Differential Geom., 44, (1996) [4] Bolton, J., Woodward, L. M.: Congruence theorems for harmonic maps from a Riemann surface into CP n and S n. J. London Math. Soc., 45, (1992) [5] Bolton, J., Jensen, G. R., Rigoli, M., Woodward, L. M.: On conformal minimal immersions of S 2 into CP n. Math. Ann., 279, (1988) [6] Liao, R.: Cyclic properties of the harmonic sequence of surfaces in CP n. Math. Ann., 296, (1993) [7] Wolfson, J. G.: Harmonic sequences, harmonic maps and algebraic geometry. Harmonic mappings, twistors, and σ-models (Luminy, 1986), , Adv. Ser. Math. Phys., 4, World Sci. Publishing, Singapore, 1988 [8] Jensen, G. R., Liao, R.: Families of flat minimal tori in CP n. J. Differential Geom., 42, (1995) [9] Lemaire, L.: Harmonic nonholomorphic maps from a surface to a sphere. Proc.Amer.Math.Soc., 71, (1978) [10] Dong, Y., Shen Y.: On twistor Gauss maps of surfaces in 4-spheres. Acta Math. Sinica (N. S.), 12, (1996) [11] Chen, Q., Jost, J., Li, J., Wang, G.: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z., 251, (2005) [12] Ding, W., Tian, G.: Energy identity for a class of approximate harmonic maps from surfaces. Comm. Anal. Geom., (1995) [13] Lin, F., Wang, C.: Energy identity of harmonic map flows from surfaces at finite singular time. Calc. Var. Partial Differential Equations, 6, (1998) [14] Zhao, L.: Energy identities for Dirac-harmonic maps. Calc. Var. Partial Differential Equations, 28, (2007) [15] Jiao, X., Peng, J.: On non-isotropic harmonic maps of surfaces into complex projective spaces. Sci. China Ser. A, 44, (2001) [16] McIntosh, I.: A construction of all non-isotropic harmonic tori in complex projective space. Intern. J. Math., 6, (1995) [17] Taniguchi, T.: Non-isotropic harmonic tori in complex projective spaces and configurations of points on rational or elliptic curves. Tohoku Math. J., 52, (2000) [18] Chern, S. S., Wolfson, J. G.: Harmonic maps of the two-sphere into a complex Grassmann manifold. II. Ann. of Math., 125, (1987) [19] Wolfson, J. G.: Harmonic sequences and harmonic maps of surfaces into complex Grassmann manifolds. J. Differential Geom., 27, (1988) [20] Burstall, F. E., Wood, J. C.: The construction of harmonic maps into complex Grassmannians. J. Differential Geom., 23, (1986) [21] Eells, J., Wood, J. C.: Harmonic maps from surfaces to complex projective spaces. Adv. Math., 49, (1983) [22] Wood, J. C.: Holomorphicity of certain harmonic maps from a surface to complex projective n-space. J. London Math. Soc., 20, (1979)

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